qu.1.topic=Binomial Distributions@

qu.1.1.mode=Multiple Choice@
qu.1.1.name=26b..  μ & σ² :  find p@
qu.1.1.comment=<p>We have <em>&mu; = </em>$mu and <em>&sigma;<sup>2</sup></em> = $Var , the mean and variance values given in the question. Using the properties of the Binomial Distribution we have :&nbsp;</p>
<p><em>np = </em>$mu&nbsp; and <em>np(1-p) = </em>$Var<br />
Combining :&nbsp; $mu(1-<em>p</em>)<em> </em>= $Var so <em><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>p</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msup><mi mathvariant='normal'>&sigma;</mi><mrow><mn>2</mn></mrow></msup><mrow><mi mathvariant='normal'>&mu;</mi></mrow></mfrac></mrow></mstyle></math></em><em> </em>or <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>p</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><msup><mi mathvariant='normal'>&sigma;</mi><mrow><mn>2</mn></mrow></msup><mrow><mi mathvariant='normal'>&mu;</mi></mrow></mfrac></mrow></mstyle></math><em> = </em>$p which rounded to 3 decimals is $Ans</p>@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$Q="26b";
$mu=range(0.5,3.5,0.1);
$Var=range(0.2,$mu-0.1,0.1);
$p=1-$Var/$mu;
$Ans=decimal(3, $p);
$Alt1=decimal(4,$Ans+range(0.5,0.9,0.01)*(1-$Ans));
$Alt2=decimal(3,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(4,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.1.1.uid=4a514add-f6d7-44e3-b713-cccc5c0aa3e7@
qu.1.1.info=  Keyword=binomial;
  Course=230;
  Type=MC;
@
qu.1.1.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q">A series of <em>n</em> independent trials are run for a Binomial Process with probability of success <em>p</em>. If the mean is found to be&nbsp;$mu and the variance is $Var, what would you estimate <em>p</em> to be?</div>@
qu.1.1.answer=1@
qu.1.1.choice.1=$Ans@
qu.1.1.choice.2=$Alt1@
qu.1.1.choice.3=$Alt2@
qu.1.1.choice.4=$Alt3@
qu.1.1.fixed=@

qu.1.2.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q"><img hspace="4" align="$Align" src="__BASE_URI__DPM/Binomial/Typist$Which.gif" alt="Typist" title="Typist [IMG:Typist$Which.gif]" />A typist at a constant speed of $WPM words per minute makes a mistake in any particular word with probability $PMistake, independently from word to word. Each incorrect word must be corrected, a task which takes $CTime seconds per word. Find the variance of the time (1 decimal accuracy) it takes to finish a $Passage-word passage.</div>@
qu.1.2.answer.num=$Ans@
qu.1.2.answer.units=@
qu.1.2.showUnits=false@
qu.1.2.grading=toler_abs@
qu.1.2.err=0.1@
qu.1.2.negStyle=minus@
qu.1.2.numStyle=thousands scientific dollars arithmetic@
qu.1.2.mode=Numeric@
qu.1.2.name=24a.Typing: Variance in time@
qu.1.2.comment=<p>Let <span style="font-style: italic;">X</span> be the number of words typed wrong.  <span style="font-style: italic;">X ~ Bi(n,p)</span>  where <em>n </em>is the number of words to be typed, and<em> p</em> is the  probability of an error. Here we have X ~ Bin($Passage,$PMistake) .</p>
<p>Then <span style="font-style: italic;">E(X) = np = $ExpWrong</span> and  <span style="font-style: italic;">Var(X) = np(1 - p)   </span>= $VarWrong .</p>
<p>Notice also that the time to type each word is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>60</mn><mrow><mi mathvariant='normal'>$WPM</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$SecPerWord</mi></mrow></mstyle></math> in seconds. This is so for correct and incorrect words, incorrect words require additional time to correct.</p>
<p>The time <span style="font-style: italic;">T</span> to type the passage is ($SecPerWord second(s) for each correct word) + ($SecPerWord second(s) + $CTime seconds to correct for each incorrect word)<br />
<br />
T = $SecPerWord($Passage - X) + ($SecPerWord + $CTime)X = $SecPerWord($Passage) + $CTime*X    <br />
so</p>
<p>Var(T) = Var($SecPerWord($Passage) + $CTime*X) = ($CTime)<sup>2</sup> ($VarWrong)<font face="Times New Roman"><strong> = </strong>$Ans<strong><br />
</strong></font></p>
<p>&nbsp;</p>
<p><font face="Times New Roman"><strong><br />
</strong></font></p>
<p>&nbsp;</p>@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=$Q="24a";
$WPM=range(50,110,5);
$SecPerWord=decimal(2,60/$WPM);
$PMistake=decimal(2,range(0.02,0.09,0.01));
$CTime=range(9,18,1);
$Passage=range(400,750,25);
$ExpWrong=$Passage*$PMistake;
$VarWrong=$ExpWrong*(1-$PMistake);
$Ans=decimal(1,$CTime^2*$VarWrong);
$Which=rint(4);
$Align=switch(rint(2),"Left","Right");@
qu.1.2.uid=b363a4df-675d-4c06-bf92-a1fc19675970@
qu.1.2.info=  Difficulty=3;
  Keyword=binomial;
  Keyword=variance;
  Course=230;
@

qu.1.3.mode=Multiple Choice@
qu.1.3.name=05. B(?,?) describes dice toss@
qu.1.3.comment=<div class="shadedDiv descriptionSpan" style="margin-top: 0px; margin-bottom: 2px;">n =&nbsp;$n since we are repeating the "experiment" (dice-tossing)&nbsp;$n times. The probability of getting a&nbsp;$x&nbsp; or&nbsp;$y or $z in a toss is p = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mstyle></math>, so X ~Bi(n,p) = Bi($n,<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mstyle></math>)</div>@
qu.1.3.editing=useHTML@
qu.1.3.solution=@
qu.1.3.algorithm=$Q=5;
$n=range(10,30,5);
$x=range(1,2,1);
$y=range(3,4,1);
$z=range(5,6,1);
$d=6;@
qu.1.3.uid=daae9118-9c9f-4374-a486-876ec6a4fa5f@
qu.1.3.info=  Course=230;
  Type=MC;
  Author=Sean Scott;
@
qu.1.3.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q">A fair die is tossed $n times and the outcome of each toss is recorded. Let X be the number of times the die comes up $x, $y, or $z in the $n tosses. Then X has which of the following binomial distributions?</div>@
qu.1.3.answer=1@
qu.1.3.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Bi</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$n</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced></mrow></mstyle></math>@
qu.1.3.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Bi</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$n</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mfrac><mn>1</mn><mrow><mn>3</mn></mrow></mfrac></mrow></mfenced></mrow></mstyle></math>@
qu.1.3.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Bi</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$n</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mfrac><mn>1</mn><mrow><mn>6</mn></mrow></mfrac></mrow></mfenced></mrow></mstyle></math>@
qu.1.3.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Bi</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$d</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mfrac><mn>1</mn><mrow><mn>6</mn></mrow></mfrac></mrow></mfenced></mrow></mstyle></math>@
qu.1.3.choice.5=None of the above@
qu.1.3.fixed=4@

qu.1.4.question=<div title="University of Waterloo Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q">One third of the persons donating blood at a clinic have O<sup>+</sup> type blood. If&nbsp;$n people all donate at the same time, what is the probability that at least two have O<sup>+</sup> type blood? (4 decimals)</div>@
qu.1.4.answer.num=$Ans@
qu.1.4.answer.units=@
qu.1.4.showUnits=false@
qu.1.4.grading=toler_abs@
qu.1.4.err=0.0005@
qu.1.4.negStyle=minus@
qu.1.4.numStyle=thousands scientific dollars arithmetic@
qu.1.4.mode=Numeric@
qu.1.4.name=02. P(at least two O+ donors in {n})@
qu.1.4.comment=<p title="C6_BINOMIAL_DISTRIBUTIONS_I_1">Let<em> d</em> be the number of donors.<em> </em>Define "success" here as a donor having O<sup>+</sup> type blood. Define a r.v. X to be the number of successes amongst <em>d</em> donors. <br />
Then X ~ Bi(<span style="font-style: italic;">d,1/3</span>), that is X is Binomial with <span style="font-style: italic;"><span style="font-style: italic;">n = d=$n </span></span>and <span style="font-style: italic;">p = 1/3</span>.<br />
&nbsp;</p>
<p title="C6_BINOMIAL_DISTRIBUTIONS_I_1">P(X &ge; 2) = 1 - P(X &le; 1)<br />
= 1 - P(X = 0) - P(X = 1) =$Ans</p>@
qu.1.4.editing=useHTML@
qu.1.4.solution=@
qu.1.4.algorithm=$Q=2;
$n=range(6,16,1);
$x0=(2/3)^$n;
$x1=$n*(1/3)*(2/3)^($n-1);
$Ans=decimal(4,1-$x0-$x1);@
qu.1.4.uid=0f6edbf3-a09f-43de-8a79-50f790d4feb4@
qu.1.4.info=  Difficulty=2;
  Course=230;
@

qu.1.5.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q">Three students play a game&nbsp;$n times. The probability of winning a single game for Jimmy is $jp%, for Robert $rp%, and for Tommy $tp%. Assuming the outcomes of the&nbsp;$n games are independent of each, find the probability that $Who wins exactly two games. (Please answer to 3 decimals of accuracy.)</div>@
qu.1.5.answer.num=$Ans@
qu.1.5.answer.units=@
qu.1.5.showUnits=false@
qu.1.5.grading=toler_abs@
qu.1.5.err=0.01@
qu.1.5.negStyle=minus@
qu.1.5.numStyle=thousands scientific dollars arithmetic@
qu.1.5.mode=Numeric@
qu.1.5.name=14. P(Win exactly 2)@
qu.1.5.comment=<div class="shadedDiv descriptionSpan" style="margin-top: 0px; margin-bottom: 2px;">On a single game $Who wins with probability $WinP and loses with probability 1 - $WinP = $WinPn.<br />
<br />
If we ignore the order of wins/losses then, the probability of winning exactly 2 games is ($WinP)<sup>2</sup>($WinPn)<sup>$nM2</sup>.<br />
<br />
But order does count of course. How many ways can we arrange the&nbsp;$n games so 2 are wins for $Who?&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mi mathvariant='normal'>$n</mi></mrow></mtd></mtr><mtr><mtd><mrow><mn>2</mn></mrow></mtd></mtr></mtable></mfenced></mrow></mfenced></mrow></mstyle></math>&nbsp; !!!<br />
Thus P($Who wins exactly 2) =&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mi mathvariant='normal'>$n</mi></mrow></mtd></mtr><mtr><mtd><mrow><mn>2</mn></mrow></mtd></mtr></mtable></mfenced></mrow></mfenced></mrow><mrow><msup><mi mathvariant='normal'>$WinP</mi><mrow><mn>2</mn></mrow></msup><msup><mi mathvariant='normal'>$WinPn</mi><mrow><mi mathvariant='normal'>$nM2</mi></mrow></msup></mrow></mstyle></math> = $Ans</div>@
qu.1.5.editing=useHTML@
qu.1.5.solution=@
qu.1.5.algorithm=$Q=14;
$j=decimal(2,range(0.05,0.4,0.01));
$r=decimal(2,range(0.05,0.5,0.01));
$t=1-$j-$r;
$jp=$j*100;
$rp=$r*100;
$tp=$t*100;
$n=range(5,8,1);
$nM2=$n-2;
$jn=(1-$j);
$rn=1-$r;
$tn=1-$t;
$c=fact($n)/(fact($nM2)*fact(2));
$PickWho=rint(3);
$Who=switch($PickWho,"Jimmy","Robert","Tommy");
$WinP=switch($PickWho,$j,$r,$t);
$WinPn=1-$WinP;
$c=fact($n)/(fact($nM2)*fact(2));
$Ans=decimal(3,$c*$WinP^2*$WinPn^$nM2);@
qu.1.5.uid=ac1a3e05-a27e-4aac-93eb-ce3bc8034f3c@
qu.1.5.info=  Course=230;
  Type=numeric;
@

qu.1.6.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q"><img align="$Align" src="__BASE_URI__Chapter5/BD/Game$Which.gif" alt="" />Three students play a game 6 times. The probability of winning a single game for Jimmy is $jp%, for Robert $rp%, and for Tommy $tp%. Assuming the outcomes of the 6 games are independent of each, find the probability that Tommy wins 3 games, Robert wins two and Jimmy wins one game. (Please answer to 3 decimals of accuracy.)</div>@
qu.1.6.answer.num=$Ans@
qu.1.6.answer.units=@
qu.1.6.showUnits=false@
qu.1.6.grading=toler_abs@
qu.1.6.err=0.01@
qu.1.6.negStyle=minus@
qu.1.6.numStyle=thousands scientific dollars arithmetic@
qu.1.6.mode=Numeric@
qu.1.6.name=15. P(given win combination)@
qu.1.6.comment=<div class="shadedDiv descriptionSpan" style="margin-top: 0px; margin-bottom: 2px;">Suppose the sequence of winners is (for example) R T T J T R . The probability of this outcome is ($r)($t)($t)($j)($t)($r) = ($t)<sup>3</sup>($r)<sup>2</sup>($j)<br />
<br />
In fact that is the probability of any outcome that awards wins in that 3-2-1 proportion. How many ways can we arrange the sequence of wins? That's the same as asking how many unique strings can you make with the letters TTTRRJ . The answer is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mn>6</mn><mo lspace='0.1111111em' rspace='0.1111111em'>&amp;excl;</mo></mrow><mrow><mn>3</mn><mo lspace='0.1111111em' rspace='0.1111111em'>&amp;excl;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>2</mn><mo lspace='0.1111111em' rspace='0.1111111em'>&amp;excl;</mo></mrow></mfrac></mrow></mstyle></math>. <br />
<br />
Thus the probability of T-R-J winning 3-2-1 is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mn>6</mn><mo lspace='0.1111111em' rspace='0.1111111em'>&amp;excl;</mo></mrow><mrow><mn>3</mn><mo lspace='0.1111111em' rspace='0.1111111em'>&amp;excl;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>2</mn><mo lspace='0.1111111em' rspace='0.1111111em'>&amp;excl;</mo></mrow></mfrac></mrow></mstyle></math>($t)<sup>3</sup>($r)<sup>2</sup>($j) = $Ans</div>@
qu.1.6.editing=useHTML@
qu.1.6.solution=@
qu.1.6.algorithm=$Q=15;
$Which=rint(7)+1;
$Align=switch(rint(2),"Left","Right");
$j=decimal(2,range(0.1,0.3,0.01));
$r=decimal(2,range($j+0.01,0.5,0.01));
$t=1-$j-$r;
$jp=$j*100;
$rp=$r*100;
$tp=$t*100;
$jn=1-$j;
$rn=1-$r;
$tn=1-$t;
$Ans=decimal(3,60*$t^3*$r^2*$j);@
qu.1.6.uid=bad7f70a-490f-4f0d-92b1-48b5143a54cc@
qu.1.6.info=  Course=230;
  Type=numeric;
@

qu.1.7.mode=Multiple Choice@
qu.1.7.name=12. P(1 players wins all)@
qu.1.7.comment=<div style="margin-top: 0px; margin-bottom: 2px;" class="shadedDiv descriptionSpan">The events are independent. The probability that $Who wins any particular game is $Base, so P($Who wins all $n games) = ($Base)<sup>$n</sup>.</div>@
qu.1.7.editing=useHTML@
qu.1.7.solution=@
qu.1.7.algorithm=$Q=12;
$PickHim=rint(3);
$Who=switch($PickHim,"Jimmy","Robert","Tommy");

$j=decimal(2,range(0.05,0.3,0.01));
$r=decimal(2,range(0.05,0.5,0.01));
$t=1-$j-$r;
$Base=switch($PickHim,$j,$r,$t);
$jp=$j*100;
$rp=$r*100;
$tp=$t*100;
$n=range(5,15,1);
$nk=$n-2;
$b=decimal(2,$n*$j);
$nj=(1-$j);
$nt=1-$t;
$Which=rint(7)+1;
$Align=switch(rint(2),"Left","Right");@
qu.1.7.uid=4a76856b-5629-457b-9ac4-d7e924cde719@
qu.1.7.info=  Course=230;
  Type=MC;
@
qu.1.7.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q"><img hspace="4" align="$Align" alt="Game" title="Game [IMG:Game$Which.gif]" src="__BASE_URI__DPM/Binomial/Game$Which.gif" />Three students play a game&nbsp;$n times. The probability of winning a single game for Jimmy is $jp%, for Robert $rp%, and for Tommy $tp%. Assuming the outcomes of the $n games are independent of each, find the probability that $Who wins all the games.</div>@
qu.1.7.answer=1@
qu.1.7.choice.1=$Base<sup>$n</sup>@
qu.1.7.choice.2=$j×$r<sup>$nk</sup>×$t@
qu.1.7.choice.3=$b@
qu.1.7.choice.4=$r<sup>$n</sup>×$nj<sup>$n</sup>×$nt<sup>$n</sup>@
qu.1.7.choice.5=None of the above@
qu.1.7.fixed=4@

qu.1.8.mode=Multiple Choice@
qu.1.8.name=13. P(does not win any game)@
qu.1.8.comment=<div class="shadedDiv descriptionSpan" style="margin-top: 0px; margin-bottom: 2px">
<div class="shadedDiv descriptionSpan" style="margin-top: 0px; margin-bottom: 2px">The events are independent. The probability that Jimmy does not win any particular game is 1 -&nbsp;$j = $jn, so P(Jim loses all $n) = ($jn)<sup>$n</sup>.</div>
</div>@
qu.1.8.editing=useHTML@
qu.1.8.solution=@
qu.1.8.algorithm=$Q=13;
$Which=1+rint(7);
$Align=switch(rint(2),"Left","Right");
$j=decimal(2,range(0.05,0.3,0.01));
$r=decimal(2,range(0.05,0.5,0.01));
$t=1-$j-$r;
$jp=$j*100;
$rp=$r*100;
$tp=$t*100;
$n=range(5,15,1);
$nk=$n-2;
$jn=(1-$j);
$rn=1-$r;
$tn=1-$t;
$a='$jn*$r^$nk*$t';
$ANSWER='$jn^$n';
$b='$j^$n';
$c='$jn^$n*$t^$n*$r^$n';@
qu.1.8.uid=b75004e9-fd94-472c-9d60-e09487c0ff2a@
qu.1.8.info=  Course=230;
  Type=MC;
@
qu.1.8.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q"><img align="$Align" src="__BASE_URI__Chapter5/BD/Game$Which.gif" alt="" />Three students play a game&nbsp;$n times. The probability of winning a single game for Jimmy is $jp%, for Robert $rp%, and for Tommy $tp%. Assuming the outcomes of the $n games are independent of each, find the probability that Jimmy does not win any game.</div>@
qu.1.8.answer=1@
qu.1.8.choice.1=$jn<sup>$n</sup>@
qu.1.8.choice.2=$jn × $r<sup>$nk</sup> × $t@
qu.1.8.choice.3=$j<sup>$n</sup>@
qu.1.8.choice.4=$jn<sup>$n</sup> × $t<sup>$n</sup> × $r<sup>$n</sup>@
qu.1.8.choice.5=None of the above@
qu.1.8.fixed=4@

qu.1.9.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q"><img hspace="4" align="$Align" src="__BASE_URI__DPM/Binomial/Derrick$Which.gif" alt="Oil Derrick" title="Oil Derrick [Derrick$Which.gif]" /> An oil exploration firm is formed with enough capital to finance $n explorations. The probability of a particular exploration being successful is $p. Assume that explorations are independent of each other. The firm has fixed costs of  \\$$FC prior to carrying out its first exploration. Each successful exploration costs \\$$SC and each unsuccessful one costs \\$$UC. Find, <strong>to the nearest dollar,</strong> the Mean of the total cost to the firm of its $n explorations.</div>@
qu.1.9.answer.num=$ExpCost@
qu.1.9.answer.units=@
qu.1.9.showUnits=false@
qu.1.9.grading=toler_abs@
qu.1.9.err=1@
qu.1.9.negStyle=minus@
qu.1.9.numStyle=thousands scientific dollars arithmetic@
qu.1.9.mode=Numeric@
qu.1.9.name=21a. Oil Drilling: Mean of total cost@
qu.1.9.comment=<p>First let X be the number of successful wells. Notice that this is a Binomial Distribution with p = $p and n = $n, that is X ~ Bin($n,$p).</p>
<p>Then the mean is &mu;<sub>X</sub> = np = $MeanIs.<br />
<br />
The cost to the firm is: <br />
<br />
C = (<span style="font-weight: bold;">F</span>ixed <span style="font-weight: bold;">C</span>osts) + (<span style="font-weight: bold;">S</span>uccess <span style="font-weight: bold;">C</span>ost)*(# successes) + (<span style="font-weight: bold;">U</span>nsuccessful <span style="font-weight: bold;">C</span>ost)*(#attempts - # successes)<br />
= $FC + $SC*X + $UC*($n - X)&nbsp;&nbsp;&nbsp;&nbsp; (in \\$)<br />
<br />
The Mean of costs then is found by substituting the mean &mu;<sub>X </sub>= $MeanIs for X:<br />
<br />
E(C) = $FC + $SC*$MeanIs + $UC)*($n - $MeanIs)</p>
<p>= \\$$ExpCost</p>@
qu.1.9.editing=useHTML@
qu.1.9.solution=@
qu.1.9.algorithm=$Q="21a";
$n=range(1,25,1);
$FC=range(15000,45000,5000);
$SC=range(25000,100000,5000);
$UC=range(25000,$SC,5000);
$p=decimal(2,range(.10,.55,.05));
$MeanIs = $n*$p;
$ExpCost = $FC + ($MeanIs)*$SC + ($n-$MeanIs)*$UC;
$Which=rint(5)+1;
$Align=switch(rint(2),"Left","Right");@
qu.1.9.uid=d6292ac2-92f6-449a-8e1a-ef290272dc7d@
qu.1.9.info=  Difficulty=3;
  Keyword=binomial;
  Keyword=mean;
  Keyword=expected value;
  Course=230;
@

qu.1.10.mode=Multiple Choice@
qu.1.10.name=28. Super Bowl@
qu.1.10.comment=<p>This is a binomial distribution with <font size="3" face="Times New Roman"><em>n</em> = $N</font> and <font size="3" face="Times New Roman"><em>p</em> = $P</font> . Then:</p>
<p>Mean = <em><font size="3" face="Times New Roman">np</font></em> <font size="3" face="Times New Roman">=</font> <font size="3" face="Times New Roman">$U</font><br />
Variance = <font size="3" face="Times New Roman"><em>np</em>(1 -<em> p</em>) = $V</font></p>@
qu.1.10.editing=useHTML@
qu.1.10.solution=@
qu.1.10.algorithm=$Q=28;
$Which=rint(6);
$Align=switch(rint(2),"Left","Right");
$P = range(0.3,0.35,0.001);
$N = range(180,185,1);
$U = decimal(2,$P*$N);
$V = decimal(2,$N*$P*(1-$P));
$MeanAlt1 = decimal(2,range(0.4,0.9,0.01)*$U);
$MeanAlt2 = decimal(2,range(1.2,1.8,0.01)*$U);
$MeanAlt3=decimal(2,0.5*($U+switch(rint(2),$MeanAlt1,$MeanAlt2)));
$VarAlt1 = decimal(2,range(0.4,0.9,0.01)*$V);
$VarAlt2 = decimal(2,range(1.2,1.8,0.01)*$V);
$VarAlt3=decimal(2,0.5*($V+switch(rint(2),$VarAlt1,$VarAlt2)));
$PER = 100*$P;@
qu.1.10.uid=498c2192-2c1a-4ad8-a3c5-8d169dbd9aa2@
qu.1.10.info=  Type=MC;
  Course=202;
@
qu.1.10.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q"><img hspace="4" align="$Align" src="__BASE_URI__DPM/Binomial/Football$Which.gif" alt="Football" title="Football [IMG:Football$Which.gif]" />A television station estimates that $PER% of college students watch the Super Bowl. For a sample of $N students selected at random, what is the mean and variance of the number of students who watch this game?</div>@
qu.1.10.answer=4@
qu.1.10.choice.1=Mean = $MeanAlt1, Variance = $VarAlt1@
qu.1.10.choice.2=Mean = $MeanAlt2, Variance = $VarAlt2@
qu.1.10.choice.3=Mean = $MeanAlt3, Variance = $VarAlt3@
qu.1.10.choice.4=Mean = $U, Variance = $V@
qu.1.10.fixed=@

qu.1.11.mode=Inline@
qu.1.11.name=16.SD(# homes)@
qu.1.11.comment=<p>Model this as a Binomial Distribution with n = $n and p = $p. The <em>variance</em> of such a distribution is given by <em>np(1-p)</em> so the Standard Deviation = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msqrt><mrow><mi mathvariant='normal'>$n</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$p</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$p</mi></mrow></mfenced></mrow></msqrt></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math> = $Ans</p>@
qu.1.11.editing=useHTML@
qu.1.11.solution=@
qu.1.11.algorithm=$Q=16;
$City=switch(rint(4),"Guelph","Waterloo","Kitchener","London");
$Which=rint(5)+1;
$Align=switch(rint(2),"Left","Right");
$pp=range(10,90);
$p=$pp/100;
$n=range(100,500,5);
$Ans=decimal(3,($n*$p*(1-$p))^0.5);@
qu.1.11.uid=e70fe589-3688-427a-9923-d329cfd7edd3@
qu.1.11.info=  Course=230;
  Course=202;
  Type=numeric;
@
qu.1.11.weighting=1@
qu.1.11.numbering=alpha@
qu.1.11.part.1.name=sro_id_1@
qu.1.11.part.1.answer.units=@
qu.1.11.part.1.numStyle=thousands scientific  arithmetic@
qu.1.11.part.1.editing=useHTML@
qu.1.11.part.1.showUnits=false@
qu.1.11.part.1.err=0.01@
qu.1.11.part.1.question=(Unset)@
qu.1.11.part.1.mode=Numeric@
qu.1.11.part.1.grading=toler_abs@
qu.1.11.part.1.negStyle=minus@
qu.1.11.part.1.answer.num=$Ans@
qu.1.11.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q"><img hspace="4" align="$Align" src="__BASE_URI__DPM/Binomial/House$Which.gif" alt="House" title="House [IMG:House$Which.gif]" />According to Statistics Canada, $pp % new residential units built  in the city of $City are single detached homes. If there are $n new residential units in the city, what is the standard deviation of the number of single detached homes in all these new residential units?<p>&nbsp;</p><p>Ans:&nbsp;<span>&nbsp;</span><1><span>&nbsp;</span> <em>(answer to 3 decimals)</em></p></div>@

qu.1.12.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q">A series of <em>n</em> independent trials are run for a Binomial Process with probability of success <em>p</em>. If the mean is found to be&nbsp;$mu and the variance is $Var, what would you estimate <em>p</em> to be? (Please answer to 3 decimals of accuracy.)
<p>&nbsp;</p>
</div>@
qu.1.12.answer.num=$Ans@
qu.1.12.answer.units=@
qu.1.12.showUnits=false@
qu.1.12.grading=toler_abs@
qu.1.12.err=0.01@
qu.1.12.negStyle=minus@
qu.1.12.numStyle=thousands scientific dollars arithmetic@
qu.1.12.mode=Numeric@
qu.1.12.name=26a. μ & σ² :  find p@
qu.1.12.comment=<p>We have <em>&mu; = </em>$mu and <em>&sigma;<sup>2</sup></em> = $Var , the mean and variance values given in the question. Using the properties of the Binomial Distribution we have :&nbsp;</p>
<p><em>np = </em>$mu&nbsp; and <em>np(1-p) = </em>$Var</p>
<p>Combining :&nbsp; $mu(1-<em>p</em>)<em> </em>= $Var so <em><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>p</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msup><mi mathvariant='normal'>$Var</mi><mrow><mi></mi></mrow></msup></mrow><mrow><mi mathvariant='normal'>$mu</mi></mrow></mfrac></mrow></mstyle></math></em><em> </em>or <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>p</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><msup><mi mathvariant='normal'>$Var</mi><mrow><mi></mi></mrow></msup></mrow><mrow><mi mathvariant='normal'>$mu</mi></mrow></mfrac></mrow></mstyle></math><em> = </em>$p which rounded to 3 decimals is $Ans</p>@
qu.1.12.editing=useHTML@
qu.1.12.solution=@
qu.1.12.algorithm=$Q="26a";
$mu=range(0.5,3.5,0.1);
$Var=range(0.2,$mu-0.1,0.1);
$p=1-$Var/$mu;
$Ans=decimal(3, $p);@
qu.1.12.uid=e4dc0dc7-7c7d-4f64-9b09-d53aaae9806f@
qu.1.12.info=  Keyword=binomial;
  Course=230;
  Type=numeric;
@

qu.1.13.mode=Multiple Choice@
qu.1.13.name=20b. Oil drilling: Variance@
qu.1.13.comment=<p>This is a binomial distribution so the mean is np and the variance is np(1-p) . <br />
In this case that is $n*$p*(1-$p) = $Ans .</p>@
qu.1.13.editing=useHTML@
qu.1.13.solution=@
qu.1.13.algorithm=$Q="20b";
$n=range(12,35,1);
$p=decimal(2,range(.04,.40,.01));
$Which=rint(5)+1;
$Align=switch(rint(2),"Left","Right");
$Ans = decimal(3,$n*$p*(1-$p));
$Alt1=decimal(3,range(1.1,1.9,0.01)*$Ans);
$Alt2=decimal(3,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(2,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.1.13.uid=9e35c7bc-0a9f-44ea-a257-bc980dbf62f4@
qu.1.13.info=  Difficulty=1;
  Keyword=binomial;
  Keyword=variance;
  Course=230;
@
qu.1.13.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q"><img hspace="4" align="$Align" title="Oil Derrick [Derrick$Which.gif]" alt="Oil Derrick" src="__BASE_URI__DPM/Binomial/Derrick$Which.gif" />An oil exploration firm is going to make $n explorations. The probability of a particular exploration being successful is $p. Assume that explorations are independent of each other. The variance of the number of successful explorations is:</div>@
qu.1.13.answer=1@
qu.1.13.choice.1=$Ans@
qu.1.13.choice.2=$Alt1@
qu.1.13.choice.3=$Alt2@
qu.1.13.choice.4=$Alt3@
qu.1.13.fixed=@

qu.1.14.mode=Multiple Choice@
qu.1.14.name=06. Dice as Bi(n,p)@
qu.1.14.comment=<div style="margin-top: 0px; margin-bottom: 2px;" class="shadedDiv descriptionSpan">n =&nbsp;$n since we are repeating the "experiment" (dice-tossing)&nbsp;$n times. The probability of getting a&nbsp;$x&nbsp; or&nbsp;$y in a toss is p = 1/3, so X ~Bi(n,p) = Bi($n,1/3)</div>@
qu.1.14.editing=useHTML@
qu.1.14.solution=@
qu.1.14.algorithm=$Q=6;
$Which=rint(4)+1;
$Align=switch(rint(2),"Left","Right");
$n=range(10,30,5);
$x=range(1,3,1);
$y=range(4,6,1);@
qu.1.14.uid=cca0c083-51c5-4c23-9a32-f0e7fc42b975@
qu.1.14.info=  Course=230;
  Type=MC;
@
qu.1.14.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q"><img align="$Align" src="__BASE_URI__DPM/Binomial/Dice$Which.gif" alt="Dice" title="Dice [IMG:Dice$Which.gif]" />A fair die is tossed $n times and the outcome of each toss is recorded. Let X be the number of $x's and $y's that appear in the $n tosses. Then X has which of the following binomial distributions?&nbsp;</div>@
qu.1.14.answer=2@
qu.1.14.choice.1=Bi($n,1/2)@
qu.1.14.choice.2=Bi($n,1/3)@
qu.1.14.choice.3=Bi($n,1/6)@
qu.1.14.choice.4=Bi(6,1/6)@
qu.1.14.choice.5=None of the above@
qu.1.14.fixed=4@

qu.1.15.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q"><img hspace="4" align="$Align" title="Oil Derrick [Derrick$Which.gif]" alt="Oil Derrick" src="__BASE_URI__DPM/Binomial/Derrick$Which.gif" />An oil exploration firm is going to make $n explorations. The probability of a particular exploration being successful is $p. Assume that explorations are independent of each other. The variance of the number of successful explorations is: (Please answer to 3 decimals of accuracy.)</div>@
qu.1.15.answer.num=$VarIs@
qu.1.15.answer.units=@
qu.1.15.showUnits=false@
qu.1.15.grading=toler_abs@
qu.1.15.err=.01@
qu.1.15.negStyle=minus@
qu.1.15.numStyle=thousands scientific dollars arithmetic@
qu.1.15.mode=Numeric@
qu.1.15.name=20a. Oil drilling: Variance@
qu.1.15.comment=This is a binomial distribution so the mean is np and the variance is np(1-p) . <br>In this case that is $n*$p*(1-$p) = $VarIs .<br>@
qu.1.15.editing=useHTML@
qu.1.15.solution=@
qu.1.15.algorithm=$Q="20a";
$n=range(12,35,1);
$p=decimal(2,range(.04,.40,.01));
$VarIs = decimal(4,$n*$p*(1-$p));
$Which=rint(5)+1;
$Align=switch(rint(2),"Left","Right");@
qu.1.15.uid=c2128972-cf90-4feb-a506-7854bb5f5de8@
qu.1.15.info=  Difficulty=1;
  Keyword=binomial;
  Keyword=variance;
  Course=230;
@

qu.1.16.mode=True False@
qu.1.16.name=03. Bin equal values?@
qu.1.16.comment=<p>True, since &nbsp;<sub>n</sub>C<sub>r </sub>= <sub>n</sub>C<sub>(n-r)</sub>&nbsp; and p=0.5, so we have<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mi>n</mi><mo lspace='0.1111111em' rspace='0.1111111em'>&amp;excl;</mo></mrow><mrow><mi>k</mi><mo lspace='0.1111111em' rspace='0.1111111em'>&amp;excl;</mo><mfenced open='(' close=')' separators=','><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>k</mi></mrow></mfenced><mo lspace='0.1111111em' rspace='0.1111111em'>&amp;excl;</mo></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mn>0.5</mn><mrow><mi>k</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mn>0.5</mn><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>k</mi></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mo lspace='0.0em' rspace='0.0em'> </mo><mfrac><mrow><mi>n</mi><mo lspace='0.1111111em' rspace='0.1111111em'>&amp;excl;</mo></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>k</mi></mrow></mfenced><mi>k</mi><mo lspace='0.1111111em' rspace='0.1111111em'>&amp;excl;</mo></mrow></mfrac><msup><mn>0.5</mn><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>k</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mn>0.5</mn><mrow><mi>k</mi></mrow></msup></mrow></mstyle></math></p>
<p>where n=$n, k=$x, and n-k=$y</p>@
qu.1.16.editing=useHTML@
qu.1.16.solution=@
qu.1.16.algorithm=$Q=3;
$p=0.5;
$n=range(3,19,2);
$z=($n-1)/2;
$x=range(1,$z,1);
$y=$n-$x;@
qu.1.16.uid=f10bb33b-9340-47b6-8fb3-f99aa518525e@
qu.1.16.info=  Difficulty=1;
  Course=230;
@
qu.1.16.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q">Let X have a binomial distribution with n=$n and p=0.5 . Then (T or F):<br />
P(X=$x) = P(X=$y)</div>@
qu.1.16.answer=1@
qu.1.16.choice.1=True@
qu.1.16.choice.2=False@
qu.1.16.fixed=@

qu.1.17.mode=Multiple Choice@
qu.1.17.name=29. Mean/var for sample population@
qu.1.17.comment=<p>This is a binomial distribution with <font size="3" face="Times New Roman"><em>n</em> = $SS</font> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>p</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mo lspace='0.1111111em' rspace='0.1111111em'>&num;</mo><mi>males</mi></mrow><mrow><mi>Total</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$p</mi></mrow></mstyle></math></p>
<p><font size="3" face="Times New Roman"><em>Mean</em> = <em>np</em> = $SS($p) = $Mean, </font></p>
<p><font size="3" face="Times New Roman"><em>Var</em> = <em>np</em>(1 - <em>p</em>) = $Var</font></p>@
qu.1.17.editing=useHTML@
qu.1.17.solution=@
qu.1.17.algorithm=$Q="29";
$M=range(2500,5500,100);
$F = range(2500,5500,100);
$T = $M + $F;
$SS=range(25,45,5);
$p=decimal(2,$M/$T);
$Mean=$p*$SS;
$Var = $p*(1-$p)*$SS;
$Ans1=$Mean;
$Ans2=$Var;
$Alt11=decimal(4,range(1.1,1.9,0.01)*$Ans1);
$Alt12=decimal(4,range(1.1,1.9,0.01)*$Ans2);
$Alt21=decimal(4,range(0.5,0.9,0.01)*$Ans1);
$Alt22=decimal(4,range(0.5,0.9,0.01)*$Ans2);
$Alt31=decimal(4,0.5*($Ans1+switch(rint(2),$Alt11,$Alt21)));
$Alt32=decimal(4,0.5*($Ans2+switch(rint(2),$Alt12,$Alt22)));
$Which=rint(4);
$Align=switch(rint(2),"Left","Right");@
qu.1.17.uid=eda6af19-2fb2-4164-ae3d-73724def591b@
qu.1.17.info=  Type=MC;
  Course=202;
@
qu.1.17.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q"><img hspace="4" align="$Align" title="Students [IMG:Students$Which.gif]" alt="Students" src="__BASE_URI__DPM/Binomial/Students$Which.gif" />A university has $T students of which $M are male and $F are female. If a class of $SS students is chosen at random from the university population, find the mean and variance of the number of male students.</div>@
qu.1.17.answer=1@
qu.1.17.choice.1=Mean = $Mean, Variance = $Var@
qu.1.17.choice.2=Mean = $Alt11, Variance =$Alt12@
qu.1.17.choice.3=Mean = $Alt21, Variance = $Alt22@
qu.1.17.choice.4=Mean = $Alt31, Variance = $Alt32@
qu.1.17.fixed=@

qu.1.18.mode=Multiple Choice@
qu.1.18.name=24b.Typing: Variance in time@
qu.1.18.comment=<p>Let <span style="font-style: italic;">X</span> be the number of words typed wrong.  <span style="font-style: italic;">X ~ Bi(n,p)</span>  where <em>n </em>is the number of words to be typed, and<em> p</em> is the  probability of an error. Here we have X ~ Bin($Passage,$PMistake) .</p>
<p>Then <span style="font-style: italic;">E(X) = np = $ExpWrong</span> and  <span style="font-style: italic;">Var(X) = np(1 - p)   </span>= $VarWrong .</p>
<p>Notice also that the time to type each word is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>60</mn><mrow><mi mathvariant='normal'>$WPM</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$SecPerWord</mi></mrow></mstyle></math> in seconds. This is so for correct and incorrect words, incorrect words require additional time to correct.</p>
<p>The time <span style="font-style: italic;">T</span> to type the passage is ($SecPerWord second(s) for each correct word) + ($SecPerWord second(s) + $CTime seconds to correct for each incorrect word)<br />
<br />
T = $SecPerWord($Passage - X) + ($SecPerWord + $CTime)X = $SecPerWord($Passage) + $CTime*X    <br />
so</p>
<p>Var(T) = Var($SecPerWord($Passage) + $CTime*X) = ($CTime)<sup>2</sup> ($VarWrong)<font face="Times New Roman"><strong> = </strong>$Ans<strong><br />
</strong></font></p>@
qu.1.18.editing=useHTML@
qu.1.18.solution=@
qu.1.18.algorithm=$Q="24b";
$WPM=range(50,110,5);
$SecPerWord=decimal(2,60/$WPM);
$PMistake=decimal(2,range(0.02,0.09,0.01));
$CTime=range(9,18,1);
$Passage=range(400,750,25);
$ExpWrong=$Passage*$PMistake;
$VarWrong=$ExpWrong*(1-$PMistake);
$Ans=decimal(1,$CTime^2*$VarWrong);
$Alt1=decimal(1,range(1.1,1.9,0.01)*$Ans);
$Alt2=decimal(1,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(1,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));
$Which=rint(5)+1;
$Align=switch(rint(2),"Left","Right");@
qu.1.18.uid=1f657b96-0f00-446c-a3f1-2b72b8af438c@
qu.1.18.info=  Difficulty=3;
  Keyword=binomial;
  Keyword=variance;
  Course=230;
  Type=MC;
@
qu.1.18.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q"><img hspace="4" align="$Align" src="__BASE_URI__DPM/Binomial/Typist$Which.gif" alt="Typist" title="Typist [IMG:Typist$Which.gif]" />A typist at a constant speed of $WPM words per minute makes a mistake in any particular word with probability $PMistake, independently from word to word. Each incorrect word must be corrected, a task which takes $CTime seconds per word. Find the variance of the time it takes to finish a $Passage-word passage.</div>@
qu.1.18.answer=1@
qu.1.18.choice.1=$Ans@
qu.1.18.choice.2=$Alt1@
qu.1.18.choice.3=$Alt2@
qu.1.18.choice.4=$Alt3@
qu.1.18.fixed=@

qu.1.19.mode=Multiple Choice@
qu.1.19.name=25b. μ & σ² :  find n@
qu.1.19.comment=<p>We have <em>&mu; = </em>$mu and <em>&sigma;<sup>2</sup></em> = $Var .&nbsp; Using the properties of the Binomial Distribution we have :</p>
<p><strong>[1]</strong><em>&nbsp; np = </em>$mu&nbsp; and <br />
<strong>[2]</strong><em> np(1-p) = </em>$Var<em> </em></p>
<p>Combining :&nbsp; <font size="3" face="Times New Roman">$mu</font><em>(1-p) = </em>$Var so <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>p</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><msup><mi>&sigma;</mi><mrow><mn>2</mn></mrow></msup><mrow><mi mathvariant='normal'>&mu;</mi></mrow></mfrac></mrow></mrow></mstyle></math><em> </em>or <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>p</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><msup><mi mathvariant='normal'>&sigma;</mi><mrow><mn>2</mn></mrow></msup><mrow><mi mathvariant='normal'>&mu;</mi></mrow></mfrac></mrow></mstyle></math>= <font size="3" face="Times New Roman">$p</font></p>
<p>Substitute this <em>p</em> value in <strong>[1] and solve:</strong>:&nbsp; <em><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi mathvariant='normal'>&mu;</mi><mrow><mi>p</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Ans</mi></mrow></mstyle></math></em></p>
<p>For <em>n</em> you really should round UP to the next integer, but "normal" roundoff is accepted.</p>@
qu.1.19.editing=useHTML@
qu.1.19.solution=@
qu.1.19.algorithm=$Q="25b";
$mu=range(0.5,3.5,0.1);
$Var=range(0.2,$mu-0.1,0.1);
$p=1-$Var/$mu;
$n=$mu/$p;
$Ans=1+int($n);
$Alt1=range(1,$Ans)+$Ans;
$Alt3=$Alt1+range(2,5);
$Alt2=if(lt($Ans,3),$Alt3+2+range(-1,1),range(1,$Ans-1));@
qu.1.19.uid=3a57446d-4dbc-4b37-8cfc-12956d68e47f@
qu.1.19.info=  Keyword=binomial;
  Course=230;
  Type=MC;
@
qu.1.19.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q">A series of <em>n</em> independent trials are run for a Binomial Process with probability of success <em>p</em>. If the mean is found to be&nbsp;$mu and the variance is $Var, what would you estimate <em>n</em> to be?</div>@
qu.1.19.answer=1@
qu.1.19.choice.1=$Ans@
qu.1.19.choice.2=$Alt1@
qu.1.19.choice.3=$Alt2@
qu.1.19.choice.4=$Alt3@
qu.1.19.fixed=@

qu.1.20.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q">A series of <em>n</em> independent trials are run for a Binomial Process with probability of success <em>p</em>. If the mean is found to be&nbsp;$mu and the variance is $Var, what would you estimate <em>n</em> to be (round off to an integer)?</div>@
qu.1.20.answer.num=$Ans@
qu.1.20.answer.units=@
qu.1.20.showUnits=false@
qu.1.20.grading=toler_abs@
qu.1.20.err=1@
qu.1.20.negStyle=minus@
qu.1.20.numStyle=thousands scientific dollars arithmetic@
qu.1.20.mode=Numeric@
qu.1.20.name=25a. μ & σ² :  find n@
qu.1.20.comment=<p>We have <em>&mu; = </em>$mu and <em>&sigma;<sup>2</sup></em> = $Var .&nbsp; Using the properties of the Binomial Distribution we have :</p>
<p><strong>[1]</strong><em>&nbsp; np = </em>$mu&nbsp; and <br />
<strong>[2]</strong><em> np(1-p) = </em>$Var<em> </em></p>
<p>Combining :&nbsp; <font size="3" face="Times New Roman">$mu</font><em>(1-p) = </em>$Var so <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>p</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi mathvariant='normal'>$Var</mi></mrow><mrow><mi mathvariant='normal'>$mu</mi></mrow></mfrac></mrow></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math><em> </em>or <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>p</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><msup><mi mathvariant='normal'>$Var</mi><mrow><mi></mi></mrow></msup></mrow><mrow><mi mathvariant='normal'>$mu</mi></mrow></mfrac></mrow></mstyle></math>= <font size="3" face="Times New Roman">$p</font></p>
<p>Substitute this <em>p</em> value in <strong>[1] and solve:</strong>:&nbsp; <em><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi mathvariant='normal'>$mu</mi></mrow><mrow><mi mathvariant='normal'>$p</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></em></p>
<p>For <em>n</em> you really should round UP to the next integer, but "normal" roundoff is accepted.</p>@
qu.1.20.editing=useHTML@
qu.1.20.solution=@
qu.1.20.algorithm=$Q=31;
$mu=range(0.5,3.5,0.1);
$Var=range(0.2,$mu-0.1,0.1);
$p=decimal(4,1-$Var/$mu);
$n=$mu/$p;
$Ans=1+int($n);@
qu.1.20.uid=cb9dada6-2b35-4fb2-be0d-e48fd56ab115@
qu.1.20.info=  Keyword=binomial;
  Course=230;
  Type=numeric;
@

qu.1.21.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q"><img vspace="4" hspace="4" align="$Align" alt="Baseball picture" title="Baseball picture [IMG:Baseball$Which.gif]" src="__BASE_URI__DPM/Binomial/Baseball$Which.gif" />A baseball player has a $pp% chance of hitting the ball each time at bat, with successive times at bat being independent. Calculate the probability that he gets exactly 2 hits in $n times at bat. Answer to 3 decimals please.</div>@
qu.1.21.answer.num=$Ans@
qu.1.21.answer.units=@
qu.1.21.showUnits=false@
qu.1.21.grading=toler_abs@
qu.1.21.err=0.005@
qu.1.21.negStyle=minus@
qu.1.21.numStyle=thousands scientific dollars arithmetic@
qu.1.21.mode=Numeric@
qu.1.21.name=08. P(2 hits|n times at bat)@
qu.1.21.comment=<div style="margin-top: 0px; margin-bottom: 2px;" class="shadedDiv descriptionSpan">This is an example of a <span style="font-weight: bold;">Binomial</span> distribution with p = $p, n = $n.&nbsp; We want P(X&nbsp;= 2) = P(X&nbsp;= 2) =&nbsp; <sub><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mi mathvariant='normal'>$n</mi></mrow></mtd></mtr><mtr><mtd><mrow><mn>2</mn></mrow></mtd></mtr></mtable></mfenced></mrow></mstyle></math></sub>($p)<sup>2</sup>(1-$p)<sup>$n-2</sup> = $A which rounds to&nbsp;$Ans .</div>@
qu.1.21.editing=useHTML@
qu.1.21.solution=@
qu.1.21.algorithm=$Q=8;
$Align=switch(rint(2),"Left","Right");
$Which=1+rint(4);
$p=decimal(1,range(0.2,0.7,0.1));
$pp=$p*100;
$n=range(5,12,1);
$A=maple("with(Statistics);
X := RandomVariable(Binomial($n, $p));
ProbabilityFunction(X, u);
ProbabilityFunction(X, 2)");
$Ans=decimal(3,$A);@
qu.1.21.uid=247f9df7-22a5-4279-b5fe-275a486c5235@
qu.1.21.info=  Course=230;
  Type=numeric;
@

qu.1.22.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q"><img hspace="4" align="$Align" src="__BASE_URI__DPM/Binomial/Camera$Which.gif" title="Camera [IMG:Camera$Which.gif]" alt="Camera" />A camera's flash mechanism fails on $pAp% of shots. If&nbsp;$n shots are taken using the flash, what is the probability that the flash fails exactly twice? (Please answer to 4 decimals of accuracy.)</div>@
qu.1.22.answer.num=$ANSWER@
qu.1.22.answer.units=@
qu.1.22.showUnits=false@
qu.1.22.grading=toler_abs@
qu.1.22.err=0.001@
qu.1.22.negStyle=minus@
qu.1.22.numStyle=thousands scientific dollars arithmetic@
qu.1.22.mode=Numeric@
qu.1.22.name=11. P(Flash fails twice)@
qu.1.22.comment=<p title="C6_BINOMIAL_DISTRIBUTIONS_II_2">Let X<sub> </sub>&nbsp;be the number of times the flash fails in n=10 shots. We have <font size="3" face="Times New Roman">$pA</font> as the probability that the flash fails (divide the given % by 100 to get this).</p>
<p>Then X ~ Bi(10,$pA) and</p>
<p>P(X = 2) =&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'></mi></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mi mathvariant='normal'>$n</mi></mrow></mtd></mtr><mtr><mtd><mrow><mn>2</mn></mrow></mtd></mtr></mtable></mfenced></mrow></mfenced></mrow></mstyle></math><font size="3" face="Times New Roman">($pA)<sup>2</sup>(1 - $pA)<sup>$n-2</sup></font> = $ans which rounds up to $ANSWER</p>@
qu.1.22.editing=useHTML@
qu.1.22.solution=@
qu.1.22.algorithm=$Q=11;
$Which=rint(5)+1;
$Align=switch(rint(2),"Left","Right");
$pA=decimal(2,range(0.05,0.35,0.01));
$n=range(5,20,5);
$pAp=$pA*100;
$npA=1-$pA;
$nn=$n-2;
$c=fact($n)/(fact($nn)*fact(2));
$ans=$c*$pA^2*$npA^$nn;
$ANSWER=decimal(4,$ans);@
qu.1.22.uid=3e51deaf-09a1-4e66-b692-e1b2c8f2c7d9@
qu.1.22.info=  Course=230;
  Type=numeric;
@

qu.1.23.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q"><img hspace="4" align="$Align" title="Camera [IMG:Camera$Which.gif]" alt="Camera" src="__BASE_URI__DPM/Binomial/Camera$Which.gif" />Two identical cameras, call them A and B, have their flashes fail $pAp% and $pBp% of the time respectively. A photographer selects a camera at random and uses it to take $Flashes shots with the flash. If the flash failed exactly $Fails times, what is the probability that the photographer used Camera B? (Please answer to 4 decimals of accuracy.)</div>@
qu.1.23.answer.num=$pBUsed@
qu.1.23.answer.units=@
qu.1.23.showUnits=false@
qu.1.23.grading=toler_abs@
qu.1.23.err=.001@
qu.1.23.negStyle=minus@
qu.1.23.numStyle=thousands scientific dollars arithmetic@
qu.1.23.mode=Numeric@
qu.1.23.name=01. Cameras fail rate, which was used?@
qu.1.23.comment=<p>Let A (B) be the event "camera A (B) is chosen".</p>
<p>Let X<sub>A</sub>  (X<sub>B</sub>)  be the number of times Camera A's (B's) flash fails in 10 shots.</p>
<p>Let p<sub>A</sub> (p<sub>B</sub>) be the probability that Camera A's (B's) flash fails, so we have p<sub>A</sub> = $pA and p<sub>B</sub> = $pB.</p>
<p>Then X<sub>A</sub> ~ Bi($Flashes,$pA) and X<sub>B</sub> ~ Bi($Flashes,$pB)</p>
<p>Finally, let F be the event "flash fails exactly $Fails times". Then you want :</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>P</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&ApplyFunction;</mo><mfenced><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>B</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='verythinmathspace' rspace='verythinmathspace' stretchy='true' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&verbar;</mo><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>F</mi></mrow></mfenced></mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&equals;</mo><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>P</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&ApplyFunction;</mo><mfenced><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>F</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='verythinmathspace' rspace='verythinmathspace' stretchy='true' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&verbar;</mo><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>B</mi></mrow></mfenced></mrow><mfrac linethickness='1' denomalign='center' numalign='center' bevelled='false'><mrow><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>P</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&ApplyFunction;</mo><mfenced><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>B</mi></mrow></mfenced></mrow></mrow><mrow><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>P</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&ApplyFunction;</mo><mfenced><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>F</mi></mrow></mfenced></mrow></mrow></mfrac></mrow></math><br />
So first we find P(F) (we'll find P(F|B) as part of this):<br />
P(F) = P(F|A)P(A) + P(F|B)P(B)    <em>(this is an example of the Partition rule)</em><br />
= P(X<sub>A</sub> = $Fails)(0.5) + P(X<sub>B</sub> = $Fails)(0.5)  <em>(the 0.5 is because the camera is chosen at random)<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&equals;</mo><mrow><mfenced><mrow><mrow><munder accentunder='false'><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$Flashes</mi><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$Fails</mi></munder></mrow></mrow></mfenced><msup superscriptshift='0'><mfenced><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$pA</mi></mrow></mfenced><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$Fails</mi></msup><msup superscriptshift='0'><mfenced><mrow><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>1</mn><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='mediummathspace' rspace='mediummathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&#8722;</mo><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$pA</mi></mrow></mfenced><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$Successes</mi></mrow></msup><mfenced><mrow><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>0&period;5</mn></mrow></mfenced><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='mediummathspace' rspace='mediummathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&plus;</mo><mfenced><mrow><mrow><munder accentunder='false'><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$Flashes</mi><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$Fails</mi></munder></mrow></mrow></mfenced><msup superscriptshift='0'><mfenced><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$pB</mi></mrow></mfenced><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$Fails</mi></msup><msup superscriptshift='0'><mfenced><mrow><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>1</mn><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&#8722;</mo><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$pB</mi></mrow></mfenced><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$Successes</mi></mrow></msup><mfenced><mrow><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>0&period;5</mn></mrow></mfenced></mrow></mrow></math><br />
So:<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>B</mi><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mi>F</mi></mrow></mfenced></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><munder><mi mathvariant='normal'>$Flashes</mi><mi mathvariant='normal'>$Fails</mi></munder></mrow></mfenced><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$pB</mi></mrow></mfenced><mi mathvariant='normal'>$Fails</mi></msup><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mi mathvariant='normal'>$pB</mi></mrow></mfenced><mi mathvariant='normal'>$Successes</mi></msup><mfenced open='(' close=')' separators=','><mrow><mn>0.5</mn></mrow></mfenced></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><munder><mi mathvariant='normal'>$Flashes</mi><mi mathvariant='normal'>$Fails</mi></munder></mrow></mfenced><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$pA</mi></mrow></mfenced><mi mathvariant='normal'>$Fails</mi></msup><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mi mathvariant='normal'>$pA</mi></mrow></mfenced><mi mathvariant='normal'>$Successes</mi></msup><mfenced open='(' close=')' separators=','><mrow><mn>0.5</mn></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><munder><mi mathvariant='normal'>$Flashes</mi><mi mathvariant='normal'>$Fails</mi></munder></mrow></mfenced><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$pB</mi></mrow></mfenced><mi mathvariant='normal'>$Fails</mi></msup><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mi mathvariant='normal'>$pB</mi></mrow></mfenced><mi mathvariant='normal'>$Successes</mi></msup><mfenced open='(' close=')' separators=','><mrow><mn>0.5</mn></mrow></mfenced></mrow></mfrac></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&equals;</mo><mfrac linethickness='1' denomalign='center' numalign='center' bevelled='false'><mrow><msup superscriptshift='0'><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$pB</mi><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$Fails</mi></msup><msup superscriptshift='0'><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$OneMinuspB</mi><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$Successes</mi></msup></mrow><mrow><msup superscriptshift='0'><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$pA</mi><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$Fails</mi></msup><msup superscriptshift='0'><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$OneMinuspA</mi><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$Successes</mi></msup><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='mediummathspace' rspace='mediummathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&plus;</mo><msup superscriptshift='0'><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$pB</mi><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$Fails</mi></msup><msup superscriptshift='0'><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$OneMinuspB</mi><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$Successes</mi></msup></mrow></mfrac></mrow></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&equals;</mo><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$pBUsed</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&period;</mo></mrow></math></em></p>@
qu.1.23.editing=useHTML@
qu.1.23.solution=@
qu.1.23.algorithm=$Q=1;
$Which=rint(5)+1;
$Align=switch(rint(2),"Left","Right");
$pA	=	decimal(2,range(0.05,0.25,.01));
$OneMinuspA = 1 - $pA;
$pAp	=	100*$pA;
$pB	=	decimal(2,range(0.05,0.25,.01));
$OneMinuspB = 1 - $pB;
$pBp	=	100*$pB;
$Flashes=	range(5,25,5);
$Holder	=	$Flashes/5;
$Fails	=	range(2,2+$Holder,1);
$Successes = $Flashes - $Fails;
$pA1	=	$pA^$Fails;
$pA2	=	(1-$pA)^($Flashes - $Fails);
$pB1	=	$pB^($Fails);
$pB2	=	(1-$pB)^(($Flashes) - ($Fails));
$pBUsed =	$pB1*$pB2/($pA1*$pA2 + $pB1*$pB2);@
qu.1.23.uid=464f0533-07e4-4722-a584-2cb8996aa62c@
qu.1.23.info=  Keyword=partition rule;
  Difficulty=3;
  Course=230;
  Type=numeric;
  Author=Sean Scott;
@

qu.1.24.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q"><img hspace="4" align="$Align" src="__BASE_URI__DPM/Binomial/Typist$Which.gif" alt="Typist" title="Typist [IMG:Typist$Which.gif]" />A typist at a constant speed of $WPM words per minute makes a mistake in any particular word with probability $PMistake, independently from word to word. Each incorrect word must be corrected, a task which takes $CTime seconds per word. Find the mean of the time (in seconds, round-off to an integer) it takes to finish a $Passage-word passage.</div>@
qu.1.24.answer.num=$Ans@
qu.1.24.answer.units=@
qu.1.24.showUnits=false@
qu.1.24.grading=toler_abs@
qu.1.24.err=1@
qu.1.24.negStyle=minus@
qu.1.24.numStyle=thousands scientific dollars arithmetic@
qu.1.24.mode=Numeric@
qu.1.24.name=23a. Typing: Mean@
qu.1.24.comment=<p>Let <span style="font-style: italic;">X</span> be the number of words typed wrong.  <span style="font-style: italic;">X ~ Bi(n,p)</span>  where <em>n </em>is the number of words to be typed, and<em> p</em> is the  probability of an error. Here we have X ~ Bin($Passage,$PMistake) .</p>
<p>Then <span style="font-style: italic;">E(X) = np = $ExpWrong</span> and  <span style="font-style: italic;">Var(X) = np(1 - p)   </span>= $VarWrong .</p>
<p>Notice also that the time to type each word is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>60</mn><mrow><mo>$WPM</mo></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>$SecPerWord</mo></mrow></mstyle></math> in seconds. This is so for correct and incorrect words, incorrect words require additional time to correct.</p>
<p>The time <span style="font-style: italic;">T</span> to type the passage is ($SecPerWord second(s) for each correct word) + ($SecPerWord second(s) + $CTime seconds to correct for each incorrect word)<br />
<br />
<em>T</em> = $SecPerWord($Passage - <em>X</em>) + ($SecPerWord + $CTime)<em>X</em> = $SecPerWord($Passage) + $CTime*<em>X</em>    <br />
so</p>
<p><span style="font-style: italic;">E(T)</span> = $SecPerWord($Passage) + $CTime*E(X) = $SecPerWord($Passage) + $CTime($ExpWrong) = $Ans</p>@
qu.1.24.editing=useHTML@
qu.1.24.solution=@
qu.1.24.algorithm=$Q="23a";
$WPM=range(50,110,5);
$SecPerWord=decimal(2,60/$WPM);
$PMistake=decimal(2,range(0.02,0.09,0.01));
$CTime=range(9,18,1);
$Passage=range(400,750,25);
$ExpWrong=$Passage*$PMistake;
$VarWrong=$ExpWrong*(1-$PMistake);
$Ans=decimal(0,$SecPerWord*$Passage + $CTime*$ExpWrong);
$Which=rint(4);
$Align=switch(rint(2),"Left","Right");@
qu.1.24.uid=134f771d-8b74-46c1-9302-5b0cc48482da@
qu.1.24.info=  Difficulty=3;
  Keyword=binomial;
  Keyword=expected value;
  Type=numeric;
@

qu.1.25.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q"><img hspace="4" align="$Align" alt="Camera" title="Camera [IMG:Camera$Which.gif]" src="__BASE_URI__DPM/Binomial/Camera$Which.gif" />Two identical cameras, call them A and B, have their flashes fail $pAp% and $pBp% of the time respectively. A photographer selects a camera at random and uses it to take 10 shots with the flash. What is the probability that the flash failed exactly twice? (Please answer to 4 decimals of accuracy.)</div>@
qu.1.25.answer.num=$ANSWER@
qu.1.25.answer.units=@
qu.1.25.showUnits=false@
qu.1.25.grading=toler_abs@
qu.1.25.err=.001@
qu.1.25.negStyle=minus@
qu.1.25.numStyle=thousands scientific dollars arithmetic@
qu.1.25.mode=Numeric@
qu.1.25.name=10. Pick camera, P(flash fails twice)@
qu.1.25.comment=<p title="C6_BINOMIAL_DISTRIBUTIONS_II_2">Let X<sub>A </sub>&nbsp;(X<sub>B</sub>)&nbsp; be the number of times Camera A's (B's) flash fails in 10 shots.</p>
<p>Let <em>p<sub>A</sub></em> (<em>p<sub>B</sub>) </em>be the probability that Camera A's (B's) flash fails.</p>
<p>Then X<sub>A</sub> ~ Bi(10,<em>p<sub>A</sub></em>) and X<sub>B</sub> ~ Bi(10,<em>p<sub>B</sub></em>)</p>
<p>Finally,&nbsp; let F be the event "flash fails exactly twice". Then:</p>
<p>P(F) = P(F|A)P(A) + P(F|B)P(B)&nbsp;&nbsp;&nbsp; (<em>this is an example of the Partition rule)</em><br />
= P(X<sub>A</sub> = 2)(0.5) + P(X<sub>B</sub> = 2)(0.5)<br />
&nbsp;=&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><munder><mrow><mn>10</mn></mrow><mrow><mn>2</mn></mrow></munder></mrow></mfenced></mrow></mstyle></math>(<em>p<sub>A</sub></em>)<sup>2</sup>(1 - <em>p<sub>A</sub></em>)<sup>8</sup>(0.5) + <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><munder><mrow><mn>10</mn></mrow><mrow><mn>2</mn></mrow></munder></mrow></mfenced></mrow></mstyle></math>(<em>p<sub>B</sub></em>)<sup>2</sup>(1 - <em>p<sub>B</sub></em>)<sup>8</sup>(0.5)<br />
=0.5 <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><munder><mrow><mn>10</mn></mrow><mrow><mn>2</mn></mrow></munder></mrow></mfenced></mrow></mstyle></math>((<em>p<sub>A</sub></em>)<sup>2</sup>(1 - <em>p<sub>A</sub></em>)<sup>8</sup> + (<em>p<sub>B</sub></em>)<sup>2</sup>(1 - <em>p<sub>B</sub></em>)<sup>8</sup>) =$ans which rounds to $ANSWER</p>@
qu.1.25.editing=useHTML@
qu.1.25.solution=@
qu.1.25.algorithm=$Q=10;
$Which=rint(5)+1;
$Align=switch(rint(2),"Left","Right");
$pA=decimal(2,range(0.05,0.35,0.01));
$pB=decimal(2,range(0.05,0.25,0.01));
$pAp=$pA*100;
$pBp=$pB*100;
$npA=1-$pA;
$npB=1-$pB;
$c=fact(10)/(fact(8)*fact(2));
$ans=0.5*$c*($pA^2*$npA^8+$pB^2*$npB^8);
$ANSWER=decimal(4,$ans);@
qu.1.25.uid=bcbf2926-340e-44de-8502-001307b44f0d@
qu.1.25.info=  Course=230;
  Type=numeric;
@

qu.1.26.mode=Inline@
qu.1.26.name=17. Most likely outcome@
qu.1.26.comment=<p>The brute force approach is to just work out the probability of each possible outcome and see which is biggest. Use the formula: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>P</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>n</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>6</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi mathvariant='normal'>n</mi></mrow></mtd></mtr></mtable></mfenced><msup><mi mathvariant='normal'>$p</mi><mrow><mi>n</mi></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$p</mi></mrow></mfenced><mrow><mn>6</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>n</mi></mrow></msup></mrow></mstyle></math> to get (to 4 decimals):</p>
<p>
<table cellspacing="0" cellpadding="2" bordercolor="#111111" border="1" style="border-collapse: collapse;" id="AutoNumber1">
    <tbody>
        <tr>
            <td>n</td>
            <td>P(x=n)</td>
        </tr>
        <tr>
            <td>0</td>
            <td align="center">$x0</td>
        </tr>
        <tr>
            <td>1</td>
            <td align="center">$x1</td>
        </tr>
        <tr>
            <td>2</td>
            <td align="center">$x2</td>
        </tr>
        <tr>
            <td>3</td>
            <td align="center">$x3</td>
        </tr>
        <tr>
            <td>4</td>
            <td align="center">$x4</td>
        </tr>
        <tr>
            <td>5</td>
            <td align="center">$x5</td>
        </tr>
        <tr>
            <td>6</td>
            <td align="center">$x6</td>
        </tr>
    </tbody>
</table>
</p>
<p>From which you can read the most likely outcome is x = $ans</p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.1.26.editing=useHTML@
qu.1.26.solution=@
qu.1.26.algorithm=$Q=17;
$p=decimal(1,range(.1,.9,.1));
$q=1-$p;
$px0=decimal(4,$q^6);
$x0=if(gt($px0,0.0001),$px0,0);
$x1=decimal(4,6*$p*($q^5));
$x2=decimal(4,15*($p^2)*($q^4));
$x3=decimal(4,20*($p^3)*($q^3));
$x4=decimal(4,15*($p^4)*($q^2));
$x5=decimal(4,6*($p^5)*$q);
$px6=decimal(4,$p^6);
$x6=if(gt($px6,0.0001),$px6,0);
$max=max($x0,$x1,$x2,$x3,$x4,$x5,$x6);
$ans=if(eq($max,$x1),1,if(eq($max,$x2),2,if(eq($max,$x3),3,if(eq($max,$x4),4,if(eq($max,$x5),5,if(eq($max,$x6),6,0))))));
$m=int(7*$p);@
qu.1.26.uid=6e3204e6-4937-49f4-b890-6863a1449e47@
qu.1.26.info=  Course=202;
  Type=numeric;
@
qu.1.26.weighting=1@
qu.1.26.numbering=alpha@
qu.1.26.part.1.name=sro_id_1@
qu.1.26.part.1.answer.units=@
qu.1.26.part.1.numStyle=   @
qu.1.26.part.1.editing=useHTML@
qu.1.26.part.1.showUnits=false@
qu.1.26.part.1.question=(Unset)@
qu.1.26.part.1.mode=Numeric@
qu.1.26.part.1.grading=exact_value@
qu.1.26.part.1.negStyle=both@
qu.1.26.part.1.answer.num=$ans@
qu.1.26.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q">Suppose that X has a binomial distribution with parameters n=6 and p=$p. The most likely outcome is X = <1><span>&nbsp;</span></div>@

qu.1.27.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q"><img hspace="4" align="$Align" src="__BASE_URI__DPM/Binomial/Derrick$Which.gif" alt="Oil Derrick" title="Oil Derrick [Derrick$Which.gif]" />An oil exploration firm is going to make $n explorations. The probability of a particular exploration being successful is $p. Assume that explorations are independent of each other. The mean of the number of successful explorations is: (Please answer to 3 decimals of accuracy.)</div>@
qu.1.27.answer.num=$MeanIs@
qu.1.27.answer.units=@
qu.1.27.showUnits=false@
qu.1.27.grading=toler_abs@
qu.1.27.err=.01@
qu.1.27.negStyle=minus@
qu.1.27.numStyle=thousands scientific dollars arithmetic@
qu.1.27.mode=Numeric@
qu.1.27.name=19a. Oil drilling: Mean (Numeric)@
qu.1.27.comment=This is a binomial distribution so the mean is np . In this case that is $n*$p = $MeanIs .<br>@
qu.1.27.editing=useHTML@
qu.1.27.solution=@
qu.1.27.algorithm=$Q="19a";
$n=range(12,35);
$p=decimal(2,range(.04,.40,.01));
$Which=rint(5)+1;
$Align=switch(rint(2),"Left","Right");
$MeanIs = $n*$p;@
qu.1.27.uid=a4719781-f329-4d61-a1af-fc117888ab47@
qu.1.27.info=  Difficulty=1;
  Keyword=binomial;
  Keyword=mean;
  Course=230;
@

qu.1.28.mode=Multiple Choice@
qu.1.28.name=27.  n & p, find μ & σ²@
qu.1.28.comment=<p><em>n</em> = $n is the number sampled, <em>p</em> = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi mathvariant='normal'>$Per</mi><mrow><mn>100</mn></mrow></mfrac></mrow></mstyle></math>.&nbsp; Then:</p>
<p>Mean = &mu; = <em>np</em> = $mu</p>
<p>Variance = &sigma;&sup2; = <em>np(1 - p) </em>= $Var</p>@
qu.1.28.editing=useHTML@
qu.1.28.solution=@
qu.1.28.algorithm=$Q=27;
$p = range(0.3,0.35,0.001);
$n = range(180,185,1);
$X1 = range(50,53,1);
$X2 = range(65,67,1);
$mu = decimal(2,$p*$n);
$Var = decimal(2,$n*$p*(1-$p));
$Alt1mu = decimal(2,range(1.1,1.9,0.01)*$mu);
$Alt1Var= decimal(2,range(1.1,1.9,0.01)*$Var);
$Alt2mu = decimal(2,range(0.6,0.9,0.01)*$mu);
$Alt2Var= decimal(2,range(0.6,0.9,0.01)*$Var);
$Alt3mu = $Alt1mu;
$Alt3Var=$Var;
$Alt4mu=$mu;
$Alt4Var=$Alt2Var;
$Per = 100*$p;@
qu.1.28.uid=85f29faf-b4b0-42e9-92db-f5b009b9924d@
qu.1.28.info=  Course=202;
  Type=MC;
@
qu.1.28.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q">A television station estimates that $Per% of college students watch the Super Bowl. For a sample of $n students selected at random, what is the mean and variance of the number of students who watch this game?</div>@
qu.1.28.answer=1@
qu.1.28.choice.1=Mean = $mu, Variance = $Var@
qu.1.28.choice.2=Mean = $Alt1mu, Variance = $Alt1Var@
qu.1.28.choice.3=Mean = $Alt2mu, Variance = $Alt2Var@
qu.1.28.choice.4=Mean = $Alt3mu, Variance = $Alt3Var@
qu.1.28.choice.5=Mean = $Alt4mu, Variance = $Alt4Var@
qu.1.28.fixed=@

qu.1.29.mode=Multiple Choice@
qu.1.29.name=22b. Oil Drilling: SD Total Cost@
qu.1.29.comment=<p>First let X be the number of successful wells. Notice that this is a Binomial Distribution with p = $p and n = $n, that is X ~ Bin($n,$p).</p>
<p>Then the variance is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&sigma;</mi><msubsup><mi></mi><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>np</mi><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>p</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$VarIs</mi></mrow></mstyle></math> meaning the Standard Deviation is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&sigma;</mi><mrow><mi>X</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mi mathvariant='normal'>$VarIs</mi></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$SDIs</mi></mrow></mstyle></math><br />
<br />
The cost to the firm is: <br />
<br />
C = (<span style="font-weight: bold;">F</span>ixed <span style="font-weight: bold;">C</span>osts) + (<span style="font-weight: bold;">S</span>uccess <span style="font-weight: bold;">C</span>ost)*(# successes) + (<span style="font-weight: bold;">U</span>nsuccessful <span style="font-weight: bold;">C</span>ost)*(#attempts - # successes)<br />
= $FC + $SC*X + $UC*($n - X)&nbsp; <em>which we can rewrite as:</em><br />
<em>=($SC-$UC)X + ($FC+$UC*$n)</em></p>
<p>Notice that this has the form aX + b, a linear function of the r.v. X. The Standard Deviation of costs then is found by using the formula: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&sigma;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>a</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><msub><mi>&sigma;</mi><mrow><mi>X</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$SC</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$UC</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$SDIs</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p>&nbsp;</p>@
qu.1.29.editing=useHTML@
qu.1.29.solution=@
qu.1.29.algorithm=$Q="22b";
$n=range(1,25,1);
$FC=range(15000,45000,5000);
$SC=range(25000,100000,5000);
$UC=range(25000,$SC,5000);
$p=decimal(2,range(.10,.55,.05));
$VarIs = $n*$p*(1-$p);
$SDIs=sqrt($VarIs);
$Ans = decimal(2,($SC-$UC)*$SDIs);
$Alt1=decimal(2,range(1.1,1.9,0.01)*$Ans);
$Alt2=decimal(2,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(2,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));
$Which=rint(5)+1;
$Align=switch(rint(2),"Left","Right");@
qu.1.29.uid=37fd27e4-a06e-452f-964c-c3fa230592c9@
qu.1.29.info=  Difficulty=3;
  Keyword=binomial;
  Keyword=variance;
  Keyword=standard deviation;
  Course=230;
@
qu.1.29.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q"><img hspace="4" align="$Align" title="Oil Derrick [Derrick$Which.gif]" alt="Oil Derrick" src="__BASE_URI__DPM/Binomial/Derrick$Which.gif" />An oil exploration firm is formed with enough capital to finance $n explorations. The probability of a particular exploration being successful is $p. Assume that explorations are independent of each other. The firm has fixed costs of  \\$$FC prior to carrying out its first exploration. Each successful exploration costs \\$$SC and each unsuccessful one costs \\$$UC. The Standard Deviation of the total cost to the firm of its $n explorations is:.</div>@
qu.1.29.answer=1@
qu.1.29.choice.1=$Ans@
qu.1.29.choice.2=$Alt1@
qu.1.29.choice.3=$Alt2@
qu.1.29.choice.4=$Alt3@
qu.1.29.fixed=@

qu.1.30.mode=Multiple Choice@
qu.1.30.name=09. P(Series ends in game n)@
qu.1.30.comment=<div style="margin-top: 0px; margin-bottom: 2px;" class="shadedDiv descriptionSpan">
<p>We have a binomial situation. Define success as team A winning a game. Then p =&nbsp;$pml = 1 - p and n =&nbsp;$n . The correct calculation (for the answer $AnsML) is shown in <font color="#ff0000">red</font> below:<br />
<br />
<table>
    <tbody>
        <tr>
            <td>$R1$HR1            P(Series terminates at the end of 4 games) = P(A wins all 4 or B wins all 4)<br />
            Consider the case where B wins all 4. Then x = 0 and:<br />
            P(series ends in 4 with B the winner) = P(X = 0; n = 4, p = 0.5) = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>4</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd></mtr></mtable></mfenced></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced><mrow><mn>0</mn></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced><mrow><mn>4</mn></mrow></msup></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mn>1</mn><mrow><mn>16</mn></mrow></mfrac></mrow></mrow></mstyle></math><br />
            By symmetry, P(series ends in 4 with A the winner) = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>16</mn></mrow></mfrac></mrow></mstyle></math>, so<br />
            <br />
            P(series ends in 4) = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>16</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mn>1</mn><mrow><mn>8</mn></mrow></mfrac></mrow></mrow></mstyle></math>$HR1            $EndSpan</td>
        </tr>
        <tr>
            <td>$R2$HR2            P(Series terminates at the end of 5 games) = 2P(A wins in 5)&nbsp;&nbsp; <span style="font-style: italic;">by the symmetry of A and B</span><br />
            = 2 P(A wins exactly 3 of the first 4, then fifth)<br />
            = 2 P(X = 3; n = 4, p = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mstyle></math>)P(A wins 5th)<br />
            = 2<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>4</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>3</mn></mrow></mtd></mtr></mtable></mfenced></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced><mrow><mn>3</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>8</mn><mrow><mn>32</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mn>1</mn><mrow><mn>4</mn></mrow></mfrac></mrow></mrow></mstyle></math>$HR2            $EndSpan</td>
        </tr>
        <tr>
            <td>$R3$HR3
            P(Series ends after 6th game)= P(A wins in 6 OR B wins in 6)= 2P(A wins in 6)&nbsp; <span style="font-style: italic;">by the symmetry of A and B</span><br />
            = 2P(A wins 3 of the first 5)P(A wins 6th)<br />
            = 2 P(X = 3; n = 5, p = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mstyle></math>)<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced></mrow></mstyle></math><br />
            = 2<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>5</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>3</mn></mrow></mtd></mtr></mtable></mfenced></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced><mrow><mn>3</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>20</mn><mrow><mn>64</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mn>5</mn><mrow><mn>16</mn></mrow></mfrac></mrow></mrow></mstyle></math>$HR3
            $EndSpan</td>
        </tr>
        <tr>
            <td>$R4$HR4
            P(Series runs 7 games)= P(even after 6 games)<br />
            = P(X = 3; n = 6, p = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mstyle></math>)<br />
            =<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>6</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>3</mn></mrow></mtd></mtr></mtable></mfenced></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced><mrow><mn>3</mn></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced><mrow><mn>3</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mn>5</mn><mrow><mn>16</mn></mrow></mfrac></mrow></mrow></mstyle></math>.$HR4
            $EndSpan</td>
        </tr>
    </tbody>
</table>
</p>
</div>@
qu.1.30.editing=useHTML@
qu.1.30.solution=@
qu.1.30.algorithm=$Q=9;
$p=1/2;
$pml=mathml("1/2");
$n=range(4,7,1);
$R1=if(eq($n,4),"<span style='color:red'>","<span style='color:#999999'>");
$R2=if(eq($n,5),"<span style='color:red'>","<span style='color:#999999'>");
$R3=if(eq($n,6),"<span style='color:red'>","<span style='color:#999999'>");
$R4=if(eq($n,7),"<span style='color:red'>","<span style='color:#999999'>");
$HR1=if(eq($n,4),"<hr style='color:red'>","<hr style='color:#999999'>");
$HR2=if(eq($n,5),"<hr style='color:red'>","<hr style='color:#999999'>");
$HR3=if(eq($n,6),"<hr style='color:red'>","<hr style='color:#999999'>");
$HR4=if(eq($n,7),"<hr style='color:red'>","<hr style='color:#999999'>");
$EndSpan="</span>";
$Pick=if(eq($n,7),2,$n-4);
$AnsML=switch($Pick,mathml("1/8"),mathml("1/4"),mathml("5/16"));
$Alt1ML=switch($Pick,mathml("1/4"),mathml("5/16"),mathml("1/8"));
$Alt2ML=switch($Pick,mathml("5/16"),mathml("1/8"),mathml("1/4"));
$Alt3ML=switch(rint(2),mathml("7/16"),mathml("3/4"));
$Alt4ML=switch(rint(2),mathml("3/7"),mathml("5/7"));@
qu.1.30.uid=f62d46de-d186-425e-8582-5ac4ca873e89@
qu.1.30.info=  Course=230;
  Type=MC;
@
qu.1.30.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q"><img width="130" hspace="4" height="124" align="right" alt="World Series Crest" src="__BASE_URI__DPM/Binomial/WorldSeries.gif" />The World Series terminates when one team wins its fourth game. Suppose the two teams are evenly matched, so each has probability 1/2 of winning any one game. What is the probability that the series will take&nbsp;$n games?</div>@
qu.1.30.answer=1@
qu.1.30.choice.1=$AnsML@
qu.1.30.choice.2=$Alt1ML@
qu.1.30.choice.3=$Alt2ML@
qu.1.30.choice.4=$Alt3ML@
qu.1.30.choice.5=$Alt4ML@
qu.1.30.fixed=4@

qu.1.31.mode=Multiple Choice@
qu.1.31.name=19b. Oil drilling: Mean@
qu.1.31.comment=<p>This is a binomial distribution so the mean is np . In this case that is $n*$p = $Ans .</p>@
qu.1.31.editing=useHTML@
qu.1.31.solution=@
qu.1.31.algorithm=$Q="19b";
$n=range(12,35,1);
$p=decimal(2,range(.04,.40,.01));
$Which=rint(5)+1;
$Align=switch(rint(2),"Left","Right");
$Ans = $n*$p;
$Alt1=decimal(2,range(1.1,1.9,0.01)*$Ans);
$Alt2=decimal(2,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(2,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.1.31.uid=5af5c034-3a04-468b-bf3b-56bdc5fe7abc@
qu.1.31.info=  Difficulty=1;
  Keyword=binomial;
  Keyword=mean;
  Course=230;
@
qu.1.31.question=<div title="STAT230/Chapter 7/Binomial Distribution/Q1 C71001"><img hspace="4" align="$Align" title="Oil Derrick [Derrick$Which.gif]" alt="Oil Derrick" src="__BASE_URI__DPM/Binomial/Derrick$Which.gif" />An oil exploration firm is going to make $n explorations. The probability of a particular exploration being successful is $p. Assume that explorations are independent of each other. The mean of the number of successful explorations is:</div>@
qu.1.31.answer=1@
qu.1.31.choice.1=$Ans@
qu.1.31.choice.2=$Alt1@
qu.1.31.choice.3=$Alt2@
qu.1.31.choice.4=$Alt3@
qu.1.31.fixed=@

qu.1.32.mode=Multiple Choice@
qu.1.32.name=23b. Typing: Mean@
qu.1.32.comment=<p>Let <span style="font-style: italic;">X</span> be the number of words typed wrong.  <span style="font-style: italic;">X ~ Bi(n,p)</span>  where <em>n </em>is the number of words to be typed, and<em> p</em> is the  probability of an error. Here we have X ~ Bin($Passage,$PMistake) .</p>
<p>Then <span style="font-style: italic;">E(X) = np = $ExpWrong</span> and  <span style="font-style: italic;">Var(X) = np(1 - p)   </span>= $VarWrong .</p>
<p>Notice also that the time to type each word is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>60</mn><mrow><mo>$WPM</mo></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>$SecPerWord</mo></mrow></mstyle></math> in seconds. This is so for correct and incorrect words, incorrect words require additional time to correct.</p>
<p>The time <span style="font-style: italic;">T</span> to type the passage is ($SecPerWord second(s) for each correct word) + ($SecPerWord second(s) + $CTime seconds to correct for each incorrect word)<br />
<br />
<em>T</em> = $SecPerWord($Passage - <em>X</em>) + ($SecPerWord + $CTime)<em>X</em> = $SecPerWord($Passage) + $CTime*<em>X</em>    <br />
so</p>
<p><span style="font-style: italic;">E(T)</span> = $SecPerWord($Passage) + $CTime*E(X) = $SecPerWord($Passage) + $CTime($ExpWrong) = $Ans</p>@
qu.1.32.editing=useHTML@
qu.1.32.solution=@
qu.1.32.algorithm=$Q="23b";
$WPM=range(50,110,5);
$SecPerWord=decimal(2,60/$WPM);
$PMistake=decimal(2,range(0.02,0.09,0.01));
$CTime=range(9,18,1);
$Passage=range(400,750,25);
$ExpWrong=$Passage*$PMistake;
$VarWrong=$ExpWrong*(1-$PMistake);
$Ans=decimal(0,$SecPerWord*$Passage + $CTime*$ExpWrong);
$Alt1=int(range(1.1,1.9,0.01)*$Ans);
$Alt2=int(range(0.5,0.9,0.01)*$Ans);
$Alt3=int(0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));
$Which=rint(4);
$Align=switch(rint(2),"Left","Right");@
qu.1.32.uid=346007dc-3851-4fdf-81f4-dd8c64173715@
qu.1.32.info=  Difficulty=3;
  Keyword=binomial;
  Keyword=expected value;
  Type=MC;
@
qu.1.32.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q"><img hspace="4" align="$Align" src="__BASE_URI__DPM/Binomial/Typist$Which.gif" alt="Typist" title="Typist [IMG:Typist$Which.gif]" />A typist at a constant speed of $WPM words per minute makes a mistake in any particular word with probability $PMistake, independently from word to word. Each incorrect word must be corrected, a task which takes $CTime seconds per word. Find the mean of the time (in seconds, round-off to an integer) it takes to finish a $Passage-word passage.</div>@
qu.1.32.answer=1@
qu.1.32.choice.1=$Ans@
qu.1.32.choice.2=$Alt1@
qu.1.32.choice.3=$Alt2@
qu.1.32.choice.4=$Alt3@
qu.1.32.fixed=@

qu.1.33.mode=Multiple Choice@
qu.1.33.name=21b. Mean of Total Cost (MC)@
qu.1.33.comment=<p>First let X be the number of successful wells. Notice that this is a Binomial Distribution with p = $p and n = $n, that is X ~ Bin($n,$p).</p>
<p>Then the mean is &mu;<sub>X</sub> = np = $MeanIs.<br />
<br />
The cost to the firm is: <br />
<br />
C = (<span style="font-weight: bold;">F</span>ixed <span style="font-weight: bold;">C</span>osts) + (<span style="font-weight: bold;">S</span>uccess <span style="font-weight: bold;">C</span>ost)*(# successes) + (<span style="font-weight: bold;">U</span>nsuccessful <span style="font-weight: bold;">C</span>ost)*(#attempts - # successes)<br />
= $FC + $SC*X + $UC*($n - X)&nbsp;&nbsp;&nbsp;&nbsp; (in \\$)<br />
<br />
The Mean of costs then is found by substituting the mean &mu;<sub>X </sub>= $MeanIs for X:<br />
<br />
E(C) = $FC + $SC*$MeanIs + $UC)*($n - $MeanIs)</p>
<p>= \\$$Ans</p>@
qu.1.33.editing=useHTML@
qu.1.33.solution=@
qu.1.33.algorithm=$Q="21b";
$n=range(1,25,1);
$FC=range(15000,45000,5000);
$SC=range(25000,100000,5000);
$UC=range(25000,$SC,5000);
$p=decimal(2,range(.10,.55,.05));
$MeanIs = $n*$p;
$Ans = $FC + ($MeanIs)*$SC + ($n-$MeanIs)*$UC;
$Alt1=int(range(1.1,1.9,0.01)*$Ans);
$Alt2=int(range(0.5,0.9,0.01)*$Ans);
$Alt3=int(0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));
$Which=rint(5)+1;
$Align=switch(rint(2),"Left","Right");@
qu.1.33.uid=2bbc371d-6bd0-4c84-bed4-cb16b5ccdcea@
qu.1.33.info=  Difficulty=3;
  Keyword=binomial;
  Keyword=mean;
  Keyword=expected value;
  Course=230;
@
qu.1.33.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q"><img hspace="4" align="$Align" title="Oil Derrick [Derrick$Which.gif]" alt="Oil Derrick" src="__BASE_URI__DPM/Binomial/Derrick$Which.gif" /> An oil exploration firm is formed with enough capital to finance $n explorations. The probability of a particular exploration being successful is $p. Assume that explorations are independent of each other. The firm has fixed costs of  \\$$FC prior to carrying out its first exploration. Each successful exploration costs \\$$SC and each unsuccessful one costs \\$$UC. Find, to the nearest dollar, the Mean of the total cost to the firm of its $n explorations.</div>@
qu.1.33.answer=1@
qu.1.33.choice.1=$Ans@
qu.1.33.choice.2=$Alt1@
qu.1.33.choice.3=$Alt2@
qu.1.33.choice.4=$Alt3@
qu.1.33.fixed=@

qu.1.34.mode=Inline@
qu.1.34.name=18. Properties of Binomial Distribution@
qu.1.34.comment=@
qu.1.34.editing=useHTML@
qu.1.34.solution=@
qu.1.34.algorithm=$Q = 18;@
qu.1.34.uid=604c11bc-7c2f-4c9b-89ba-a3a5724158dd@
qu.1.34.info=  Course=202;
  Type=MS;
@
qu.1.34.weighting=1@
qu.1.34.numbering=alpha@
qu.1.34.part.1.name=sro_id_1@
qu.1.34.part.1.editing=useHTML@
qu.1.34.part.1.fixed=@
qu.1.34.part.1.choice.4=There are only two outcomes in each trial<br>@
qu.1.34.part.1.question=null@
qu.1.34.part.1.choice.3=Trials are independent<br>@
qu.1.34.part.1.choice.2=Trials are repeated@
qu.1.34.part.1.choice.1=Trials have identical probability of success@
qu.1.34.part.1.mode=Multiple Selection@
qu.1.34.part.1.display=vertical@
qu.1.34.part.1.answer=1,2,3,4@
qu.1.34.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q">What are the properties of Binomial distribution? <em>(Select all correct choices)</em><p><span>&nbsp;</span><1><span>&nbsp;</span></p></div>@

qu.1.35.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q"><img hspace="4" align="$Align" title="Oil Derrick [Derrick$Which.gif]" alt="Oil Derrick" src="__BASE_URI__DPM/Binomial/Derrick$Which.gif" />An oil exploration firm is formed with enough capital to finance $n explorations. The probability of a particular exploration being successful is $p. Assume that explorations are independent of each other. The firm has fixed costs of  \\$$FC prior to carrying out its first exploration. Each successful exploration costs \\$$SC and each unsuccessful one costs \\$$UC. Find, <strong>to the nearest cent,</strong> the Standard Deviation of the total cost to the firm of its $n explorations.</div>@
qu.1.35.answer.num=$Ans@
qu.1.35.answer.units=@
qu.1.35.showUnits=false@
qu.1.35.grading=toler_abs@
qu.1.35.err=1@
qu.1.35.negStyle=minus@
qu.1.35.numStyle=thousands scientific dollars arithmetic@
qu.1.35.mode=Numeric@
qu.1.35.name=22a. Oil Drilling: SD Total Cost@
qu.1.35.comment=<p>First let X be the number of successful wells. Notice that this is a Binomial Distribution with p = $p and n = $n, that is X ~ Bin($n,$p).</p>
<p>Then the variance is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&sigma;</mi><msubsup><mi></mi><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>np</mi><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>p</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$VarIs</mi></mrow></mstyle></math> meaning the Standard Deviation is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&sigma;</mi><mrow><mi>X</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mi mathvariant='normal'>$VarIs</mi></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$SDIs</mi></mrow></mstyle></math><br />
<br />
The cost to the firm is: <br />
<br />
C = (<span style="font-weight: bold;">F</span>ixed <span style="font-weight: bold;">C</span>osts) + (<span style="font-weight: bold;">S</span>uccess <span style="font-weight: bold;">C</span>ost)*(# successes) + (<span style="font-weight: bold;">U</span>nsuccessful <span style="font-weight: bold;">C</span>ost)*(#attempts - # successes)<br />
= $FC + $SC*X + $UC*($n - X)&nbsp; <em>which we can rewrite as:</em><br />
<em>=($SC-$UC)X + ($FC+$UC*$n)</em></p>
<p>Notice that this has the form aX + b, a linear function of the r.v. X. The Standard Deviation of costs then is found by using the formula: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&sigma;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>a</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><msub><mi>&sigma;</mi><mrow><mi>X</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$SC</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$UC</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$SDIs</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p>&nbsp;</p>@
qu.1.35.editing=useHTML@
qu.1.35.solution=@
qu.1.35.algorithm=$Q="22a";
$n=range(1,25,1);
$FC=range(15000,45000,5000);
$SC=range(25000,100000,5000);
$UC=range(25000,$SC,5000);
$p=decimal(2,range(.10,.55,.05));
$VarIs = $n*$p*(1-$p);
$SDIs=sqrt($VarIs);
$Ans = decimal(2,($SC-$UC)*$SDIs);
$Which=rint(5)+1;
$Align=switch(rint(2),"Left","Right");@
qu.1.35.uid=16872051-1aad-452e-ade3-6021ae342848@
qu.1.35.info=  Difficulty=3;
  Keyword=binomial;
  Keyword=variance;
  Keyword=standard deviation;
  Course=230;
  Type=numeric;
@

qu.1.36.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q"><img hspace="4" align="$Align" title="Baseball [IMG:Baseball$Which.gif]" alt="Baseball" src="__BASE_URI__DPM/Binomial/Baseball$Which.gif" />A baseball player has a $pp% chance of hitting the ball each time at bat, with succesive times at bat being independent. Calculate the probability that he gets at least 2 hits in&nbsp;$n times at bat. Answer to 3 decimals please.</div>@
qu.1.36.answer.num=$Ans@
qu.1.36.answer.units=@
qu.1.36.showUnits=false@
qu.1.36.grading=toler_abs@
qu.1.36.err=0.001@
qu.1.36.negStyle=minus@
qu.1.36.numStyle=thousands scientific dollars arithmetic@
qu.1.36.mode=Numeric@
qu.1.36.name=07. P(>=2 hits|n at bats)@
qu.1.36.comment=<div style="margin-top: 0px; margin-bottom: 2px;" class="shadedDiv descriptionSpan">Let X be the number of hits in&nbsp;$n at bats. This is an example of a <span style="font-weight: bold;">Binomial</span> distribution with p = $p, n = $n.&nbsp; We want P(X&nbsp;&ge; 2) = 1 - P(X < 2) = 1 - P(X = 0) - P(X = 1) = 1 - <sub><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mi>$n</mi></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd></mtr></mtable></mfenced></mrow></mstyle></math></sub>($p)<sup>0</sup>(1-$p)<sup>$n</sup> - <sub><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mi>$n</mi></mrow></mtd></mtr><mtr><mtd><mrow><mn>1</mn></mrow></mtd></mtr></mtable></mfenced></mrow></mstyle></math></sub>($p)<sup>1</sup>(1-$p)<sup>$n-1</sup> = $A which rounds to&nbsp;$Ans .</div>@
qu.1.36.editing=useHTML@
qu.1.36.solution=@
qu.1.36.algorithm=$Q=7;
$Which=rint(4)+1;
$Align=switch(rint(2),"Left","Right");
$p=decimal(1,range(0.2,0.7,0.1));
$pp=$p*100;
$n=range(5,12,1);
$A=maple("with(Statistics);
X := RandomVariable(Binomial($n, $p));
ProbabilityFunction(X, u);
1-ProbabilityFunction(X, 0)-ProbabilityFunction(X, 1)");
$Ans=decimal(3,$A);@
qu.1.36.uid=408627d7-af98-4b5b-948e-b684246bf7ad@
qu.1.36.info=  Course=230;
  Type=numeric;
@

qu.1.37.mode=Multiple Choice@
qu.1.37.name=04. Given n,p, P(#succ. gt x)@
qu.1.37.comment=<div class="shadedDiv descriptionSpan" style="margin-top: 0px; margin-bottom: 2px">Use the binomial formula to find and add together P($n) and P($x). Alternately, just use "logic":<br />
<br />
P($n successes) = $p<sup>$n</sup>, P($x successes) = ($n ways to select failure)P(fail)P(success)<sup>$x</sup> = $n($np)$p<sup>$x</sup> . Add these to get $ANSWER.</div>@
qu.1.37.editing=useHTML@
qu.1.37.solution=@
qu.1.37.algorithm=$Q=4;
$p=decimal(2,range(0.5,0.9,0.2));
$n=range(10,25,1);
$n=17;
$x=$n-1;
$np=1-$p;
$ns=$p^$n;
$xs=$n*(1-$p)*$p^$x;
$ANSWER=decimal(3,$ns+$xs);
$a=decimal(3,range(0.35,0.65,0.01)*$ANSWER);
$b=decimal(3,($ANSWER+1)/2);
$c=decimal(3,($ANSWER+$a)/2);
condition:lt($ANSWER,1);
condition:gt($ANSWER,0);@
qu.1.37.uid=1a7c4dd1-944c-43b6-b571-f0d402a4f818@
qu.1.37.info=  Course=230;
  Difficulty=2;
@
qu.1.37.question=<div title="UW Statistics Bank/Discrete Probability Models/Binomial Distributions/Q$Q">Given a binomial distribution in which the probability of success is $p and the number of trials is $n, what is the approximate probability of getting at least $x successes?</div>@
qu.1.37.answer=1@
qu.1.37.choice.1=$ANSWER@
qu.1.37.choice.2=$c@
qu.1.37.choice.3=$a@
qu.1.37.choice.4=$b@
qu.1.37.choice.5=None of the above@
qu.1.37.fixed=4@

qu.2.topic=Poisson Distribution@

qu.2.1.question=<div title="UW Statistics Bank/Discrete Probability Models/Poisson Distribution/Q$Q"><img width="100" hspace="4" height="72" align="right" title="Roll of Tape [IMG:tape_roll.gif]" alt="Roll of Tape" src="__BASE_URI__DPM/Poisson/tape_roll.gif" />Defects occur in a certain manufactured tape on the average of 1 per 1,000 m. Assuming a Poisson distribution for the number of defects in a given length of tape, what is the probability that in a box of five&nbsp;$n m rolls two have just one defect each and three have none? (Answer to 3 decimal accuracy, for example 0.217)</div>@
qu.2.1.answer.num=$ANSWER@
qu.2.1.answer.units=@
qu.2.1.showUnits=false@
qu.2.1.grading=toler_abs@
qu.2.1.err=.001@
qu.2.1.negStyle=minus@
qu.2.1.numStyle=thousands scientific dollars arithmetic@
qu.2.1.mode=Numeric@
qu.2.1.name=13.  Good & bad tapes@
qu.2.1.comment=<div style="margin-top: 0px; margin-bottom: 2px;" class="shadedDiv descriptionSpan">Let X = number of defects in a&nbsp;$n m roll, so X ~&nbsp; Poisson($lambda). P(X = 0) = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mn>1.2</mn><mrow><mn>0</mn></mrow></msup><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1.2</mn></mrow></msup></mrow><mrow><mn>0</mn><mo lspace='0.1111111em' rspace='0.1111111em'>&excl;</mo></mrow></mfrac></mrow></mstyle></math> =&nbsp;$p0 and P(X = 1) = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mn>1.2</mn><mrow><mn>1</mn></mrow></msup><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1.2</mn></mrow></msup></mrow><mrow><mn>1</mn><mo lspace='0.1111111em' rspace='0.1111111em'>&excl;</mo></mrow></mfrac></mrow></mstyle></math> = $p1<br />
<br />
Now we can select 3 rolls of the 5 in&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>5</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>3</mn></mrow></mtd></mtr></mtable></mfenced></mrow></mstyle></math> ways. Then by an argument similar to that which leads to the binomial distribution, <br />
P(3 rolls with no defects and 2 with one defect in a box of 5 rolls) = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>5</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>3</mn></mrow></mtd></mtr></mtable></mfenced></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$p0</mi></mrow></mfenced><mrow><mn>3</mn></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><mi>$p1</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math> = $ANSWER.</div>@
qu.2.1.editing=useHTML@
qu.2.1.solution=@
qu.2.1.algorithm=$Q=13;
$n=range(1000,3000,100);
$lambda=$n/1000;
$p0x=maple("stats[statevalf, pf, poisson[$lambda]](0)");
$p1x=maple("stats[statevalf, pf, poisson[$lambda]](1)");
$p0=decimal(3,$p0x);
$p1=decimal(3,$p1x);
$ans=maple("10*$p0^3*$p1^2");
$ANSWER=decimal(3,$ans);@
qu.2.1.uid=fe03a319-8b6d-472b-9c8e-f44c1e74efa7@
qu.2.1.info=  Course=230;
@

qu.2.2.question=<img width="150" hspace="4" height="101" align="right" title="Geiger Counter [IMG:Gieger.gif]" alt="Geiger Counter" src="__BASE_URI__DPM/Poisson/Gieger.gif" />
<div title="UW Statistics Question Bank/Discrete Probability Models/Poisson/Q$Q">Pulses arrive at a Geiger counter in accordance with a Poisson Process. If &lambda; = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi mathvariant='normal'>$Lambda</mi><mrow><mi mathvariant='normal'>hour</mi></mrow></mfrac></mrow></mstyle></math> what is the probability that 3 or more pulses arrive to the counter between 6 p.m. and 8:30 p.m. of the same day? (4 decimal accuracy)</div>@
qu.2.2.answer.num=$Ans@
qu.2.2.answer.units=@
qu.2.2.showUnits=false@
qu.2.2.grading=toler_abs@
qu.2.2.err=0.001@
qu.2.2.negStyle=minus@
qu.2.2.numStyle=thousands scientific dollars arithmetic@
qu.2.2.mode=Numeric@
qu.2.2.name=01. Geiger Counter I@
qu.2.2.comment=<p>Let Y = number of pulses that arrive between 6 and 8:30 p.m. of the same day.  Then t = 2.5, so Y ~ Poisson(&mu; = &lambda;t = 2.5&lambda;).&nbsp;  Then:</p>
<p>P(Y &ge; 3) = 1 - P(Y = 0) - P(Y = 1) - P(Y = 2)</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><msup><mi>&ExponentialE;</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2.5</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$Lambda</mi></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><mn>2.5</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$Lambda</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>2</mn><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><msup><mi>&ExponentialE;</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2.5</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$Lambda</mi></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><mn>2.5</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$Lambda</mi></mrow></mfenced><mrow><mn>1</mn></mrow></msup></mrow><mrow><mn>1</mn><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mrow><msup><mi>&ExponentialE;</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2.5</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$Lambda</mi></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><mn>2.5</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$Lambda</mi></mrow></mfenced><mrow><mn>0</mn></mrow></msup></mrow><mrow><mn>0</mn><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac></mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>&ExponentialE;</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2.5</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$Lambda</mi></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mn>25</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$Lambda</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>8</mn></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mrow><mn>5</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$Lambda</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.2.2.editing=useHTML@
qu.2.2.solution=@
qu.2.2.algorithm=$Q=1;
$Lambda=decimal(2,range(1,5,0.25));
$Ans=1-exp(-2.5*$Lambda)*(25*($Lambda)^2/8+5*$Lambda/2+1);@
qu.2.2.uid=ab3a9980-c66c-47e4-bb76-95da9222db1a@
qu.2.2.info=  Keyword=poisson;
  Course=230;
  Type=numeric;
@

qu.2.3.question=<div title="UW Statistics Bank/Discrete Probability Models/Poisson Distribution/Q$Q"><img hspace="3" border="0" align="left" src="__BASE_URI__DPM/Poisson/tape_roll.gif" title="Roll of tape [IMG:tape_roll.gif]" alt="Roll of tape" />Defects occur in a certain manufactured tape on the average of 1 per 1,000 m. Assuming a Poisson distribution for the number of defects in a given length of tape, what is the probability that a&nbsp;$n m roll will have at most 2 defects? (3 decimal accuracy)&nbsp;</div>@
qu.2.3.answer.num=$Ans@
qu.2.3.answer.units=@
qu.2.3.showUnits=false@
qu.2.3.grading=toler_abs@
qu.2.3.err=0.01@
qu.2.3.negStyle=minus@
qu.2.3.numStyle=thousands scientific dollars arithmetic@
qu.2.3.mode=Numeric@
qu.2.3.name=09a. P(<= defects)@
qu.2.3.comment=<div class="shadedDiv descriptionSpan" style="margin-top: 0px; margin-bottom: 2px;">
<div class="shadedDiv descriptionSpan" style="margin-top: 0px; margin-bottom: 2px;">
<p>Let X = number of defects in a&nbsp;$n m roll. The average number of defects in a&nbsp;$n m roll is&nbsp;$lambda and thus X ~ Poisson($lambda). <br />
P(X &le; 2) = P(X = 0) + P(X = 1) + P(X = 2)<br />
= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mi mathvariant='normal'>$lambda</mi><mrow><mn>0</mn></mrow></msup><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$lambda</mi></mrow></msup></mrow><mrow><mn>0</mn><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mrow><msup><mi mathvariant='normal'>$lambda</mi><mrow><mn>1</mn></mrow></msup><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$lambda</mi></mrow></msup></mrow><mrow><mn>1</mn><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mrow><msup><mi mathvariant='normal'>$lambda</mi><mrow><mn>2</mn></mrow></msup><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$lambda</mi></mrow></msup></mrow><mrow><mn>2</mn><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac></mrow></mstyle></math><br />
=&nbsp;$Ans .</p>
</div>
</div>@
qu.2.3.editing=useHTML@
qu.2.3.solution=@
qu.2.3.algorithm=$Q="09a";
$n=range(1000,4000,100);
$x=2;
$lambda=$n/1000;
$Ans=decimal(4,exp(-$lambda)*(1+$lambda+$lambda^2/2));@
qu.2.3.uid=a3cc3558-135f-4951-94b6-0878eed7f56e@
qu.2.3.info=  Course=230;
  Type=numeric;
  Keyword=poisson;
@

qu.2.4.mode=Multiple Choice@
qu.2.4.name=25. Chips in cookies@
qu.2.4.comment=<p>P(no more than 2 chips) = P(0 chips) + P(1 chip) + P(2 chips)</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msup><mi mathvariant='normal'>$Lambda</mi><mrow><mn>0</mn></mrow></msup><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Lambda</mi></mrow></msup></mrow><mrow><mn>0</mn><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mrow><msup><mi mathvariant='normal'>$Lambda</mi><mrow><mn>1</mn></mrow></msup><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Lambda</mi></mrow></msup></mrow><mrow><mn>1</mn><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mrow><msup><mi mathvariant='normal'>$Lambda</mi><mrow><mn>2</mn></mrow></msup><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Lambda</mi></mrow></msup></mrow><mrow><mn>2</mn><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$p0</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$p1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$p2</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.2.4.editing=useHTML@
qu.2.4.solution=@
qu.2.4.algorithm=$Q="25";
$Name=switch(rint(4),"Mary","Joan","Lucy","Robert");
$Cookie=switch(rint(4),"chocolate chip","pecan","M&M","chocolate chunk");
$Lambda=range(3,7.5,0.5);
$p0=exp(-$Lambda)/fact(0);
condition:gt($p0,0.001);
$p1=$Lambda*exp(-$Lambda)/fact(1);
$p2=$Lambda^2*exp(-$Lambda)/fact(2);
$Ans=decimal(4,$p0+$p1+$p2);
$Alt1=decimal(4,range(0.4,0.8,0.05)*$Ans);
$Alt2=decimal(4,$Ans+range(0.4,0.8,0.05)*(1-$Ans));
$Alt3=decimal(4,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));
$Which=rint(4);
$Align=switch(rint(2),"Left","Right");@
qu.2.4.uid=9706936d-a40b-42a1-9216-665978b92529@
qu.2.4.info=  Type=MC;
  Course=202;
  Keyword=poisson;
@
qu.2.4.question=<div title="UW Statistics Bank/Discrete Probability Models/Poisson Distribution/Q$Q"><img hspace="4" align="$Align" alt="Cookies" title="Cookies [IMG:Cookie$Which]" src="__BASE_URI__DPM/Poisson/Cookie$Which.gif" title="Cookies [IMG:Cookie$Which.gif]" />A food processing plant bakes $Name's $Cookie cookies.&nbsp; Once the $Cookie\\s are placed in the dough and mixed the dough is sent to extruders which shape the cookies.&nbsp; Even though the dough is thoroughly mixed, random factors or chance affect the process and as a result not all cookies have exactly the same number of $Cookie\\s.&nbsp; If the design specifications for the cookies call for an average of $Lambda $Cookie\\s per cookie what is the probability that a cookie has no more than 2 $Cookie\\s?</div>@
qu.2.4.answer=1@
qu.2.4.choice.1=$Ans@
qu.2.4.choice.2=$Alt1@
qu.2.4.choice.3=$Alt2@
qu.2.4.choice.4=$Alt3@
qu.2.4.fixed=@

qu.2.5.mode=Multiple Choice@
qu.2.5.name=14. P(2 bad cheques/day)@
qu.2.5.comment=<p>This is a Poisson process with t = 1 day, &lambda; =&nbsp;$Lambda so if X = # bad cheques/day, the probability distribution for X is: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msup><mi>&lambda;</mi><mrow><mi>x</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&lambda;</mi></mrow></mrow></msup></mrow><mrow><mi>x</mi><mo lspace='0.1111111em' rspace='0.1111111em'>&excl;</mo></mrow></mfrac></mrow></mstyle></math>, so f($x) = $Ans.</p>@
qu.2.5.editing=useHTML@
qu.2.5.solution=@
qu.2.5.algorithm=$Q=14;
$Lambda=range(3,10,1);
$x=range(2,$Lambda,1);
$PreAns=maple("stats[statevalf,pf,poisson[$Lambda]]($x);
");
$Ans=decimal(4,$PreAns);
$Alt1=decimal(4,range(0.4,0.8,0.05)*$Ans);
$Alt2=decimal(4,$Ans+range(0.4,0.8,0.05)*(1-$Ans));
$Alt3=decimal(4,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.2.5.uid=9ec6cb8b-400b-4d5e-a4f5-1c48092224fc@
qu.2.5.info=  Course=230;
  Type=MC;
@
qu.2.5.question=<div title="UW Statistics Bank/Discrete Probability Models/Poisson Distribution/Q$Q">Using the Poisson distribution and assuming a bank averages&nbsp;$Lambda bad checks a day, what is the approximate probability that the bank receives&nbsp;$x bad checks on a given day?</div>@
qu.2.5.answer=1@
qu.2.5.choice.1=$Ans@
qu.2.5.choice.2=$Alt1@
qu.2.5.choice.3=$Alt2@
qu.2.5.choice.4=$Alt3@
qu.2.5.choice.5=None of the above@
qu.2.5.fixed=4@

qu.2.6.mode=Multiple Choice@
qu.2.6.name=03. Geiger counter 3@
qu.2.6.comment=<div style="margin-top: 0px; margin-bottom: 2px;" class="shadedDiv descriptionSpan">Quite easy actually, just multiply the number of hours in a week (168) by the rate given to get 168($Lambda) = $Ans.</div>@
qu.2.6.editing=useHTML@
qu.2.6.solution=@
qu.2.6.algorithm=$Q=3;
$Lambda=range(1,20,1);
$Ans=$Lambda*168;
$Alt1=int(range(1.1,1.9,0.01)*$Ans);
$Alt2=int(range(0.5,0.9,0.01)*$Ans);
$Alt3=int(0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.2.6.uid=be833925-d7d2-447d-9489-95cdc2b7bbd9@
qu.2.6.info=  Course=230;
  Keyword=poisson;
  Type=MC;
  Difficulty=0;
@
qu.2.6.question=<div title="UW Statistics Bank/Discrete Probability Models/Poisson Distribution/Q$Q">Pulses arrive at a Geiger counter in accordance with a Poisson process. Assuming &lambda; = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi mathvariant='normal'>$Lambda</mi><mrow><mi mathvariant='normal'>hour</mi></mrow></mfrac></mrow></mstyle></math>, how many pulses are expected to arrive at the counter during a week?</div>@
qu.2.6.answer=1@
qu.2.6.choice.1=$Ans@
qu.2.6.choice.2=$Alt1@
qu.2.6.choice.3=$Alt2@
qu.2.6.choice.4=$Alt3@
qu.2.6.fixed=4@

qu.2.7.question=<div title="UW Statistics Bank/Discrete Probability Models/Poisson Distribution/Q$Q">At a particular location on a river the number of fish caught per man hour of fishing has a Poisson distribution with &lambda;= $lambda. If a man fishes there for one hour, what is the probability he will catch at least one fish? (Please answer to 3 decimals of accuracy.)</div>@
qu.2.7.answer.num=$Ans@
qu.2.7.answer.units=@
qu.2.7.showUnits=false@
qu.2.7.grading=toler_abs@
qu.2.7.err=0.005@
qu.2.7.negStyle=minus@
qu.2.7.numStyle=thousands scientific dollars arithmetic@
qu.2.7.mode=Numeric@
qu.2.7.name=12. P(>=1 fish)@
qu.2.7.comment=<div style="margin-top: 0px; margin-bottom: 2px;" class="shadedDiv descriptionSpan">P(catching at least one) = 1 - P(catching none) = 1 - e<sup>-$lambda</sup> =$ANSWER.</div>@
qu.2.7.editing=useHTML@
qu.2.7.solution=@
qu.2.7.algorithm=$Q=12;
$lambda=decimal(1,range(0.8,3,0.1));
$ans=maple("1-stats[statevalf, pf, poisson[$lambda]](0)");
$Ans=decimal(3, $ans);
condition:lt($Ans,0.97);@
qu.2.7.uid=314f82f3-e62e-49d3-8e2f-695aaee700f9@
qu.2.7.info=  Course=230;
  Type=numeric;
  Keyword=poisson;
@

qu.2.8.question=<div title="UW Statistics Bank/Discrete Probability Models/Poisson Distribution/Q$Q"><img vspace="4" hspace="4" align="$Align" title="Cake [IMG:Cake$Which.gif]" alt="Cake" src="__BASE_URI__DPM/Poisson/Cake$Which.gif" />A merchant sells on the average $mu cakes on a round of his route. Cakes are packaged two to a box but are sold individually. At the end of the round, the salesman eats any cake left over in a box that has been opened. He only opens a box after emptying the previous box. If the number of cakes sold has a Poisson distribution, then on what proportion of rounds does he have a cake to eat? (4 decimal accuracy).</div>@
qu.2.8.answer.num=$Ans@
qu.2.8.answer.units=@
qu.2.8.showUnits=false@
qu.2.8.grading=toler_abs@
qu.2.8.err=.001@
qu.2.8.negStyle=minus@
qu.2.8.numStyle=thousands scientific dollars arithmetic@
qu.2.8.mode=Numeric@
qu.2.8.name=20. Cakes@
qu.2.8.comment=<p>Let &mu; represent the average number of cakes sold in a day. He eats a cake whenever the number of cakes sold is odd so we need to evaluate&nbsp; <br />
P(X = 1) + P(X = 3) + P(X = 5) +&nbsp; &hellip;. = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&mu;</mi></mrow></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>&mu;</mi><mrow><mn>1</mn></mrow></msup></mrow><mrow><mn>1</mn><mo lspace='0.1111111em' rspace='0.1111111em'>&excl;</mo></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mrow><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&mu;</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>&mu;</mi><mrow><mn>3</mn></mrow></msup></mrow><mrow><mn>3</mn><mo lspace='0.1111111em' rspace='0.1111111em'>&excl;</mo></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mrow><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&mu;</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>&mu;</mi><mrow><mn>5</mn></mrow></msup></mrow><mrow><mn>5</mn><mo lspace='0.1111111em' rspace='0.1111111em'>&excl;</mo></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&mu;</mi></mrow></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><msup><mi>&mu;</mi><mrow><mn>1</mn></mrow></msup></mrow><mrow><mn>1</mn><mo lspace='0.1111111em' rspace='0.1111111em'>&excl;</mo></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mrow><msup><mi>&mu;</mi><mrow><mn>3</mn></mrow></msup></mrow><mrow><mn>3</mn><mo lspace='0.1111111em' rspace='0.1111111em'>&excl;</mo></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mrow><msup><mi>&mu;</mi><mrow><mn>5</mn></mrow></msup></mrow><mrow><mn>5</mn><mo lspace='0.1111111em' rspace='0.1111111em'>&excl;</mo></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mfenced></mrow></mstyle></math></p>
<p>The problem is evaluating this infinite sum! However recall the series expansion for e<sup>x</sup>:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>x</mi></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><msup><mi>x</mi><mrow><mn>1</mn></mrow></msup><mrow><mn>1</mn><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mrow><mn>2</mn><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><msup><mi>x</mi><mrow><mn>3</mn></mrow></msup><mrow><mn>3</mn><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math></p>
<p>Now replace x with -x:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>x</mi></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><msup><mi>x</mi><mrow><mn>1</mn></mrow></msup><mrow><mn>1</mn><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mrow><mn>2</mn><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><msup><mi>x</mi><mrow><mn>3</mn></mrow></msup><mrow><mn>3</mn><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math></p>
<p>Now subtract the second series from the first, divide through by 2 to get:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>x</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mn>2</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><msup><mi>x</mi><mrow><mn>3</mn></mrow></msup><mrow><mn>3</mn><mo lspace='0.1111111em' rspace='0.1111111em'>&excl;</mo></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><msup><mi>x</mi><mrow><mn>5</mn></mrow></msup><mrow><mn>5</mn><mo lspace='0.1111111em' rspace='0.1111111em'>&excl;</mo></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math></p>
<p>Substitute &mu; for x and the RHS of that equation is just the bracketed part of our expression above, so the proportion is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&mu;</mi></mrow></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><msup><mi>e</mi><mrow><mi>&mu;</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&mu;</mi></mrow></mrow></msup></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&mu;</mi></mrow></msup></mrow></mfenced></mrow></mstyle></math></p>
<p>So P(Gets to eat a cake) = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$mu</mi></mrow></mfenced></mrow></msup></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>@
qu.2.8.editing=useHTML@
qu.2.8.solution=@
qu.2.8.algorithm=$Q="20";
$Align=switch(rint(2),"Left","Right");
$Which=rint(5);
$mu=decimal(1,range(1,5,0.1));
$ans=maple("0.5*(1-exp(-2*$mu))");
$Ans=decimal(4,$ans);@
qu.2.8.uid=9de825ba-bc65-4cfe-8c8b-ee3f17066312@
qu.2.8.info=  Course=230;
  Type=numeric;
  Keyword=poisson;
@

qu.2.9.question=<div title="UW Statistics Bank/Discrete Probability Models/Poisson Distribution/Q$Q">A bank knows that arrival of customers between 8 a.m. (opening time) and 9 a.m. is a Poisson Process, with on average $lambda customers arriving during that hour. Find the probability that exactly $k $Cust arrive during a given 1 minute interval during that hour (4 decimal accuracy).</div>@
qu.2.9.answer.num=$Ans@
qu.2.9.answer.units=@
qu.2.9.showUnits=false@
qu.2.9.grading=toler_abs@
qu.2.9.err=.001@
qu.2.9.negStyle=minus@
qu.2.9.numStyle=thousands scientific dollars arithmetic@
qu.2.9.mode=Numeric@
qu.2.9.name=07a. Arrival of bank customers.@
qu.2.9.comment=<div style="margin-top: 0px; margin-bottom: 2px;" class="shadedDiv descriptionSpan">The trick here is to think of the rate as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi mathvariant='normal'>$lambda</mi><mrow><mi mathvariant='normal'>hour</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi mathvariant='normal'>$mu</mi><mrow><mi mathvariant='normal'>minute</mi></mrow></mfrac></mrow></mstyle></math>. Then &mu; = $mu, x = $k and <br />
P(X = $k) = f($k) = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mi mathvariant='normal'>$mu</mi><mrow><mi mathvariant='normal'>$k</mi></mrow></msup><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$mu</mi></mrow></msup></mrow><mrow><mi mathvariant='normal'>$k</mi><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac></mrow></mstyle></math> = $Ans</div>@
qu.2.9.editing=useHTML@
qu.2.9.solution=@
qu.2.9.algorithm=$Q="07a";
$k=range(0,4,1);
$Cust=if(ne(1,$k),"customers","customer");
$lambda=range(60,360,60);
$mu=$lambda/60;
$Ans=decimal(4,$mu^$k*exp(-$mu)/fact($k));@
qu.2.9.uid=9cec5aa6-a60c-40e3-bd2b-c13bbd53f2ad@
qu.2.9.info=  Course=230;
  Type=numeric;
  Keyword=poisson;
@

qu.2.10.question=<div title="UW Statistics Bank/Discrete Probability Models/Poisson Distribution/Q$Q">Pulses arrive at a Geiger counter in accordance with a Poisson process. Assume that &lambda; = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi mathvariant='normal'>$Lambda</mi><mrow><mi mathvariant='normal'>hour</mi></mrow></mfrac></mrow></mstyle></math>. What is the probability that no pulses arrive at the counter between 8 a.m. and 11 a.m. if it is known that&nbsp;$n pulses have arrived at the counter prior to 8 a.m.? (Please, answer to 4 decimals of accuracy)</div>@
qu.2.10.answer.num=$Ans@
qu.2.10.answer.units=@
qu.2.10.showUnits=false@
qu.2.10.grading=toler_abs@
qu.2.10.err=0.001@
qu.2.10.negStyle=minus@
qu.2.10.numStyle=thousands scientific dollars arithmetic@
qu.2.10.mode=Numeric@
qu.2.10.name=04. Geiger counter 4@
qu.2.10.comment=<p><strong>The correct answer is $Ans</strong></p>
<hr />
<p>By the definition of a Poisson Process, what happens before 8 a.m. has no affect on what happens between 8 and 11 a.m. since these are non-overlapping and independent intervals! The question collapses to a sham, you just need to evaluate the probability function at x = 0. In other words the answer is just <font size="3" face="Times New Roman">e<sup>-3&lambda;</sup> = e<sup>-3($Lambda)</sup> = $Ans</font>!</p>
<p align="center"><em><font size="1"><br />
</font></em></p>@
qu.2.10.editing=useHTML@
qu.2.10.solution=@
qu.2.10.algorithm=$Q=4;
$Lambda=decimal(1,range(1,2.4,0.1));
$n=range(1,10,1);
$Ans=decimal(4,exp(-3*$Lambda));@
qu.2.10.uid=d8f93480-315f-4c44-8c9d-a609be6eadb8@
qu.2.10.info=  Course=230;
  Keyword=poisson;
  Type=numeric;
@

qu.2.11.mode=Multiple Choice@
qu.2.11.name=15. Telephone calls arival@
qu.2.11.comment=<p>Let X = number of calls in a&nbsp;$t minute period. Since the average rate is&nbsp;$m per hour the average number per&nbsp;$t&nbsp;minutes is&nbsp;$lambda and thus X ~ Poisson($lambda).<br />
Thus P(X = 0) = e<sup>&minus;$lambda</sup> = $Ans</p>@
qu.2.11.editing=useHTML@
qu.2.11.solution=@
qu.2.11.algorithm=$Q=15;
$t=range(10,40,10);
$m=range(5,20,1);
$lambda=decimal(2,$t/60*$m);
$ans=maple("stats[statevalf,pf,poisson[$lambda]](0);
");
condition:gt($ans,0.0009);
$Ans=decimal(4,$ans);
$Alt1=decimal(4,$Ans+range(0.5,0.9,0.01)*(1-$Ans));
$Alt2=decimal(4,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(4,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.2.11.uid=0a67c414-0ea8-4688-acb2-d9d62453fb9f@
qu.2.11.info=  Course=230;
  Type=MC;
  Keyword=poisson;
@
qu.2.11.question=<div title="UW Statistics Bank/Discrete Probability Models/Poisson Distribution/Q$Q">Telephone calls arrive at a village exchange at an average rate of $m per hour. Determine the probability that no calls arrive during a $t minute period.</div>@
qu.2.11.answer=1@
qu.2.11.choice.1=$Ans@
qu.2.11.choice.2=$Alt1@
qu.2.11.choice.3=$Alt2@
qu.2.11.choice.4=$Alt3@
qu.2.11.fixed=4@

qu.2.12.question=<div title="UW Statistics Bank/Discrete Probability Models/Poisson Distribution/Q$Q"><img vspace="4" hspace="4" align="$Align" src="__BASE_URI__DPM/Poisson/Fishing$Which.gif" title="Fishing [IMG:Fishing$Which]" alt="Fishing" /> At a particular location on a river the number of fish caught per man hour of fishing has a Poisson distribution with &lambda;=$lambda. If a man fishes there for one hour, what is the probability he will catch exactly $NumFish fish? Answer to 4 decimal accuracy please.</div>@
qu.2.12.answer.num=$ANSWER@
qu.2.12.answer.units=@
qu.2.12.showUnits=false@
qu.2.12.grading=toler_abs@
qu.2.12.err=0.001@
qu.2.12.negStyle=minus@
qu.2.12.numStyle=thousands scientific dollars arithmetic@
qu.2.12.mode=Numeric@
qu.2.12.name=05. P(n fish /hr)@
qu.2.12.comment=<p>Let X be the r.v. representing the number of fish caught in an hour. Then X ~ Poisson($lambda) and</p>
<p>P(X = $NumFish) =<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$lambda</mi></mrow></msup><msup><mi mathvariant='normal'>$lambda</mi><mrow><mi mathvariant='normal'>$NumFish</mi></mrow></msup></mrow><mrow><mi mathvariant='normal'>$NumFish</mi><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac></mrow></mstyle></math>&nbsp;= $ANSWER&nbsp;</p>
<p align="center"><em><font size="1"><br />
</font></em></p>@
qu.2.12.editing=useHTML@
qu.2.12.solution=@
qu.2.12.algorithm=$Q=5;
$Which=1+rint(5);
$Align=switch(rint(2),"Left","Right");
$lambda=decimal(1,range(0.2,2.5,0.1));
$NumFish=if(le($lambda,0.7),if(le($lambda,0.4),range(1,3),range(1,4)),range(1,5));
$ans=exp(-$lambda)*$lambda^$NumFish/fact($NumFish);
$ANSWER=decimal(4,$ans);@
qu.2.12.uid=a36bfd07-b41a-49d7-a822-db3e9d9f21ba@
qu.2.12.info=  Course=230;
  Type=numeric;
@

qu.2.13.question=<div title="UW Statistics Bank/Discrete Probability Models/Poisson Distribution/Q$Q"><img width="50" vspace="4" hspace="4" height="50" align="right" src="__BASE_URI__Tools/TestGuy.gif" title="This question is drawn from a STAT 230 test or exam. [IMG:TestGuy.gif]" alt="This question is drawn from a STAT 230 test or exam." />Power outages in a particular region follow a Poisson process with an average rate of $Lambda outages per year. For simplicity assume that all months are exactly 1/12 of a year. Find the probability that in a given calendar year, exactly $Period months have no outages. (4 decimal accuracy)</div>@
qu.2.13.answer.num=$Ans@
qu.2.13.answer.units=@
qu.2.13.showUnits=false@
qu.2.13.grading=toler_abs@
qu.2.13.err=.001@
qu.2.13.negStyle=minus@
qu.2.13.numStyle=thousands scientific dollars arithmetic@
qu.2.13.mode=Numeric@
qu.2.13.name=23.  P(exactly n months with no outage)@
qu.2.13.comment=<p>With time t in years we have <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mn>1</mn><mrow><mn>12</mn></mrow></mfrac></mrow></mrow></mstyle></math>. From the question &lambda; = $Lambda and X = # of outages in a t-year period ~ Poisson (&mu; = $Lambda t) so &mu; = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi mathvariant='normal'>$Lambda</mi><mrow><mn>12</mn></mrow></mfrac></mrow></mstyle></math>.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&mu;</mi></mrow></mrow></msup><msup><mi>&mu;</mi><mrow><mi>x</mi></mrow></msup></mrow><mrow><mi>x</mi><mo lspace='0.1111111em' rspace='0.1111111em'>&excl;</mo></mrow></mfrac></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msup><mi>e</mi><mrow><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi mathvariant='normal'>$Lambda</mi><mrow><mn>12</mn></mrow></mfrac></mrow></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mi mathvariant='normal'>$Lambda</mi><mrow><mn>12</mn></mrow></mfrac></mrow></mfenced><mrow><mn>0</mn></mrow></msup></mrow><mrow><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mi mathvariant='normal'>$Lambda</mi><mrow><mn>12</mn></mrow></mfrac></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$PXeq0</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p>Let Y = # months in a year with no outages ~ Bi(n=12,p) where p = P(X=0).</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Period</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>12</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi mathvariant='normal'>$Period</mi></mrow></mtd></mtr></mtable></mrow></mfenced><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>e</mi><mrow><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi mathvariant='normal'>$Lambda</mi><mrow><mn>12</mn></mrow></mfrac></mrow></mrow></msup></mrow></mfenced><mrow><mi mathvariant='normal'>$Period</mi></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>e</mi><mrow><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi mathvariant='normal'>$Lambda</mi><mrow><mn>12</mn></mrow></mfrac></mrow></mrow></msup></mrow></mfenced><mrow><mfenced open='(' close=')' separators=','><mrow><mn>12</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Period</mi></mrow></mfenced></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow></mrow></mtd></mtr></mtable></mrow></mstyle></math></p>
<p>&nbsp;</p>@
qu.2.13.editing=useHTML@
qu.2.13.solution=@
qu.2.13.algorithm=$Q="23";
$Lambda=range(2,8,1);
$Period=range(int($Lambda/2),$Lambda-1,1);
$mu=$Lambda/12;
$PXeq0=exp(-$mu);
$PXne0 = 1 - $PXeq0;
$NCM=fact(12)/(fact($Period)*fact(12-$Period));
$Ans=decimal(4,$NCM*($PXeq0^$Period)*($PXne0^(12-$Period)));
condition:gt($Ans,0.0001);@
qu.2.13.uid=1cd01907-32c4-4c0c-8e9b-6640df6046fd@
qu.2.13.info=  Difficulty=2;
  Keyword=poisson;
  Course=230;
  Type=numeric;
@

qu.2.14.question=<div title="UW Statistics Bank/Discrete Probability Models/Poisson Distribution/Q$Q"><img vspace="4" hspace="4" align="$Align" src="__BASE_URI__DPM/Poisson/$WhichName$Which.gif" alt="Car(s)" title="Car(s) [IMG:$WhichName$Which.gif]" />Car accidents at a certain intersection are randomly distributed in time according to a Poisson process, with $n accidents per week on average. If there were $x accidents in a $m-week period, what is the probability there were $y accidents in the first $w of these $m weeks? (4 decimal accuracy please)</div>@
qu.2.14.answer.num=$Ans@
qu.2.14.answer.units=@
qu.2.14.showUnits=false@
qu.2.14.grading=toler_abs@
qu.2.14.err=.001@
qu.2.14.negStyle=minus@
qu.2.14.numStyle=thousands scientific dollars arithmetic@
qu.2.14.mode=Numeric@
qu.2.14.name=18. P(x-1 in m/2 weeks|x in m weeks)@
qu.2.14.comment=<p><strong>The correct answer is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi mathvariant='normal'>$x</mi><mrow><msup><mn>2</mn><mrow><mi mathvariant='normal'>$x</mi></mrow></msup></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math>.</strong></p>
<p>Let A be the event "$y accidents in first&nbsp;$w of the $m weeks", B the event "$x accidents in&nbsp;$m weeks".</p>
<p>Notice that &lambda; = $n*$m = $lambda1</p>
<p>We want <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>A</mi><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mi>B</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>AB</mi></mrow></mfenced></mrow><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.&nbsp;</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$lambda1</mi></mrow></msup><msup><mi mathvariant='normal'>$lambda1</mi><mrow><mi>$x</mi></mrow></msup></mrow><mrow><mi>$x</mi><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac></mrow></mrow></mstyle></math>&nbsp;</p>
<p>(<em>leave P(B) like this, our answer will be more accurate if we work with the algebraic expression instead of the numeric evaluation).</em></p>
<p><br />
P(AB) = P($y accidents in first $w week(s))P(1 more accident in the last $w week(s))</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$lambda2</mi></mrow></msup><msup><mi mathvariant='normal'>$lambda2</mi><mrow><mi mathvariant='normal'>$y</mi></mrow></msup></mrow><mrow><mi>$y</mi><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mfrac><mrow><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$lambda2</mi></mrow></msup><msup><mi mathvariant='normal'>$lambda2</mi><mrow><mn>1</mn></mrow></msup></mrow><mrow><mn>1</mn><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac></mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$lambda2</mi></mrow></msup><msup><mi>$lambda2</mi><mrow><mi>$y</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></msup></mrow><mrow><mi>$y</mi><mo lspace='0.1111111em' rspace='0.1111111em'>&excl;</mo></mrow></mfrac></mrow></mstyle></math><br />
<br />
Before doing anything numeric, let's setup the solution and simplify:</p>
<p>P(A|B) =<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mfrac><mrow><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$lambda1</mi></mrow></msup><msup><mi mathvariant='normal'>$lambda2</mi><mrow><mi mathvariant='normal'>$x</mi></mrow></msup></mrow><mrow><mi mathvariant='normal'>$y</mi><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac></mrow><mrow><mfrac><mrow><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$lambda1</mi></mrow></msup><msup><mi mathvariant='normal'>$lambda1</mi><mrow><mi mathvariant='normal'>$x</mi></mrow></msup></mrow><mrow><mi mathvariant='normal'>$x</mi><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac></mrow></mfrac></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi mathvariant='normal'>$x</mi><mrow><msup><mn>2</mn><mrow><mi mathvariant='normal'>$x</mi></mrow></msup></mrow></mfrac></mrow></mrow></mstyle></math>=$Ans</p>
<p>In other words, the answer only depends on how many accidents occurred in the time period, not how they were distributed!</p>@
qu.2.14.editing=useHTML@
qu.2.14.solution=@
qu.2.14.algorithm=$Q=18;
$Which=rint(5);
$Align=switch(rint(2),"left","right");
$WhichName=switch(rint(2),"Car","CarAccident");
$n=range(1,9);
$m=range(2,6,2);
$x=range(3,9);
$lambda1=$n*$m;
$Prep1=maple("stats[statevalf,pf,poisson[$lambda1]]($x)");
$p1=decimal(5,$Prep1);
$w=$m/2;
$y=$x-1;
$lambda2=$w*$n;
$pk=maple("stats[statevalf,pf,poisson[$lambda2]]($y)");
$pz=maple("stats[statevalf,pf,poisson[$lambda2]](1)");
$p2=$pk*$pz;
$Ans=decimal(4,$p2/$Prep1);
condition:gt($Ans,0.001);@
qu.2.14.uid=eaf971d0-ed1a-4d0b-afe4-b0f870c44c12@
qu.2.14.info=  Type=numeric;
  Course=230;
  Keyword=poisson;
@

qu.2.15.mode=Multiple Choice@
qu.2.15.name=09b. P(<= defects) - clone@
qu.2.15.comment=<div class="shadedDiv descriptionSpan" style="margin-top: 0px; margin-bottom: 2px;">
<div class="shadedDiv descriptionSpan" style="margin-top: 0px; margin-bottom: 2px;">
<p>Let X = number of defects in a&nbsp;$n m roll. The average number of defects in a&nbsp;$n m roll is&nbsp;$lambda and thus X ~ Poisson($lambda). <br />
P(X &le; 2) = P(X = 0) + P(X = 1) + P(X = 2)<br />
= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mi mathvariant='normal'>$lambda</mi><mrow><mn>0</mn></mrow></msup><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$lambda</mi></mrow></msup></mrow><mrow><mn>0</mn><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mrow><msup><mi mathvariant='normal'>$lambda</mi><mrow><mn>1</mn></mrow></msup><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$lambda</mi></mrow></msup></mrow><mrow><mn>1</mn><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mrow><msup><mi mathvariant='normal'>$lambda</mi><mrow><mn>2</mn></mrow></msup><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$lambda</mi></mrow></msup></mrow><mrow><mn>2</mn><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac></mrow></mstyle></math><br />
=&nbsp;$Ans .</p>
</div>
</div>@
qu.2.15.editing=useHTML@
qu.2.15.solution=@
qu.2.15.algorithm=$Q="09b";
$n=range(1000,4000,100);
$x=2;
$lambda=$n/1000;
$Ans=decimal(4,exp(-$lambda)*(1+$lambda+$lambda^2/2));
$Alt1=decimal(4,$Ans+range(0.5,0.9,0.01)*(1-$Ans));
$Alt2=decimal(4,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(4,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.2.15.uid=ddedee45-6697-451a-aeda-84cb947a7258@
qu.2.15.info=  Course=230;
  Type=MC;
  Keyword=poisson;
@
qu.2.15.question=<div title="UW Statistics Bank/Discrete Probability Models/Poisson Distribution/Q$Q"><img hspace="3" border="0" align="left" alt="Roll of tape" title="Roll of tape [IMG:tape_roll.gif]" src="__BASE_URI__DPM/Poisson/tape_roll.gif" />Defects occur in a certain manufactured tape on the average of 1 per 1,000 m. Assuming a Poisson distribution for the number of defects in a given length of tape, what is the probability that a&nbsp;$n m roll will have at most 2 defects? (3 decimal accuracy)&nbsp;</div>@
qu.2.15.answer=1@
qu.2.15.choice.1=$Ans@
qu.2.15.choice.2=$Alt1@
qu.2.15.choice.3=$Alt2@
qu.2.15.choice.4=$Alt3@
qu.2.15.fixed=@

qu.2.16.question=<div title="UW Statistics Bank/Discrete Probability Models/Poisson Distribution/Q$Q"><img width="50" vspace="4" hspace="4" height="50" align="right" src="__BASE_URI__Tools/TestGuy.gif" title="This question is drawn from a STAT 230 test or exam. [IMG:TestGuy.gif]" alt="This question is drawn from a STAT 230 test or exam." />Power outages in a particular region follow a Poisson process with an average rate of $Lambda outages per year. For simplicity assume that all months are exactly <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>12</mn></mrow></mfrac></mrow></mstyle></math> of a year. Find the probability that there is <em>at least  </em>1 outage in a given $Period month period. (4 decimal accuracy)</div>@
qu.2.16.answer.num=$Ans@
qu.2.16.answer.units=@
qu.2.16.showUnits=false@
qu.2.16.grading=toler_abs@
qu.2.16.err=.001@
qu.2.16.negStyle=minus@
qu.2.16.numStyle=thousands scientific dollars arithmetic@
qu.2.16.mode=Numeric@
qu.2.16.name=22. P(>= 1 outage in a given period)@
qu.2.16.comment=<p>With time t in years we have <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi mathvariant='normal'>$Period</mi><mrow><mn>12</mn></mrow></mfrac></mrow></mrow></mstyle></math>. From the question &lambda; = $Lambda and X = # of outages in a t-year period ~ Poisson (&mu; = $Lambda t) so &mu; = $mu.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&mu;</mi></mrow></mrow></msup><msup><mi>&mu;</mi><mrow><mi>x</mi></mrow></msup></mrow><mrow><mi>x</mi><mo lspace='0.1111111em' rspace='0.1111111em'>&excl;</mo></mrow></mfrac></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&ge;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi mathvariant='normal'>$mut</mi><mrow><mn>12</mn></mrow></mfrac></mrow></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mrow><mfenced open='(' close=')' separators=','><mfrac><mrow><mi mathvariant='normal'>$mut</mi></mrow><mrow><mn>12</mn></mrow></mfrac></mfenced></mrow><mrow><mn>0</mn></mrow></msup></mrow><mrow><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p>&nbsp;</p>@
qu.2.16.editing=useHTML@
qu.2.16.solution=@
qu.2.16.algorithm=$Q="22";
$Lambda=range(2,8,1);
$Period=range(1,4,1);
$mut=$Lambda*$Period;
$mu=$mut/12;
$Ans=decimal(4,1-exp(-$mu)*($mu^0)/fact(0));@
qu.2.16.uid=a4945ba2-0112-4d66-a653-d44953d6af48@
qu.2.16.info=  Difficulty=2;
  Course=230;
  Keyword=poisson;
  Type=numeric;
@

qu.2.17.question=<p><img hspace="4" height="101" align="right" width="150" title="Geiger Counter [IMG:Gieger.gif]" alt="Geiger Counter" src="__BASE_URI__DPM/Poisson/Gieger.gif" /></p>
<div title="UW Statistics Question Bank/Discrete Probability Models/Poisson/Q$Q">Pulses arrive at a Geiger counter in accordance with a Poisson Process. If &lambda; = $Lambda/hour what is the probability that 3 or more pulses arrive to the counter between 6 p.m. and 8:30 p.m. of the same day? (3 decimals)</div>@
qu.2.17.answer.num=$Ans1@
qu.2.17.answer.units=@
qu.2.17.showUnits=false@
qu.2.17.grading=toler_abs@
qu.2.17.err=0.005@
qu.2.17.negStyle=minus@
qu.2.17.numStyle=thousands scientific dollars arithmetic@
qu.2.17.mode=Numeric@
qu.2.17.name=1. Geiger Counter 1@
qu.2.17.comment=<p>Let Y = number of pulses that arrive between 6 and 8:30 p.m. of the same day.  Then t = 2.5, so Y ~ Poisson(&mu; = &lambda;t = 2.5&lambda;).&nbsp;  Then:</p>
<p>P(Y &ge; 3) = 1 - P(Y = 0) - P(Y = 1) - P(Y = 2)=1-exp(2.5*Lambda)*(1+2.5*Lambda+.5*(2.5*Lambda)^2)</p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.2.17.editing=useHTML@
qu.2.17.solution=@
qu.2.17.algorithm=$Q=1;
$Lambda=decimal(2,range(1,5,0.25));
$Ans=1-exp(-2.5*$Lambda)*(25*($Lambda)^2/8+5*$Lambda/2+1);
$Ans1=1-exp(-2.5*$Lambda)*(1+2.5*$Lambda+.5*(2.5*$Lambda)^2);@
qu.2.17.uid=cacbbab9-9688-4da9-b709-cd0a18c04102@
qu.2.17.info=  Course=230;
@

qu.2.18.mode=Multiple Choice@
qu.2.18.name=06. Babies a day born with one or more teeth@
qu.2.18.comment=<p>This is a Poisson process with t = 1 day, &lambda; =&nbsp;$lambda so if X = # babies born each day with teeth, the probability distribution for X is: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msup><mi>&lambda;</mi><mrow><mi>x</mi></mrow></msup><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&lambda;</mi></mrow></mrow></msup></mrow><mrow><mi>x</mi><mo lspace='0.1111111em' rspace='0.1111111em'>&excl;</mo></mrow></mfrac></mrow></mstyle></math> so <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$k</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msup><mrow><mi mathvariant='normal'>$lambda</mi></mrow><mrow><mi mathvariant='normal'>$k</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$lambda</mi></mrow></msup></mrow><mrow><mi mathvariant='normal'>$k</mi><mo lspace='0.1111111em' rspace='0.1111111em'>&excl;</mo></mrow></mfrac></mrow></mstyle></math>=$Ans.</p>@
qu.2.18.editing=useHTML@
qu.2.18.solution=@
qu.2.18.algorithm=$Q=6;
$lambda=range(2,6);
$k=range(0,6);
$Baby=if(eq($k,1),"baby","babies");
$Ans=decimal(4,$lambda^$k*exp(-$lambda)/fact($k));
$Alt1=decimal(4,range(0.3,0.8,0.05)*$Ans);
$Alt2=decimal(4,$Ans+range(0.3,0.8,0.05)*(1-$Ans));
$Alt3=decimal(4,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.2.18.uid=dafc9117-fd26-4e47-af87-2e3ca32b45dd@
qu.2.18.info=  Course=230;
  Type=MC;
@
qu.2.18.question=<div title="UW Statistics Bank/Discrete Probability Models/Poisson Distribution/Q$Q">In a certain hospital on average $lambda babies a day are born with one or more teeth. Assuming such births can be modelled with a Poisson Process, find the probability that in one day $k $Baby are born with teeth.</div>@
qu.2.18.answer=1@
qu.2.18.choice.1=$Ans@
qu.2.18.choice.2=$Alt1@
qu.2.18.choice.3=$Alt2@
qu.2.18.choice.4=$Alt3@
qu.2.18.choice.5=None of the above@
qu.2.18.fixed=4@

qu.2.19.mode=Multiple Choice@
qu.2.19.name=16b. P(x accidents in n-weeks)@
qu.2.19.comment=<p>Let X = the number of accidents in a $m-week period.</p>
<p>Then X ~ Poisson(&lambda;) where &mu; = $n, &lambda;=&mu;t = $n($m)&nbsp; = $lambda.<br />
<br />
P(X = 2) =&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&lambda;</mi></mrow></mrow></msup><mrow><msup><mi mathvariant='normal'>&lambda;</mi><mrow><mi>x</mi></mrow></msup></mrow></mrow><mrow><mi>x</mi><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$lambda</mi></mrow></msup><msup><mi mathvariant='normal'>$lambda</mi><mrow><mi>$x</mi></mrow></msup></mrow><mrow><mi>$x</mi><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.2.19.editing=useHTML@
qu.2.19.solution=@
qu.2.19.algorithm=$Q="16b";
$n=range(1,9,1);
$Word=if($n-1,"accidents","accident");
$m=range(2,6,1);
$x=range(0,9,1);
$lambda=$n*$m;
$ans=maple("stats[statevalf, pf, poisson[$lambda]]($x)");
$Ans=decimal(4,$ans);
condition:gt($ans,0.001);
$Alt1=decimal(4,$Ans+range(0.5,0.9,0.01)*(1-$Ans));
$Alt2=decimal(4,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(4,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));
$Align=switch(rint(2),"Left","Right");
$Which=rint(5);
$WhichName=switch(rint(2),"Car","CarAccident");@
qu.2.19.uid=7e55a3fb-adbe-49a7-9e0d-e837f18d3f38@
qu.2.19.info=  Course=230;
  Type=MC;
  Keyword=poisson;
@
qu.2.19.question=<div title="UW Statistics Bank/Discrete Probability Models/Poisson Distribution/Q$Q"><img vspace="4" hspace="4" align="$Align" title="Car(s) [IMG:$WhichName$Which.gif]" alt="Car(s)" src="__BASE_URI__DPM/Poisson/$WhichName$Which.gif" /> Car accidents at a certain intersection are randomly distributed in time according to a Poisson process, with $n $Word per week on average. What is the probability of exactly $x accidents in a $m-week period?</div>@
qu.2.19.answer=1@
qu.2.19.choice.1=$Ans@
qu.2.19.choice.2=$Alt1@
qu.2.19.choice.3=$Alt2@
qu.2.19.choice.4=$Alt3@
qu.2.19.fixed=@

qu.2.20.mode=Inline@
qu.2.20.name=24.  P(<= k calls in an hour)@
qu.2.20.comment=<p>P(at most <em>k</em> calls in a shift)</p>
<p>= P(0 calls) + P(1 call) + ... + P(<em>k</em> calls)</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msup><mi mathvariant='normal'>$Lambda</mi><mrow><mn>0</mn></mrow></msup><mrow><mn>0</mn><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Lambda</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><msup><mi mathvariant='normal'>$Lambda</mi><mrow><mn>1</mn></mrow></msup><mrow><mn>1</mn><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Lambda</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><msup><mi mathvariant='normal'>$Lambda</mi><mrow><mi>k</mi></mrow></msup><mrow><mi>k</mi><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Lambda</mi></mrow></msup></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow><mrow><mi>k</mi></mrow></munderover><mfrac><mrow><msup><mi mathvariant='normal'>$Lambda</mi><mrow><mi>i</mi></mrow></msup></mrow><mrow><mi>i</mi><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Lambda</mi></mrow></msup></mrow></mstyle></math></p>@
qu.2.20.editing=useHTML@
qu.2.20.solution=@
qu.2.20.algorithm=$Q=24;
$Lambda=range(2,6,1);@
qu.2.20.uid=70fefbd4-d44d-4483-8cfa-987142bd6ba5@
qu.2.20.info=  Difficulty=3;
  Keyword=poisson;
  Type=MC;
  Course=230;
@
qu.2.20.weighting=1@
qu.2.20.numbering=alpha@
qu.2.20.part.1.comment.3=@
qu.2.20.part.1.comment.2=@
qu.2.20.part.1.name=sro_id_1@
qu.2.20.part.1.comment.1=@
qu.2.20.part.1.editing=useHTML@
qu.2.20.part.1.choice.5=None of the above<br>@
qu.2.20.part.1.fixed=4@
qu.2.20.part.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Lambda</mi></mrow></msup><mrow><mi>k</mi><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><msup><mi>k</mi><mrow><mi mathvariant='normal'>$Lambda</mi></mrow></msup></mrow></mstyle></math><br>@
qu.2.20.part.1.question=null@
qu.2.20.part.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>k</mi></mrow></msup><mrow><mi>k</mi><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><msup><mi mathvariant='normal'>$Lambda</mi><mrow><mi>k</mi></mrow></msup></mrow></mstyle></math><br>@
qu.2.20.part.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$Lambda</mi></mrow></munderover><mfrac><mrow><msup><mi>k</mi><mrow><mi>i</mi></mrow></msup></mrow><mrow><mi>i</mi><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>k</mi></mrow></msup></mrow></mstyle></math><br>@
qu.2.20.part.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow><mrow><mi>k</mi></mrow></munderover><mfrac><msup><mi mathvariant='normal'>$Lambda</mi><mrow><mi>i</mi></mrow></msup><mrow><mi>i</mi><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Lambda</mi></mrow></msup></mrow></mstyle></math>@
qu.2.20.part.1.mode=Multiple Choice@
qu.2.20.part.1.display=vertical@
qu.2.20.part.1.comment.5=@
qu.2.20.part.1.comment.4=@
qu.2.20.part.1.answer=1@
qu.2.20.question=<div title="UW Statistics Bank/Discrete Probability Models/Poisson Distribution/Q$Q"><img width="50" vspace="4" hspace="4" height="50" align="right" alt="This question is drawn from a STAT 230 test or exam." title="This question is drawn from a STAT 230 test or exam. [IMG:TestGuy.gif]" src="__BASE_URI__Tools/TestGuy.gif" />Calls arrive at a telephone crisis centre according to the conditions for a Poisson process, at an average rate of $Lambda per hour. Which of the following is an expression for the probability that  there are at most <em>k</em> calls in a one hour shift?<p>&nbsp;&nbsp;</p><p><span> </span><1><span> </span></p></div>@

qu.2.21.question=<div title="UW Statistics Bank/Discrete Probability Models/Poisson Distribution/Q$Q">On average $pp % of a certain type of TV tube burn out before their guarantee has expired. If a merchant sells&nbsp;$n such tubes, find&nbsp; the probability that he will be forced to replace at least 3 of them (Use the Poisson approximation to the Binomial distribution). Answer to 2 decimal accuracy, eg. 0.12.</div>@
qu.2.21.answer.num=$ANSWER@
qu.2.21.answer.units=@
qu.2.21.showUnits=false@
qu.2.21.grading=toler_abs@
qu.2.21.err=0.01@
qu.2.21.negStyle=minus@
qu.2.21.numStyle=thousands scientific dollars arithmetic@
qu.2.21.mode=Numeric@
qu.2.21.name=10. P(>=n burn outs)@
qu.2.21.comment=<div class="shadedDiv descriptionSpan" style="margin-top: 0px; margin-bottom: 2px">P(tube has to be replaced) = p = $p.<br />
<br />
Let X = number of tubes which have to be replaced out of n =&nbsp;$n tubes. Assuming that the tubes behave independently, X ~ bin($n, $p).<br />
<br />
Since n is large and p small we can use the Poisson approximation to the binomial distribution; &lambda; = np =&nbsp;$n &times;&nbsp;$p = $lambda, i.e. X ~ Poisson($lambda) approximately and:<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&asymp;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><msup><mi>&lambda;</mi><mrow><mi>x</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>e</mi><mrow><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&lambda;</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mi>x</mi><mo lspace='0.1111111em' rspace='0.1111111em'>&excl;</mo></mrow></mfrac></mrow></mstyle></math><br />
P(at least 3 tubes are replaced) = P(X &ge; 3) = 1 &minus; P(X &le; 2) = 1&minus;P(X = 0) &minus; P(X = 1) &minus; P(X = 2) = 1 &minus;&nbsp;$nans &asymp; $ANSWER.</div>@
qu.2.21.editing=useHTML@
qu.2.21.solution=@
qu.2.21.algorithm=$Q=10;
$p=decimal(2,range(0.01,0.06,0.001));
$pp=100*$p;
$n=range(80,300,10);
$lambda=$n*$p;
$nans=maple("stats[statevalf, pf, poisson[$lambda]](0)+stats[statevalf, pf, poisson[$lambda]](1)+stats[statevalf, pf, poisson[$lambda]](2)");
$ans=1-$nans;
$ANSWER=decimal(2,$ans);
condition:lt($ANSWER,0.99);@
qu.2.21.uid=e7f0ce3b-e82c-4ea6-94be-dda3ae6f257a@
qu.2.21.info=  Course=230;
@

qu.2.22.mode=True False@
qu.2.22.name=19. mu=1, P(X=0)=P(X=1)?@
qu.2.22.comment=<p>Just evaluate the actual values using&nbsp; P(X=x) = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&mu;</mi></mrow></mrow></msup><mrow><mfrac><msup><mi>&mu;</mi><mrow><mi>x</mi></mrow></msup><mrow><mi>x</mi><mo lspace='0.1111111em' rspace='0.1111111em'>&excl;</mo></mrow></mfrac></mrow></mrow></mstyle></math>where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&mu;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></mstyle></math>:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><msup><mn>1</mn><mrow><mn>0</mn></mrow></msup><mrow><mn>0</mn><mo lspace='0.1111111em' rspace='0.1111111em'>&excl;</mo></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mn>1</mn><mrow><mi>e</mi></mrow></mfrac></mrow></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><msup><mn>1</mn><mrow><mn>1</mn></mrow></msup><mrow><mn>1</mn><mo lspace='0.1111111em' rspace='0.1111111em'>&excl;</mo></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mn>1</mn><mrow><mi>e</mi></mrow></mfrac></mrow></mrow></mstyle></math></p>
<p>&nbsp;</p>@
qu.2.22.editing=useHTML@
qu.2.22.solution=@
qu.2.22.algorithm=@
qu.2.22.uid=032bc592-44f9-4ba4-ae82-2bda4d262425@
qu.2.22.info=  Course=230;
  Type=T/F;
  Keyword=poisson;
  Difficulty=0;
@
qu.2.22.question=<div title="UW Statistics Bank/Discrete Probability Models/Poisson Distribution/Q19">
Let X have a Poisson distribution with &mu; = 1. Then:
<p>P(X = 0) = P(X = 1)</p>
</div>@
qu.2.22.answer=1@
qu.2.22.choice.1=True@
qu.2.22.choice.2=False@
qu.2.22.fixed=@

qu.2.23.mode=Multiple Choice@
qu.2.23.name=11. P(n accidents)@
qu.2.23.comment=<p>Let t be 1 day and so &lambda; = $lambda. Then&nbsp;&mu; = 1($lambda) and our probability distribution is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msup><mi mathvariant='normal'>&lambda;</mi><mrow><mi>x</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&lambda;</mi></mrow></mrow></msup></mrow><mrow><mi>x</mi><mo lspace='0.1111111em' rspace='0.1111111em'>&excl;</mo></mrow></mfrac></mrow></mstyle></math>.&nbsp; Then we want f($k) =&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mi>$lambda</mi><mrow><mi>$k</mi></mrow></msup><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$lambda</mi></mrow></msup></mrow><mrow><mi>$k</mi><mo lspace='0.1111111em' rspace='0.1111111em'>&excl;</mo></mrow></mfrac></mrow></mstyle></math>= $Ans .</p>@
qu.2.23.editing=useHTML@
qu.2.23.solution=@
qu.2.23.algorithm=$Q=11;
$Align=switch(rint(2),"Left","Right");
$Which=1+rint(6);
$lambda=decimal(1,range(0.5,4,0.1));
$k=range(1,3,1);
$Say=switch($k-1,"one accident","two accidents","three accidents");
$Ans=decimal(4,$lambda^$k*exp(-$lambda)/fact($k));
$Alt1=decimal(4,range(0.4,0.8,0.05)*$Ans);
$Alt2=decimal(4,$Ans+range(0.4,0.8,0.05)*(1-$Ans));
$Alt3=decimal(4,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.2.23.uid=c87eb8af-ef5e-4245-8115-470fec97bae5@
qu.2.23.info=  Course=230;
@
qu.2.23.question=<div title="UW Statistics Bank/Discrete Probability Models/Poisson Distribution/Q$Q"><img hspace="4" align="$Align" alt="Accident" src="__BASE_URI__DPM/Poisson/Accident$Which.gif" title="Accident [IMG:Accident$Which.gif]" />Using the Poisson distribution function, if&nbsp;$lambda accidents can be expected at a certain intersection every day, what is approximately the probability that there will be $Say at that intersection on any given day?</div>@
qu.2.23.answer=1@
qu.2.23.choice.1=$Ans@
qu.2.23.choice.2=$Alt1@
qu.2.23.choice.3=$Alt2@
qu.2.23.choice.4=$Alt3@
qu.2.23.choice.5=None of the above@
qu.2.23.fixed=4@

qu.2.24.question=<div title="UW Statistic Question Bank/Discrete Probability Models/Poisson/Q$Q"><img hspace="4" align="$Align" title="Typist [IMG:Typist$Which.gif]" alt="Typist" src="__BASE_URI__DPM/Poisson/Typist$Which.gif" />If there are $Typos typos randomly distributed in a $Pages page manuscript, find the probability that a given page contains exactly $NumOnPage errors. 4 decimal accuracy please.</div>@
qu.2.24.answer.num=$Ans@
qu.2.24.answer.units=@
qu.2.24.showUnits=false@
qu.2.24.grading=toler_abs@
qu.2.24.err=.001@
qu.2.24.negStyle=minus@
qu.2.24.numStyle=thousands scientific dollars arithmetic@
qu.2.24.mode=Numeric@
qu.2.24.name=26. Typing Errors@
qu.2.24.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi mathvariant='normal'>$Typos</mi><mrow><mi mathvariant='normal'>$Pages</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Lambda</mi></mrow></mstyle></math> so P($NumOnPage) = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mi mathvariant='normal'>$Typos</mi><mrow><mi mathvariant='normal'>$Pages</mi></mrow></mfrac></mrow></mfenced><mrow><mi mathvariant='normal'>$NumOnPage</mi></mrow></msup><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi mathvariant='normal'>$Typos</mi><mrow><mi mathvariant='normal'>$Pages</mi></mrow></mfrac></mrow></mrow></msup></mrow><mrow><mi mathvariant='normal'>$NumOnPage</mi><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>@
qu.2.24.editing=useHTML@
qu.2.24.solution=@
qu.2.24.algorithm=$Q="26";
$Typos=range(100,250,50);
$Pages=range(300,600,50);
$NumOnPage=range(2,6,1);
$Lambda=$Typos/$Pages;
$Ans=decimal(4,$Lambda^$NumOnPage*exp(-$Lambda)/fact($NumOnPage));
condition:ge($Ans,0.0009);
$Which=rint(4);
$Align=switch(rint(2),"Left","Right");@
qu.2.24.uid=ab5f225d-88ac-4510-87c2-157b74dbf26d@
qu.2.24.info=  Course=202;
  Type=numeric;
  Keyword=poisson;
@

qu.2.25.question=<div title="UW Statistics Bank/Discrete Probability Models/Poisson Distribution/Q$Q"><img width="50" vspace="4" hspace="4" height="50" align="right" src="__BASE_URI__Tools/TestGuy.gif" title="This question is drawn from a STAT 230 test or exam. [IMG:TestGuy.gif]" alt="This question is drawn from a STAT 230 test or exam." /> Power outages in a particular region follow a Poisson process with an average rate of $Lambda outages per year. For simplicity assume that all months are exactly&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>12</mn></mrow></mfrac></mrow></mstyle></math> of a year. Find the probability that there are exactly $x outages in a given $Period month period. (4 decimal accuracy)</div>@
qu.2.25.answer.num=$Ans@
qu.2.25.answer.units=@
qu.2.25.showUnits=false@
qu.2.25.grading=toler_abs@
qu.2.25.err=.001@
qu.2.25.negStyle=minus@
qu.2.25.numStyle=thousands scientific dollars arithmetic@
qu.2.25.mode=Numeric@
qu.2.25.name=21.  P(x power outages in y months)@
qu.2.25.comment=<p>With time t in years, <em>&lambda;</em> = $Lambda and<em> X</em> = # of outages in a <em>t-</em>year period ~ Poisson (<em>&mu; </em>= $Lambda <em>t</em>) with <font size="3" face="Times New Roman"><em>t </em>= $td</font>.</p>
<p>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&mu;</mi></mrow></mrow></msup><msup><mi>&mu;</mi><mrow><mi>x</mi></mrow></msup></mrow><mrow><mi>x</mi><mo lspace='0.1111111em' rspace='0.1111111em'>&excl;</mo></mrow></mfrac></mrow></mstyle></math><br />
&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi mathvariant='normal'>$mut</mi><mrow><mn>12</mn></mrow></mfrac></mrow></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$mut</mi></mrow><mrow><mn>12</mn></mrow></mfrac></mrow></mfenced><mrow><mi>$x</mi></mrow></msup></mrow><mrow><mi>$x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p>&nbsp;</p>@
qu.2.25.editing=useHTML@
qu.2.25.solution=@
qu.2.25.algorithm=$Q=21;
$Lambda=range(2,8,1);
$x=range(2,5,1);
$Period=range(2,10,1);
$mut=$Lambda*$Period;
$mu=$mut/12;
$td=mathml("$Period/12");
$Ans=exp(-$mu)*($mu^$x)/fact($x);@
qu.2.25.uid=e2a79f1d-a19a-4245-bba8-458b91516cc8@
qu.2.25.info=  Difficulty=2;
  Keyword=poisson;
  Type=numeric;
  Course=230;
@

qu.2.26.question=<div title="UW Statistics Bank/Discrete Probability Models/Poisson Distribution/Q$Q"><img hspace="3" border="0" align="left" alt="Roll of tape" title="Roll of tape [IMG:tape_roll.gif]" src="__BASE_URI__DPM/Poisson/tape_roll.gif" />Defects occur in a certain manufactured tape on the average of 1 per 1,000 m. Assuming a Poisson distribution for the number of defects in a given length of tape, what is the probability that a&nbsp;$n m roll will have no defects? (3 decimal accuracy)</div>@
qu.2.26.answer.num=$Ans@
qu.2.26.answer.units=@
qu.2.26.showUnits=false@
qu.2.26.grading=toler_abs@
qu.2.26.err=0.01@
qu.2.26.negStyle=minus@
qu.2.26.numStyle=thousands scientific dollars arithmetic@
qu.2.26.mode=Numeric@
qu.2.26.name=08. P(0 errors in n m of tape)@
qu.2.26.comment=Let X = number of defects in a&nbsp;$n m roll. X ~ Poisson($lambda). P(X = $x) = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mi mathvariant='normal'>$lambda</mi><mrow><mi mathvariant='normal'>$x</mi></mrow></msup><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$lambda</mi></mrow></msup></mrow><mrow><mi mathvariant='normal'>$x</mi><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac></mrow></mstyle></math>= $Ans .@
qu.2.26.editing=useHTML@
qu.2.26.solution=@
qu.2.26.algorithm=$Q=8;
$n=range(1000,4000,100);
$x=0;
$lambda=$n/1000;
$Ans=decimal(4,$lambda^$x*exp(-$lambda)/fact($x));@
qu.2.26.uid=ae3d5c83-b20a-4845-9955-e515b14c5b97@
qu.2.26.info=  Course=230;
  Type=numeric;
  Keyword=poisson;
@

qu.2.27.mode=Inline@
qu.2.27.name=28. Properties of Poisson@
qu.2.27.comment=<p>All of these are properties of a Poisson Distribution.</p>@
qu.2.27.editing=useHTML@
qu.2.27.solution=@
qu.2.27.algorithm=@
qu.2.27.uid=fbd698ec-b3c0-4667-ba66-bbf1fa8e1eb7@
qu.2.27.info=  Course=202;
  Type=MS;
  Keyword=poisson;
  Algorithmic=no;
@
qu.2.27.weighting=1@
qu.2.27.numbering=alpha@
qu.2.27.part.1.name=sro_id_1@
qu.2.27.part.1.editing=useHTML@
qu.2.27.part.1.fixed=@
qu.2.27.part.1.question=null@
qu.2.27.part.1.choice.3=Events involved in this experiment occur individually (i.e. one at a time)<br>@
qu.2.27.part.1.choice.2=Events involved in this experiment occur independently (i.e. one does not affect another)@
qu.2.27.part.1.choice.1=Events involved in this experiment occur homogenously (i.e. at a constant rate)@
qu.2.27.part.1.mode=Multiple Selection@
qu.2.27.part.1.display=vertical@
qu.2.27.part.1.answer=1,2,3@
qu.2.27.question=<div title="UW Statistics Bank/Discrete Probability Models/Poisson Distribution/Q28">If an experiment follows Poisson distribution, what must be true about this experiment? <em>(Select all correct choices)</em>&nbsp;<p><span>&nbsp;</span><1><span>&nbsp;</span></p></div>@

qu.2.28.question=<div title="UW Statistics Bank/Discrete Probability Models/Poisson Distribution/Q$Q"><img vspace="4" hspace="4" align="$Align" title="Car(s) [IMG:$WhichName$Which.gif]" alt="Car(s)" src="__BASE_URI__DPM/Poisson/$WhichName$Which.gif" />Car accidents at a certain intersection are randomly distributed in time according to a Poisson process, with&nbsp;$n accident$Accid per week on average. What is the probability that there&nbsp;are $x accidents in the first&nbsp;$w week period and $x more in the second $w week period.? (4 decimal accuracy please)</div>@
qu.2.28.answer.num=$Ans@
qu.2.28.answer.units=@
qu.2.28.showUnits=false@
qu.2.28.grading=toler_abs@
qu.2.28.err=.001@
qu.2.28.negStyle=minus@
qu.2.28.numStyle=thousands scientific dollars arithmetic@
qu.2.28.mode=Numeric@
qu.2.28.name=17. P(x accidents in m wks & x in next m)@
qu.2.28.comment=<p>Let X<sub>1</sub> = the number of accidents in the first $w week(s) ~ Poisson($lambda)<br />
Let X<sub>2</sub> = the number of accidents in the second $w week(s) ~ Poisson($lambda)<br />
<br />
These events are independent, since the time periods are non-overlapping.</p>
<p>Notice that &lambda; = $n($w) = $lambda</p>
<p>Thus:<br />
P(X<sub>1</sub> = $x AND X<sub>2</sub> = $x) = P(X<sub>1</sub> = $x)P(X<sub>2</sub> = $x) = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$lambda</mi></mrow></msup><msup><mi mathvariant='normal'>$lambda</mi><mrow><mi mathvariant='normal'>$x</mi></mrow></msup></mrow><mrow><mi mathvariant='normal'>$x</mi><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow><mrow><mi></mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>= $Ans</p>@
qu.2.28.editing=useHTML@
qu.2.28.solution=@
qu.2.28.algorithm=$Q="17";
$n=range(1,9);
$Accid=if(eq($n,1),"","s");
$m=range(2,6,2);
$x=range(1,9);
$w=$m/2;
$lambda=$n*$w;
$ans=maple("(stats[statevalf, pf, poisson[$lambda]]($x))^2");
$Ans=decimal(4,$ans);
condition:gt($ans,0.001);
$Align=switch(rint(2),"Left","Right");
$Which=rint(5);
$WhichName=switch(rint(2),"Car","CarAccident");@
qu.2.28.uid=6fb008e5-4e52-4699-8271-bc92a9d37cea@
qu.2.28.info=  Course=230;
  Type=numeric;
  Keyword=poisson;
@

qu.2.29.question=<div title="UW Statistics Bank/Discrete Probability Models/Poisson Distribution/Q$Q"><img vspace="4" hspace="4" align="$Align" src="__BASE_URI__DPM/Poisson/$WhichName$Which.gif" alt="Car(s)" title="Car(s) [IMG:$WhichName$Which.gif]" /> Car accidents at a certain intersection are randomly distributed in time according to a Poisson process, with $n $Word per week on average. What is the probability of exactly $x accidents in a $m-week period? (4 decimal accuracy please).</div>@
qu.2.29.answer.num=$Ans@
qu.2.29.answer.units=@
qu.2.29.showUnits=false@
qu.2.29.grading=toler_abs@
qu.2.29.err=.001@
qu.2.29.negStyle=minus@
qu.2.29.numStyle=thousands scientific dollars arithmetic@
qu.2.29.mode=Numeric@
qu.2.29.name=16A. P(x accidents in n-weeks)@
qu.2.29.comment=<p>Let X = the number of accidents in a $m-week period.</p>
<p>Then X ~ Poisson(&lambda;) where &mu; = $n, &lambda;=&mu;t = $n($m)&nbsp; = $lambda.<br />
<br />
P(X = 2) =&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&lambda;</mi></mrow></mrow></msup><mrow><msup><mi mathvariant='normal'>&lambda;</mi><mrow><mi>x</mi></mrow></msup></mrow></mrow><mrow><mi>x</mi><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$lambda</mi></mrow></msup><msup><mi mathvariant='normal'>$lambda</mi><mrow><mi>$x</mi></mrow></msup></mrow><mrow><mi>$x</mi><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.2.29.editing=useHTML@
qu.2.29.solution=@
qu.2.29.algorithm=$Q="16A";
$n=range(1,9,1);
$Word=if($n-1,"accidents","accident");
$m=range(2,6,1);
$x=range(0,9,1);
$lambda=$n*$m;
$ans=maple("stats[statevalf, pf, poisson[$lambda]]($x)");
$Ans=decimal(4,$ans);
condition:gt($ans,0.001);
$Align=switch(rint(2),"Left","Right");
$Which=rint(5);
$WhichName=switch(rint(2),"Car","CarAccident");@
qu.2.29.uid=dcf09f70-2429-4a5c-85d4-da053f56c8ac@
qu.2.29.info=  Course=230;
  Type=numeric;
  Keyword=poisson;
@

qu.2.30.mode=Multiple Choice@
qu.2.30.name=07b. Arrival of bank customers@
qu.2.30.comment=<div style="margin-top: 0px; margin-bottom: 2px;" class="shadedDiv descriptionSpan">The trick here is to think of the rate as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi mathvariant='normal'>$lambda</mi><mrow><mi mathvariant='normal'>hour</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi mathvariant='normal'>$mu</mi><mrow><mi mathvariant='normal'>minute</mi></mrow></mfrac></mrow></mstyle></math>. Then &mu; = $mu, x = $k and <br />
P(X = $k) = f($k) = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mi mathvariant='normal'>$mu</mi><mrow><mi mathvariant='normal'>$k</mi></mrow></msup><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$mu</mi></mrow></msup></mrow><mrow><mi mathvariant='normal'>$k</mi><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac></mrow></mstyle></math> = $Ans</div>@
qu.2.30.editing=useHTML@
qu.2.30.solution=@
qu.2.30.algorithm=$Q="07b";
$k=range(0,4,1);
$Cust=if(ne(1,$k),"customers","customer");
$lambda=range(60,360,60);
$mu=$lambda/60;
$Ans=decimal(4,$mu^$k*exp(-$mu)/fact($k));
$Alt1=decimal(4,range(0.4,0.8,0.05)*$Ans);
$Alt2=decimal(4,$Ans+range(0.4,0.8,0.05)*(1-$Ans));
$Alt3=decimal(4,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.2.30.uid=c265ecc3-ed04-4f80-a816-fb49012819d4@
qu.2.30.info=  Course=230;
  Type=MC;
  Keyword=poisson;
@
qu.2.30.question=<div title="UW Statistics Bank/Discrete Probability Models/Poisson Distribution/Q$Q">A bank knows that arrival of customers between 8 a.m. (opening time) and 9 a.m. is a Poisson Process, with on average $lambda customers arriving during that hour. Find the probability that exactly $k $Cust arrive during a given 1 minute interval during that hour.</div>@
qu.2.30.answer=1@
qu.2.30.choice.1=$Ans@
qu.2.30.choice.2=$Alt1@
qu.2.30.choice.3=$Alt2@
qu.2.30.choice.4=$Alt3@
qu.2.30.choice.5=None of the above@
qu.2.30.fixed=4@

qu.2.31.question=<div title="UW Statistics Bank/Discrete Probability Models/Poisson Distribution/Q$Q"><img align="$Align" title="TV [IMG:TV$Which.gif]" alt="TV" src="__BASE_URI__DPM/Poisson/TV$Which.gif" />On average $pp % of a certain type of TV tube burn out before their guarantee has expired. If a merchant sells&nbsp;$n such tubes, find the probability that he will be forced to replace at least 3 of them (Use the Poisson approximation to the Binomial distribution). Answer to 4 decimal accuracy.</div>@
qu.2.31.answer.num=$Ans@
qu.2.31.answer.units=@
qu.2.31.showUnits=false@
qu.2.31.grading=toler_abs@
qu.2.31.err=.001@
qu.2.31.negStyle=minus@
qu.2.31.numStyle=thousands scientific dollars arithmetic@
qu.2.31.mode=Numeric@
qu.2.31.name=29. P(Replace >= n TV tubes)@
qu.2.31.comment=<div style="margin-top: 0px; margin-bottom: 2px;" class="shadedDiv descriptionSpan">P(tube has to be replaced) = p = $p.<br />
<br />
Let X = number of tubes which have to be replaced out of n =&nbsp;$n tubes. Assuming that the tubes behave independently, X ~ bin($n, $p).<br />
<br />
Since n is large and p small we can use the Poisson approximation to the binomial distribution; &lambda; = np =&nbsp;$n &times;&nbsp;$p = $lambda, i.e. X ~ Poisson($lambda) approximately and:<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&asymp;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><msup><mi>&lambda;</mi><mrow><mi>x</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>e</mi><mrow><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&lambda;</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mi>x</mi><mo lspace='0.1111111em' rspace='0.1111111em'>&excl;</mo></mrow></mfrac></mrow></mstyle></math><br />
P(at least 3 tubes are replaced) = P(X &ge; 3) = 1 &minus; P(X &le; 2) = 1&minus;P(X = 0) &minus; P(X = 1) &minus; P(X = 2) = 1 &minus;&nbsp;$nans &asymp; $Ans.</div>@
qu.2.31.editing=useHTML@
qu.2.31.solution=@
qu.2.31.algorithm=$Q=29;
$p=decimal(2,range(0.01,0.06,0.001));
$pp=100*$p;
$n=range(80,300,10);
$lambda=$n*$p;
$Pnans=maple("stats[statevalf, pf, poisson[$lambda]](0)+stats[statevalf, pf, poisson[$lambda]](1)+stats[statevalf, pf, poisson[$lambda]](2)");
$nans=decimal(4,$Pnans);
$ans=1-$nans;
$Ans=decimal(4,$ans);
condition:gt($Ans,0.0009);
$Which=rint(4);
$Align=switch(rint(2),"Left","Right");@
qu.2.31.uid=e7b7c53d-6b39-4f74-87c7-4d621a066974@
qu.2.31.info=  Course=230;
  Keyword=poisson;
  Keyword=binomial;
  Type=numeric;
@

qu.2.32.mode=Inline@
qu.2.32.name=27. P(Phone Calls frequency (various))@
qu.2.32.comment=<p>First note that &lambda; = $L . Calculate the various probabilities:&nbsp;</p>
<p>
<table cellspacing="3" cellpadding="0" bordercolor="#111111" border="1" id="AutoNumber1" style="border-collapse: collapse;">
    <tbody>
        <tr>
            <td># Calls in an Hour</td>
            <td>&nbsp;Probability</td>
        </tr>
        <tr>
            <td align="center">0</td>
            <td align="center">&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mi mathvariant='normal'>$L</mi><mrow><mn>0</mn></mrow></msup><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$L</mi></mrow></msup></mrow><mrow><mn>0</mn><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$p0</mi></mrow></mstyle></math></td>
        </tr>
        <tr>
            <td align="center">1</td>
            <td align="center"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mi mathvariant='normal'>$L</mi><mrow><mn>1</mn></mrow></msup><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$L</mi></mrow></msup></mrow><mrow><mn>1</mn><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$p1</mi></mrow></mstyle></math>&nbsp;</td>
        </tr>
        <tr>
            <td align="center">2</td>
            <td align="center"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mi mathvariant='normal'>$L</mi><mrow><mn>2</mn></mrow></msup><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$L</mi></mrow></msup></mrow><mrow><mn>2</mn><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$p2</mi></mrow></mstyle></math>&nbsp;</td>
        </tr>
        <tr>
            <td align="center">3</td>
            <td align="center">$p3</td>
        </tr>
        <tr>
            <td align="center">4</td>
            <td align="center">&nbsp;$p4</td>
        </tr>
        <tr>
            <td align="center">5</td>
            <td align="center">&nbsp;$p5</td>
        </tr>
    </tbody>
</table>
<br />
The questions can be answered simply using the table:</p>
<p>a) P(at most 3 calls) = P(0 calls) + P(1) + P(2) + P(3) = $AnsA</p>
<p>b) P(At least 3) = 1 - P(at most 2) = 1 - P(0) - P(1) - P(2) = $AnsB</p>
<p>c) P(5 or more) = 1 - P(4 or less) = 1 - P(0) - P(1) - P(2) - P(3) - P(4) = $AnsC</p>@
qu.2.32.editing=useHTML@
qu.2.32.solution=@
qu.2.32.algorithm=$Q="27";
$L=rint(3,5);
$p0=exp(-$L);
$p1=$L*exp(-$L);
$p2=$L^2*exp(-$L)/fact(2);
$p3=$L^3*exp(-$L)/fact(3);
$p4=$L^4*exp(-$L)/fact(4);
$p5=$L^5*exp(-$L)/fact(5);
$AnsA=decimal(4,$p0+$p1+$p2+$p3);
$AnsB=decimal(4,1-$p0-$p1-$p2);
$AnsC=decimal(4,1-$AnsA-$p4);@
qu.2.32.uid=b4669833-37bb-4182-add7-aa9abdb3ea85@
qu.2.32.info=  Course=202;
  Type=numericx3;
  Keyword=poisson;
@
qu.2.32.weighting=1,1,1@
qu.2.32.numbering=alpha@
qu.2.32.part.1.name=sro_id_1@
qu.2.32.part.1.answer.units=@
qu.2.32.part.1.numStyle=thousands scientific  arithmetic@
qu.2.32.part.1.editing=useHTML@
qu.2.32.part.1.showUnits=false@
qu.2.32.part.1.err=0.0020@
qu.2.32.part.1.question=(Unset)@
qu.2.32.part.1.mode=Numeric@
qu.2.32.part.1.grading=toler_abs@
qu.2.32.part.1.negStyle=minus@
qu.2.32.part.1.answer.num=$AnsA@
qu.2.32.part.2.name=sro_id_2@
qu.2.32.part.2.answer.units=@
qu.2.32.part.2.numStyle=thousands scientific  arithmetic@
qu.2.32.part.2.editing=useHTML@
qu.2.32.part.2.showUnits=false@
qu.2.32.part.2.err=0.0020@
qu.2.32.part.2.question=(Unset)@
qu.2.32.part.2.mode=Numeric@
qu.2.32.part.2.grading=toler_abs@
qu.2.32.part.2.negStyle=minus@
qu.2.32.part.2.answer.num=$AnsB@
qu.2.32.part.3.name=sro_id_3@
qu.2.32.part.3.answer.units=@
qu.2.32.part.3.numStyle=thousands scientific  arithmetic@
qu.2.32.part.3.editing=useHTML@
qu.2.32.part.3.showUnits=false@
qu.2.32.part.3.err=0.0020@
qu.2.32.part.3.question=(Unset)@
qu.2.32.part.3.mode=Numeric@
qu.2.32.part.3.grading=toler_abs@
qu.2.32.part.3.negStyle=minus@
qu.2.32.part.3.answer.num=$AnsC@
qu.2.32.question=<div title="UW Statistic Question Bank/Discrete Probability Models/Poisson/Q$Q">A firm receives, on the average $L calls per hour on its toll-free number. For any given hour, find the pr. that it will receive the following: (4 decimal accuracy)<p>&nbsp;</p><p>a) At MOST 3 calls .&nbsp;<span> </span><1><span>&nbsp;</span></p><p>&nbsp;</p><p><span>b) At LEAST 3 calls . <span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p><span>c) 5 or more calls . &nbsp;<span>&nbsp;</span><3><span>&nbsp;</span>&nbsp; </span></p></div>@

qu.2.33.question=<div title="UW Statistics Question Bank/Discrete Probability Models/Poisson/Q$Q">Pulses arrive at a Geiger counter in accordance with a Poisson Process. In any one hour time period, it is known that there is a $P% chance that no pulses arrive to the counter. What is the value of &lambda;, the arrival rate of pulses per hour? (4 decimal accuracy please.)</div>@
qu.2.33.answer.num=$Ans@
qu.2.33.answer.units=@
qu.2.33.showUnits=false@
qu.2.33.grading=toler_abs@
qu.2.33.err=0.001@
qu.2.33.negStyle=minus@
qu.2.33.numStyle=thousands scientific dollars arithmetic@
qu.2.33.mode=Numeric@
qu.2.33.name=02. Geiger Counter II@
qu.2.33.comment=<p>Let X count the # pulses which arrive during any 1 hour period.&nbsp; Then X ~ Poisson(&lambda;) and we need to determine the <span style="font-style: italic; font-weight: bold;">rate</span> &lambda;. But the question tells you P(X = 0)!&nbsp;</p>
<p>So set <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&lambda;</mi></mrow></mrow></msup><mrow><msup><mi>&lambda;</mi><mrow><mn>0</mn></mrow></msup></mrow></mrow><mrow><mn>0</mn><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi mathvariant='normal'>$P</mi><mrow><mn>100</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$DecP</mi></mrow></mstyle></math></p>
<p>So <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&lambda;</mi></mrow></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$DecP</mi></mrow></mstyle></math>,&nbsp;  &lambda; = -ln($DecP) = $Ans</p>@
qu.2.33.editing=useHTML@
qu.2.33.solution=@
qu.2.33.algorithm=$Q=2;
$P=decimal(2,range(1,15,0.01));

$DecP=$P/100;
$Ans=-Ln($P/100);@
qu.2.33.uid=6479d667-fc89-4258-a4d0-246894d42c9b@
qu.2.33.info=  Course=230;
  Keyword=poisson;
  Type=numeric;
@

qu.3.topic=Other Models@

qu.3.1.mode=Multiple Choice@
qu.3.1.name=02b. P(X > a)@
qu.3.1.comment=<p>First find "c" from the pf definition:<br />
The sum of all the probabilities must be 1, so c + 2c + ... + ($M)c = 1.&nbsp; Use the summation formula:<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><munderover accent='false' accentunder='false'><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>&Sum;i</mi><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>i</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='unset' separator='unset' lspace='thickmathspace' rspace='thickmathspace' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&equals;</mo><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>1</mn></mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>n</mi></munderover><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&equals;</mo><mfrac linethickness='1' denomalign='center' numalign='center' bevelled='false'><mrow><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>n</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&ApplyFunction;</mo><mfenced><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>n</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='mediummathspace' rspace='mediummathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&plus;</mo><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>1</mn></mrow></mfenced></mrow></mrow><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>2</mn></mfrac></mrow></mrow></math><br />
here this means: <br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac linethickness='1' denomalign='center' numalign='center' bevelled='false'><mrow><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>c$M</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&ApplyFunction;</mo><mfenced><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$Mp1</mi></mrow></mfenced></mrow></mrow><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>2</mn></mfrac><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&equals;</mo><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>1</mn><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&InvisibleTimes;</mo><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>so</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&InvisibleTimes;</mo><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>c</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&equals;</mo><mfrac linethickness='1' denomalign='center' numalign='center' bevelled='false'><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>2</mn><mrow><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$M</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&ApplyFunction;</mo><mfenced><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$Mp1</mi></mrow></mfenced></mrow></mrow></mfrac><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&equals;</mo><mfrac linethickness='1' denomalign='center' numalign='center' bevelled='false'><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>1</mn><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$PreC</mi></mfrac><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&equals;</mo><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$C</mi></mrow></math><br />
P(X > $L) <br />
= 1 - P(X &le; $L)<br />
= 1 - (c + 2c + ... + ($L)c)<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&equals;</mo><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='mediummathspace' rspace='mediummathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&InvisibleTimes;</mo><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>1</mn><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='mediummathspace' rspace='mediummathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&InvisibleTimes;</mo><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='mediummathspace' rspace='mediummathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&#8722;</mo><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='mediummathspace' rspace='mediummathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&InvisibleTimes;</mo><mfrac linethickness='1' denomalign='center' numalign='center' bevelled='false'><mrow><mfenced><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$C</mi></mrow></mfenced><mfenced><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$L</mi></mrow></mfenced><mfenced><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$L</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='mediummathspace' rspace='mediummathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&plus;</mo><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>1</mn></mrow></mfenced></mrow><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>2</mn></mfrac><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&equals;</mo><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$Ans</mi></mrow></math></p>@
qu.3.1.editing=useHTML@
qu.3.1.solution=@
qu.3.1.algorithm=$Q="02b";
$M=range(8,14,2);
$Mp1=$M+1;
$L=range(2,($M)-1,1);
$PreC=($M)*(1+$M)/2;
$C = 1/$PreC;
$Ans=decimal(3,1 - ($C)($L)(($L)+1)/2);
$Alt1=decimal(3,$Ans+range(0.5,0.9,0.01)*(1-$Ans));
$Alt2=decimal(3,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(3,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.3.1.uid=1ae7392b-c115-48c2-9e53-7bf6705fbf25@
qu.3.1.info=  Type=MC;
  Difficulty=3;
  Course=230;
@
qu.3.1.question=<div title="UW Statistics Bank/Discrete Probability Models/Other Models/Q$Q">Suppose that f(x) = cx, x = 1,2,...,$M  is a pf of a discrete random variable X. Find P(X > $L).</div>@
qu.3.1.answer=1@
qu.3.1.choice.1=$Ans@
qu.3.1.choice.2=$Alt1@
qu.3.1.choice.3=$Alt2@
qu.3.1.choice.4=$Alt3@
qu.3.1.fixed=@

qu.3.2.mode=Inline@
qu.3.2.name=01a. P(changeover time) MC@
qu.3.2.comment=<p>Just substitute $x into the probability function given.</p>@
qu.3.2.editing=useHTML@
qu.3.2.solution=@
qu.3.2.algorithm=$Q="1a";
$x=range(1,4);
$Ans = $x/10;
$Alt1=decimal(2,$Ans+range(0.5,0.9,0.01)*(1-$Ans));
$Alt2=decimal(2,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(2,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.3.2.uid=69a1705f-ec48-4a77-a475-ea420a44e4d3@
qu.3.2.info=  Course=202;
  Difficulty=1;
@
qu.3.2.weighting=1@
qu.3.2.numbering=alpha@
qu.3.2.part.1.name=sro_id_1@
qu.3.2.part.1.editing=useHTML@
qu.3.2.part.1.fixed=@
qu.3.2.part.1.choice.4=$Alt3<br>@
qu.3.2.part.1.question=null@
qu.3.2.part.1.choice.3=$Alt2<br>@
qu.3.2.part.1.choice.2=$Alt1@
qu.3.2.part.1.choice.1=$Ans@
qu.3.2.part.1.mode=Non Permuting Multiple Choice@
qu.3.2.part.1.display=vertical@
qu.3.2.part.1.answer=1@
qu.3.2.question=<div title="UW Statistics Bank/Discrete Probability Models/Other/Q$Q">The changeover operation for a production system requires from 1 to 4 hours. Let x be a random variable indicating the time in hours required to make the changeover. The following probability function can be used to compute the probability associated with any changeover time x: <br /><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>x</mi><mrow><mn>10</mn></mrow></mfrac></mrow></mrow></mstyle></math> for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>2</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>3</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>4</mn></mrow></mstyle></math></p><p>What is the probability that the changeover will take $x hours?</p><p>Ans:&nbsp; <1><span>&nbsp;</span> <em>(1 decimal, or use a fraction (preferred))</em></p></div>@

qu.3.3.question=<div title="UW Statistics Bank/Discrete Probability Models/Other Models/Q$Q"><img width="79" hspace="4" height="79" align="right" alt="8 sided die" src="__BASE_URI__DPM/OtherModels/8SidedDie.gif" />An 8-sided die is loaded to give the following probabilities:<br />
<br />
<table cellspacing="2" cellpadding="2" bordercolor="#111111" border="1" style="border-collapse: collapse;">
    <tbody>
        <tr>
            <td>&nbsp;number</td>
            <td align="center">1</td>
            <td align="center">2</td>
            <td align="center">3</td>
            <td align="center">4</td>
            <td align="center">5</td>
            <td align="center">6</td>
            <td align="center">7</td>
            <td align="center">8</td>
        </tr>
        <tr>
            <td>probability</td>
            <td align="right">$D1</td>
            <td align="right">$D2</td>
            <td align="right">$D3</td>
            <td align="right">$D4</td>
            <td align="right">$D5</td>
            <td align="right">$D6</td>
            <td align="right">$D7</td>
            <td align="right">$D8</td>
        </tr>
    </tbody>
</table>
<br />
The die is rolled (independently) 4 times. Find the probability that one <u>or more</u> of these rolls results in an odd number. (4 decimal accuracy)</div>@
qu.3.3.answer.num=$Ans@
qu.3.3.answer.units=@
qu.3.3.showUnits=false@
qu.3.3.grading=toler_abs@
qu.3.3.err=.001@
qu.3.3.negStyle=minus@
qu.3.3.numStyle=thousands scientific dollars arithmetic@
qu.3.3.mode=Numeric@
qu.3.3.name=11a. The 8-sided die is rolled 4 times; probability that an odd number occurs - clone@
qu.3.3.comment=<div style="margin-top: 0px; margin-bottom: 2px;" class="shadedDiv descriptionSpan">
<div style="margin-top: 0px; margin-bottom: 2px;" class="shadedDiv descriptionSpan">
<p>Notice that for one roll, P(odd) = $POdd, P(even) = $PEven</p>
<p>P(At least one odd) = 1 - P(no odds) = 1 - P(all evens) =&nbsp; 1 - ($PEven)<sup>4</sup> = $Ans</p>
</div>
</div>@
qu.3.3.editing=useHTML@
qu.3.3.solution=@
qu.3.3.algorithm=$Q="11a";
$S=rint(8);
$P11=switch(rint(4),0.05,0.1,0.15,0.2);
$P12=0.25-$P11;
$P21=switch(rint(5),0.05,0.1,0.15,0.2,0.25);
$P22=0.3-$P21;
$D1=switch($S,$P11,$P12,0.05,0.1,0.1,0.2,$P21,$P22);
$D2=switch($S,$P12,0.05,0.1,0.1,0.2,$P21,$P22,$P11);
$D3=switch($S,0.05,0.1,0.1,0.2,$P21,$P22,$P11,$P12);
$D4=switch($S,0.1,0.1,0.2,$P21,$P22,$P11,$P12,0.05);
$D5=switch($S,0.1,0.2,$P21,$P22,$P11,$P12,0.05,0.1);
$D6=switch($S,0.2,$P21,$P22,$P11,$P12,0.05,0.1,0.1);
$D7=switch($S,$P21,$P22,$P11,$P12,0.05,0.1,0.1,0.2);
$D8=switch($S,$P22,$P11,$P12,0.05,0.1,0.1,0.2,$P21);
$POdd=$D1+$D3+$D5+$D7;
$PEven=1-$POdd;
$Ans=decimal(4,1-$PEven^4);@
qu.3.3.uid=2313921c-02e8-43ad-a574-f1e689ac196c@
qu.3.3.info=  Type=Numeric;
  Course=230;
@

qu.3.4.mode=Inline@
qu.3.4.name=01b. P(changeover time)@
qu.3.4.comment=<p>Just substitute <font size="3" face="Times New Roman"><em>x</em> = $x</font> into <font size="3" face="Times New Roman"><em>f</em>(<em>x</em>) =</font> $F , then multiply by 60.</p>@
qu.3.4.editing=useHTML@
qu.3.4.solution=@
qu.3.4.algorithm=$Q="01b";
$Pick=rint(4);
$F=switch($Pick,mathml("x/10"),mathml("1/6+x/12"),mathml("1/3+x/15"),mathml("1/2+x/20"));
$x=range(1,4);
$Ans = decimal(4,switch($Pick,$x/10,1/6+$x/12,1/3+$x/15,1/2+$x/20));
$Alt1 = decimal(4,switch($Pick,1/6+$x/12,1/3+$x/15,1/2+$x/20,$x/10));
$Alt2 = decimal(4,switch($Pick,1/3+$x/15,1/2+$x/20,$x/10,1/6+$x/12));
$Alt3 = decimal(4,switch($Pick,1/2+$x/20,$x/10,1/6+$x/12,1/3+$x/15));@
qu.3.4.uid=12375ca1-b0be-4a7b-afbc-76b9b7e6054a@
qu.3.4.info=  Course=202;
  Difficulty=1;
  Type=MC;
@
qu.3.4.weighting=1@
qu.3.4.numbering=alpha@
qu.3.4.part.1.name=sro_id_1@
qu.3.4.part.1.editing=useHTML@
qu.3.4.part.1.fixed=@
qu.3.4.part.1.choice.4=$Alt3<br>@
qu.3.4.part.1.question=null@
qu.3.4.part.1.choice.3=$Alt2<br>@
qu.3.4.part.1.choice.2=$Alt1@
qu.3.4.part.1.choice.1=$Ans@
qu.3.4.part.1.mode=Multiple Choice@
qu.3.4.part.1.display=vertical@
qu.3.4.part.1.answer=1@
qu.3.4.question=<div title="UW Statistics Bank/Discrete Probability Models/Other Models/Q$Q">The changeover operation for a production system requires from 1 to 4 hours. Let x be a random variable indicating the time in hours required to make the changeover. The following probability function can be used to compute the probability associated with any changeover time x (in hours): <br /><p><font size="3" face="Times New Roman"><em>f</em>(<em>x</em>) =&nbsp;</font> $F for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>2</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>3</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>4</mn></mrow></mstyle></math></p><p>What is the probability that the changeover will take $x hours?</p><p>&nbsp;<1><span>&nbsp;</span></p></div>@

qu.3.5.mode=Inline@
qu.3.5.name=01a. P(changeover time)@
qu.3.5.comment=<p>Just substitute <font size="3" face="Times New Roman"><em>x</em> = $x</font> into <font size="3" face="Times New Roman"><em>f</em>(<em>x</em>) =</font> $F .</p>@
qu.3.5.editing=useHTML@
qu.3.5.solution=@
qu.3.5.algorithm=$Q="01a";
$Pick=rint(4);
$F=switch($Pick,mathml("x/10"),mathml("1/6+x/12"),mathml("1/3+x/15"),mathml("1/2+x/20"));
$x=range(1,4);
$PreAns = switch($Pick,$x/10,1/6+$x/12,1/3+$x/15,1/2+$x/20);
$Ans=decimal(4,$PreAns);
$Minutes=60*$PreAns;@
qu.3.5.uid=32f84132-9637-4e21-a58a-38d45c2bc72e@
qu.3.5.info=  Course=202;
  Difficulty=1;
  Type=numeric;
@
qu.3.5.weighting=1@
qu.3.5.numbering=alpha@
qu.3.5.part.1.name=sro_id_1@
qu.3.5.part.1.answer.units=@
qu.3.5.part.1.numStyle= scientific  arithmetic@
qu.3.5.part.1.editing=useHTML@
qu.3.5.part.1.showUnits=false@
qu.3.5.part.1.err=0.01@
qu.3.5.part.1.question=(Unset)@
qu.3.5.part.1.mode=Numeric@
qu.3.5.part.1.grading=toler_abs@
qu.3.5.part.1.negStyle=minus@
qu.3.5.part.1.answer.num=$Ans@
qu.3.5.question=<div title="UW Statistics Bank/Discrete Probability Models/Other Models/Q$Q">The changeover operation for a production system requires from 1 to 4 hours. Let x be a random variable indicating the time in hours required to make the changeover. The following probability function can be used to compute the probability associated with any changeover time x: <br /><p><font size="3" face="Times New Roman"><em>f</em>(<em>x</em>) =&nbsp;</font> $F for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>2</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>3</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>4</mn></mrow></mstyle></math></p><p>What is the probability that the changeover will take $x hours?</p><p>Ans:&nbsp; <1><span>&nbsp;</span> (3 decimal accuracy)</p></div>@

qu.3.6.mode=Multiple Choice@
qu.3.6.name=11b. The 8-sided die is rolled 4 times; probability that an odd number occurs@
qu.3.6.comment=<div style="margin-top: 0px; margin-bottom: 2px;" class="shadedDiv descriptionSpan">
<div style="margin-top: 0px; margin-bottom: 2px;" class="shadedDiv descriptionSpan">
<p>Notice that for one roll, P(odd) = $POdd, P(even) = $PEven</p>
<p>P(At least one odd) = 1 - P(no odds) = 1 - P(all evens) =&nbsp; 1 - ($PEven)<sup>4</sup> = $Ans</p>
</div>
</div>@
qu.3.6.editing=useHTML@
qu.3.6.solution=@
qu.3.6.algorithm=$Q="11b";
$S=rint(8);
$P11=switch(rint(4),0.05,0.1,0.15,0.2);
$P12=0.25-$P11;
$P21=switch(rint(5),0.05,0.1,0.15,0.2,0.25);
$P22=0.3-$P21;
$D1=switch($S,$P11,$P12,0.05,0.1,0.1,0.2,$P21,$P22);
$D2=switch($S,$P12,0.05,0.1,0.1,0.2,$P21,$P22,$P11);
$D3=switch($S,0.05,0.1,0.1,0.2,$P21,$P22,$P11,$P12);
$D4=switch($S,0.1,0.1,0.2,$P21,$P22,$P11,$P12,0.05);
$D5=switch($S,0.1,0.2,$P21,$P22,$P11,$P12,0.05,0.1);
$D6=switch($S,0.2,$P21,$P22,$P11,$P12,0.05,0.1,0.1);
$D7=switch($S,$P21,$P22,$P11,$P12,0.05,0.1,0.1,0.2);
$D8=switch($S,$P22,$P11,$P12,0.05,0.1,0.1,0.2,$P21);
$POdd=$D1+$D3+$D5+$D7;
$PEven=1-$POdd;
$Ans=decimal(4,1-$PEven^4);
$Alt1=decimal(4,$Ans+range(0.5,0.9,0.01)*(1-$Ans));
$Alt2=decimal(4,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(4,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.3.6.uid=77b7d274-a43b-4797-a593-fd2a5d165d0c@
qu.3.6.info=  Type=MC;
  Course=230;
@
qu.3.6.question=<div title="UW Statistics Bank/Discrete Probability Models/Other Models/Q$Q"><img width="79" hspace="4" height="79" align="right" alt="8 sided die" src="__BASE_URI__DPM/OtherModels/8SidedDie.gif" />An 8-sided die is loaded to give the following probabilities:<br />
<br />
<table cellspacing="2" cellpadding="2" bordercolor="#111111" border="1" style="border-collapse: collapse;">
    <tbody>
        <tr>
            <td>&nbsp;number</td>
            <td align="center">1</td>
            <td align="center">2</td>
            <td align="center">3</td>
            <td align="center">4</td>
            <td align="center">5</td>
            <td align="center">6</td>
            <td align="center">7</td>
            <td align="center">8</td>
        </tr>
        <tr>
            <td>probability</td>
            <td align="right">$D1</td>
            <td align="right">$D2</td>
            <td align="right">$D3</td>
            <td align="right">$D4</td>
            <td align="right">$D5</td>
            <td align="right">$D6</td>
            <td align="right">$D7</td>
            <td align="right">$D8</td>
        </tr>
    </tbody>
</table>
<br />
The die is rolled (independently) 4 times. Find the probability that one <u>or more</u> of these rolls results in an odd number.</div>@
qu.3.6.answer=1@
qu.3.6.choice.1=$Ans@
qu.3.6.choice.2=$Alt1@
qu.3.6.choice.3=$Alt2@
qu.3.6.choice.4=$Alt3@
qu.3.6.fixed=@

qu.3.7.question=<div title="UW Statistics Bank/Discrete Probability Models/Other Models/Q$Q"><img hspace="4" align="$Align" src="__BASE_URI__DPM/OtherModels/Dice$Which.gif" alt="Two dice" title="Two dice [IMG:Dice$Which.gif]" />Suppose we define Z to be the maximum of the throws on two fair (6-sided) dice and let W be the minimum.
<p>Let R = Z - W and let f(r) = P(R = r) represent the probability function for R.</p>
<p>Find f($r) (3 decimals of accuracy.)</p>
</div>@
qu.3.7.answer.num=$Ans@
qu.3.7.answer.units=@
qu.3.7.showUnits=false@
qu.3.7.grading=toler_abs@
qu.3.7.err=.01@
qu.3.7.negStyle=minus@
qu.3.7.numStyle=thousands scientific dollars arithmetic@
qu.3.7.mode=Numeric@
qu.3.7.name=06. Probability Function on 2 dice.@
qu.3.7.comment=<p>Write out all possible outcomes (in terms of R) in a table:</p>

<div align="center">
  <center>
  <table border="1" cellpadding="$F2 2$EndSpan" cellspacing="$F0 0$EndSpan" style="border-collapse: collapse" bordercolor="#$F1 1$EndSpan$F1 1$EndSpan$F1 1$EndSpan$F1 1$EndSpan$F1 1$EndSpan$F1 1$EndSpan" id="AutoNumber$F1 1$EndSpan">
    <tr>
      <td colspan="2" rowspan="2">&nbsp;</td>
      <td align="center" colspan="6">Dice 1</td>
    </tr>
    <tr>
      <td align="center">1</td>
      <td align="center">2</td>
      <td align="center">3</td>
      <td align="center">4</td>
      <td align="center">5</td>
      <td align="center">6</td>
    </tr>
    <tr>
      <td rowspan="6">
      <p align="center">D<br>
      i<br>
      c<br>
      e<br>
      <br>
      2</td>
      <td>1</td>
      <td>$F0 0$EndSpan</td>
      <td>$F1 1$EndSpan</td>
      <td>$F2 2$EndSpan</td>
      <td>$F3 3$EndSpan</td>
      <td>$F4 4$EndSpan</td>
      <td>$F5 5$EndSpan</td>
    </tr>
    <tr>
      <td>&nbsp;2</td>
      <td>$F1 1$EndSpan</td>
      <td>$F0 0$EndSpan</td>
      <td>$F1 1$EndSpan</td>
      <td>$F2 2$EndSpan</td>
      <td>$F3 3$EndSpan</td>
      <td>$F4 4$EndSpan</td>
    </tr>
    <tr>
      <td>&nbsp;3</td>
      <td>$F2 2$EndSpan</td>
      <td>$F1 1$EndSpan</td>
      <td>$F0 0$EndSpan</td>
      <td>$F1 1$EndSpan</td>
      <td>$F2 2$EndSpan</td>
      <td>$F3 3$EndSpan</td>
    </tr>
    <tr>
      <td>&nbsp;4</td>
      <td>$F3 3$EndSpan</td>
      <td>$F2 2$EndSpan</td>
      <td>$F1 1$EndSpan</td>
      <td>$F0 0$EndSpan</td>
      <td>$F1 1$EndSpan</td>
      <td>$F2 2$EndSpan</td>
    </tr>
    <tr>
      <td>&nbsp;5</td>
      <td>$F4 4$EndSpan</td>
      <td>$F3 3$EndSpan</td>
      <td>$F2 2$EndSpan</td>
      <td>$F1 1$EndSpan</td>
      <td>$F0 0$EndSpan</td>
      <td>$F1 1$EndSpan</td>
    </tr>
    <tr>
      <td>6</td>
      <td>$F5 5$EndSpan</td>
      <td>$F4 4$EndSpan</td>
      <td>$F3 3$EndSpan</td>
      <td>$F2 2$EndSpan</td>
      <td>$F1 1$EndSpan</td>
      <td>$F0 0$EndSpan</td>
    </tr>
    <tr>
      <td colspan="2">&nbsp;</td>
      <td colspan="6">
      <p align="center">R = (maximum - minimum)</td>
    </tr>
  </table>
  </center>
</div>


<p><br />
As you can see there are <font size="3" face="Times New Roman">$AnsT</font> outcomes where <font size="3" face="Times New Roman"><em>R </em>= $r</font>, so <font size="3" face="Times New Roman"><em>f</em>($r)= $AnsML</font> or <font size="3" face="Times New Roman">$Ans</font> .</p>@
qu.3.7.editing=useHTML@
qu.3.7.solution=@
qu.3.7.algorithm=$Q=6;
$r=rint(6);
$AnsT=switch($r,6,10,8,6,4,2);
$Ans=decimal(4,$AnsT/36);
$AnsML=mathml("$AnsT/36");
$Which=rint(4);
$Align=switch(rint(2),"Left","Right");
$EndSpan="</span>";
$F0=if(eq($r,0),"<span style='color:red'>","");
$F1=if(eq($r,1),"<span style='color:red'>","");
$F2=if(eq($r,2),"<span style='color:red'>","");
$F3=if(eq($r,3),"<span style='color:red'>","");
$F4=if(eq($r,4),"<span style='color:red'>","");
$F5=if(eq($r,5),"<span style='color:red'>","");@
qu.3.7.uid=1b474e47-3a40-4528-9749-09e45ffe3917@
qu.3.7.info=  Course=230;
  Type=numeric;
@

qu.3.8.mode=Multiple Choice@
qu.3.8.name=12. P(~x^~y on 8-sided roll)@
qu.3.8.comment=<div style="margin-top: 0px; margin-bottom: 2px;" class="shadedDiv descriptionSpan">Notice that for one roll, P(neither&nbsp;$x nor $y) = 1 - P($x or $y) = 1 - ($px + $py) = $notxy<br />
For four rolls then:<br />
<br />
P(neither&nbsp;$x nor $y) = ($notxy)<sup>4</sup> = $Ans</div>@
qu.3.8.editing=useHTML@
qu.3.8.solution=@
qu.3.8.algorithm=$Q="12";
$x=range(1,8);
$y=range(1,8);
condition:not(eq($x,$y));
$S=rint(8);
$P11=switch(rint(4),0.05,0.1,0.15,0.2);
$P12=0.25-$P11;
$P21=switch(rint(5),0.05,0.1,0.15,0.2,0.25);
$P22=0.3-$P21;
$D1=switch($S,$P11,$P12,0.05,0.1,0.1,0.2,$P21,$P22);
$D2=switch($S,$P12,0.05,0.1,0.1,0.2,$P21,$P22,$P11);
$D3=switch($S,0.05,0.1,0.1,0.2,$P21,$P22,$P11,$P12);
$D4=switch($S,0.1,0.1,0.2,$P21,$P22,$P11,$P12,0.05);
$D5=switch($S,0.1,0.2,$P21,$P22,$P11,$P12,0.05,0.1);
$D6=switch($S,0.2,$P21,$P22,$P11,$P12,0.05,0.1,0.1);
$D7=switch($S,$P21,$P22,$P11,$P12,0.05,0.1,0.1,0.2);
$D8=switch($S,$P22,$P11,$P12,0.05,0.1,0.1,0.2,$P21);
$px=switch($x-1,$D1,$D2,$D3,$D4,$D5,$D6,$D7,$D8);
$py=switch($y-1,$D1,$D2,$D3,$D4,$D5,$D6,$D7,$D8);
$notxy = 1-$px-$py;
$Ans=decimal(4,$notxy^4);
$Alt1=decimal(4,$Ans+range(0.5,0.9,0.01)*(1-$Ans));
$Alt2=decimal(4,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(4,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.3.8.uid=491771a0-f776-4d33-ab6d-d9abde967146@
qu.3.8.info=  Type=MC;
  Course=230;
@
qu.3.8.question=<div title="UW Statistics Bank/Discrete Probability Models/Other Models/Q$Q">An 8-sided die is loaded to give the following probabilities:<br />
<br />
<table cellspacing="2" cellpadding="2" bordercolor="#111111" border="1" style="border-collapse: collapse;">
    <tbody>
        <tr>
            <td>&nbsp;number</td>
            <td align="center">1</td>
            <td align="center">2</td>
            <td align="center">3</td>
            <td align="center">4</td>
            <td align="center">5</td>
            <td align="center">6</td>
            <td align="center">7</td>
            <td align="center">8</td>
        </tr>
        <tr>
            <td>probability</td>
            <td align="right">$D1</td>
            <td align="right">$D2</td>
            <td align="right">$D3</td>
            <td align="right">$D4</td>
            <td align="right">$D5</td>
            <td align="right">$D6</td>
            <td align="right">$D7</td>
            <td align="right">$D8</td>
        </tr>
    </tbody>
</table>
<br />
The die is rolled (independently) 4 times. Find the probability that neither $x or $y occurs.</div>@
qu.3.8.answer=1@
qu.3.8.choice.1=$Ans@
qu.3.8.choice.2=$Alt1@
qu.3.8.choice.3=$Alt2@
qu.3.8.choice.4=$Alt3@
qu.3.8.fixed=@

qu.3.9.mode=Multiple Choice@
qu.3.9.name=16. P(x^y on 8-sided roll)@
qu.3.9.comment=<p>Let A<sub>j</sub> = &ldquo;the number j occurs&rdquo; Then:</p>
<p>P(A<sub>x</sub>A<sub>y</sub>) = 1 - P(<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mfenced open='(' close=')' separators=','><mrow><msub><mi>A</mi><mrow><mi>x</mi></mrow></msub><msub><mi>A</mi><mrow><mi>y</mi></mrow></msub></mrow></mfenced></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>____</mo></mrow></mover></mrow></mstyle></math>)&nbsp;&nbsp; <span style="font-style: italic;"><br />
</span>= 1 - P( <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><msub><mi>A</mi><mrow><mi>x</mi></mrow></msub></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&minus;</mo></mrow></mover></mrow></mstyle></math> U <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><msub><mi>A</mi><mrow><mi>y</mi></mrow></msub></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&minus;</mo></mrow></mover></mrow></mstyle></math> )&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <span style="font-style: italic;">Now use: P(XUY) = P(X) + P(Y) - P(XY) and don't forget the leading - sign !</span><br />
= 1 - P(<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><msub><mi>A</mi><mrow><mi>x</mi></mrow></msub></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&minus;</mo></mrow></mover></mrow></mstyle></math>) - P(<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><msub><mi>A</mi><mrow><mi>y</mi></mrow></msub></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&minus;</mo></mrow></mover></mrow></mstyle></math>) + P( <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><msub><mi>A</mi><mrow><mi>x</mi></mrow></msub></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&minus;</mo></mrow></mover></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><msub><mi>A</mi><mrow><mi>y</mi></mrow></msub></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&minus;</mo></mrow></mover></mrow></mstyle></math>)</p>
<p>= 1 - (1-$px)<sup>4</sup> - (1-$py)<sup>4</sup> + ($notxy)<sup>4</sup> = $Ans</p>@
qu.3.9.editing=useHTML@
qu.3.9.solution=@
qu.3.9.algorithm=$Q="16";
$x=range(1,8);
$y=range(1,8);
condition:not(eq($x,$y));
$S=rint(8);
$P11=switch(rint(4),0.05,0.1,0.15,0.2);
$P12=0.25-$P11;
$P21=switch(rint(5),0.05,0.1,0.15,0.2,0.25);
$P22=0.3-$P21;
$D1=switch($S,$P11,$P12,0.05,0.1,0.1,0.2,$P21,$P22);
$D2=switch($S,$P12,0.05,0.1,0.1,0.2,$P21,$P22,$P11);
$D3=switch($S,0.05,0.1,0.1,0.2,$P21,$P22,$P11,$P12);
$D4=switch($S,0.1,0.1,0.2,$P21,$P22,$P11,$P12,0.05);
$D5=switch($S,0.1,0.2,$P21,$P22,$P11,$P12,0.05,0.1);
$D6=switch($S,0.2,$P21,$P22,$P11,$P12,0.05,0.1,0.1);
$D7=switch($S,$P21,$P22,$P11,$P12,0.05,0.1,0.1,0.2);
$D8=switch($S,$P22,$P11,$P12,0.05,0.1,0.1,0.2,$P21);
$px=switch($x-1,$D1,$D2,$D3,$D4,$D5,$D6,$D7,$D8);
$py=switch($y-1,$D1,$D2,$D3,$D4,$D5,$D6,$D7,$D8);
$notxy = 1-$px-$py;
$Ans=decimal(4,1-(1-$px)^4-(1-$py)^4+$notxy^4);
$Alt1=decimal(4,$Ans+range(0.5,0.9,0.01)*(1-$Ans));
$Alt2=decimal(4,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(4,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));
$x=range(1,8,1);
$y=range(1,8,1);
condition:not(eq($x,$y));@
qu.3.9.uid=65f0ac9b-3deb-4702-bcb3-73e37c4c61e9@
qu.3.9.info=  Type=MC;
  Course=230;
@
qu.3.9.question=<div title="UW Statistics Bank/Discrete Probability Models/Other Models/Q$Q">An 8-sided die is loaded to give the following probabilities:<br />
<br />
<table cellspacing="2" cellpadding="2" bordercolor="#111111" border="1" style="border-collapse: collapse;">
    <tbody>
        <tr>
            <td>&nbsp;number</td>
            <td align="center">1</td>
            <td align="center">2</td>
            <td align="center">3</td>
            <td align="center">4</td>
            <td align="center">5</td>
            <td align="center">6</td>
            <td align="center">7</td>
            <td align="center">8</td>
        </tr>
        <tr>
            <td>probability</td>
            <td align="right">$D1</td>
            <td align="right">$D2</td>
            <td align="right">$D3</td>
            <td align="right">$D4</td>
            <td align="right">$D5</td>
            <td align="right">$D6</td>
            <td align="right">$D7</td>
            <td align="right">$D8</td>
        </tr>
    </tbody>
</table>
<br />
The die is rolled (independently) 4 times. Find the probability that both $x and $y occur.</div>@
qu.3.9.answer=1@
qu.3.9.choice.1=$Ans@
qu.3.9.choice.2=$Alt1@
qu.3.9.choice.3=$Alt2@
qu.3.9.choice.4=$Alt3@
qu.3.9.fixed=@

qu.3.10.mode=Multiple Choice@
qu.3.10.name=03. Largest f(x) ?@
qu.3.10.comment=<p>Let the four consecutive odd numbers be 2n+1, 2n+3, 2n+5 and 2n+7 for some n.</p>
<p>The corresponding probabilities would be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>k</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>k</mi></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>k</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>1</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>3</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>5</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>7</mn></mrow></mstyle></math>.</p>
<p>To find n recall that the sum of all probabilities must add to 1, so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>3</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>5</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>7</mn></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></mstyle></math>,</p>
<p>8n + 16 = $n,</p>
<p>n = $k.</p>
<p>Thus the highest value of f(x) is f(2($k)+7) =&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi mathvariant='normal'>$x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac></mrow></mstyle></math></p>@
qu.3.10.editing=useHTML@
qu.3.10.hint.1="four consecutive odd numbers"&nbsp; - for example 3,5,7,9 or 211,213,215,217.@
qu.3.10.hint.2=Any odd number can be written as 2n+1 where n is an integer.@
qu.3.10.hint.3=Thus the four consecutive odds can be written as 2n+1, 2n+3, 2n+5 and 2n+7 for some (unknown) n.@
qu.3.10.solution=@
qu.3.10.algorithm=$Q=3;
$n=range(16,160,8);
$k=($n-16)/8;
$x=2*$k+7;
$AnsTop=$x;
$Alt1Top=int($AnsTop/2);
$Alt2Top=$AnsTop+range(1,15,1);
$Alt3Top=int(0.5*($AnsTop+switch(rint(2),$Alt1Top,$Alt2Top)));@
qu.3.10.uid=0f8181ad-fe8b-43bc-a2ec-9a5bd4245b41@
qu.3.10.info=  Course=230;
  Type=MC;
@
qu.3.10.question=<div title="UW Statistics Bank/Discrete Probability Models/Other Models/Q$Q">Let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac></mrow></mrow></mstyle></math>represent a (discrete) probability function. If the corresponding random variable X assumes only 4 consecutive <strong>odd</strong> numbers, what is the largest value of f(x)?</div>@
qu.3.10.answer=1@
qu.3.10.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi mathvariant='normal'>$AnsTop</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac></mrow></mstyle></math>@
qu.3.10.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi mathvariant='normal'>$Alt1Top</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac></mrow></mstyle></math>@
qu.3.10.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi mathvariant='normal'>$Alt2Top</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac></mrow></mstyle></math>@
qu.3.10.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi mathvariant='normal'>$Alt3Top</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac></mrow></mstyle></math>@
qu.3.10.fixed=4@

qu.3.11.question=<div title="UW Statistics Bank/Discrete Probability Models/Other Models/Q$Q">A random variable X takes values on {1,2,3,4,&hellip;.} and has the property that P(X &ge; x) = $p P(X &ge; x-1) for all x=2,3,4,&hellip;.. Then P(X=$n)=__________? (4 decimal accuracy)</div>@
qu.3.11.answer.num=$Ans@
qu.3.11.answer.units=@
qu.3.11.showUnits=false@
qu.3.11.grading=toler_abs@
qu.3.11.err=.001@
qu.3.11.negStyle=minus@
qu.3.11.numStyle=thousands scientific dollars arithmetic@
qu.3.11.mode=Numeric@
qu.3.11.name=13. P(X ≥ x) = kP(X≥ x-1),  find P(X=#)@
qu.3.11.comment=<p>You are given P(X &ge;&nbsp; <em>x</em>) = <em>$p</em>P(X &ge; <em>x</em> - 1) and want to find P(X = <em>$n</em>)</p>
<p>P(<em>X</em> = $n) <br />
= P(<em>X</em> &ge; $n) - P(<em>X</em> &ge; $n+1) <br />
= P(<em>X</em> &ge; $n) - <em>$p</em>P(<em>X</em> &ge; $n) <br />
= (1 - $p)P(<em>X</em> &ge; $n)<br />
= (1 - $p)($p P(<em>X</em> &ge; $n-1)) <br />
= (1 - $p)($p<sup>2</sup>P(<em>X</em> &ge; $n-2)) = ... <br />
= (1 - $p)($p<em><sup>(</sup></em><sup>$n-1</sup><em><sup>)</sup></em>P(<em>X</em> &ge; 1))&nbsp;<br />
=$Ans</p>
<p>&nbsp;</p>@
qu.3.11.editing=useHTML@
qu.3.11.solution=@
qu.3.11.algorithm=$Q="13";
$p=decimal(1,range(0.1,0.6,0.1));
$n=range(3,10,1);
$Ans=decimal(4,(1-$p)*$p^($n-1));@
qu.3.11.uid=b5827edf-edf7-4568-b25a-13b38cdda142@
qu.3.11.info=  Type=numeric;
  Course=230;
@

qu.3.12.mode=Inline@
qu.3.12.name=05. Find missing probability .@
qu.3.12.comment=<p>Just add up all the other probabilities and subtract from 1,</p>
<p><font size="3" face="Times New Roman">1 - $x1 - $x2 - $x3 - $x4 - $x5 = $x6</font> .</p>@
qu.3.12.editing=useHTML@
qu.3.12.solution=@
qu.3.12.algorithm=$Q = 5;
$x1=decimal(2,rand(0,0.15));
$x2=decimal(2,rand(0,0.2));
$x3=decimal(2,rand(0,0.3));
$x4=decimal(2,rand(0,0.25));
$x5=decimal(2,rand(0,0.1));
$x6=1-$x1-$x2-$x3-$x4-$x5;@
qu.3.12.uid=fdeaf148-d8e1-4c47-bb9f-af7af15bccf0@
qu.3.12.info=  Course=202;
  Course=230;
  Difficulty=1;
  Type=numeric;
@
qu.3.12.weighting=1@
qu.3.12.numbering=alpha@
qu.3.12.part.1.name=sro_id_1@
qu.3.12.part.1.answer.units=@
qu.3.12.part.1.numStyle=thousands scientific  arithmetic@
qu.3.12.part.1.editing=useHTML@
qu.3.12.part.1.showUnits=false@
qu.3.12.part.1.err=0.01@
qu.3.12.part.1.question=(Unset)@
qu.3.12.part.1.mode=Numeric@
qu.3.12.part.1.grading=toler_abs@
qu.3.12.part.1.negStyle=minus@
qu.3.12.part.1.answer.num=$x6@
qu.3.12.question=<div title="UW Statistics Bank/Discrete Probability Models/Other Models/Q$Q">Shown below is a part of probability distribution for the ABC Company's projected profits (in $1000s) for the first year of operation (the negative value shows a loss). <br /><br /><div align="center"><table width="200" cellspacing="1" cellpadding="1" border="1">    <tbody>        <tr>            <td align="center">x</td>            <td align="center">f(x)</td>        </tr>        <tr>            <td align="center">            <p>-100</p>            </td>            <td align="center">$x1</td>        </tr>        <tr>            <td align="center">0</td>            <td align="center">$x2</td>        </tr>        <tr>            <td align="center">50</td>            <td align="center">$x3</td>        </tr>        <tr>            <td align="center">100</td>            <td align="center">$x4</td>        </tr>        <tr>            <td align="center">150</td>            <td align="center">$x5</td>        </tr>        <tr>            <td align="center">200</td>            <td align="center">?</td>        </tr>    </tbody></table></div>What is the value of f(200)?<p>&nbsp;</p><p>Ans:&nbsp;<span>&nbsp;</span><1><span>&nbsp;</span> <em>(3 decimal accuracy)</em></p></div>@

qu.3.13.question=<div title="UW Statistics Bank/Discrete Probability Models/Other Models/Q$Q">Suppose that f(x) = cx, x = 1,2,...,$M  is a pf of a discrete random variable X. Find P(X > $L). (3 decimal accuracy.)</div>@
qu.3.13.answer.num=$Ans@
qu.3.13.answer.units=@
qu.3.13.showUnits=false@
qu.3.13.grading=toler_abs@
qu.3.13.err=.01@
qu.3.13.negStyle=minus@
qu.3.13.numStyle=thousands scientific dollars arithmetic@
qu.3.13.mode=Numeric@
qu.3.13.name=02a. Discrete pdf, P(X > a)@
qu.3.13.comment=<p>First find "c" from the pf definition:<br />
The sum of all the probabilities must be 1, so c + 2c + ... + ($M)c = 1.&nbsp; Use the summation formula:<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math><br />
here this means: <br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mi>c</mi><mi mathvariant='normal'>$M</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$Mp1</mi></mrow></mfenced></mrow><mn>2</mn></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi mathvariant='normal'>so</mi><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>c</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>2</mn><mrow><mi mathvariant='normal'>$M</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$Mp1</mi></mrow></mfenced></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mi mathvariant='normal'>$PreC</mi></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$C</mi></mrow></mstyle></math><br />
P(X > $L) <br />
= 1 - P(X &le; $L)<br />
= 1 - (c + 2c + ... + ($L)c)<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&equals;</mo><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='mediummathspace' rspace='mediummathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&InvisibleTimes;</mo><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>1</mn><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='mediummathspace' rspace='mediummathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&InvisibleTimes;</mo><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='mediummathspace' rspace='mediummathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&#8722;</mo><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='mediummathspace' rspace='mediummathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&InvisibleTimes;</mo><mfrac linethickness='1' denomalign='center' numalign='center' bevelled='false'><mrow><mfenced><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$C</mi></mrow></mfenced><mfenced><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$L</mi></mrow></mfenced><mfenced><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$L</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='mediummathspace' rspace='mediummathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&plus;</mo><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>1</mn></mrow></mfenced></mrow><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>2</mn></mfrac><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&equals;</mo><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$Ans</mi></mrow></math></p>@
qu.3.13.editing=useHTML@
qu.3.13.solution=@
qu.3.13.algorithm=$Q="02a";
$M=range(8,14,2);
$Mp1=$M+1;
$L=range(2,($M)-1,1);
$PreC=($M)*(1+$M)/2;
$C = 1/$PreC;
$Ans=decimal(3,1 - ($C)($L)(($L)+1)/2);@
qu.3.13.uid=279724b8-92bb-4887-92a5-d443d6abfaa5@
qu.3.13.info=  Type=numeric;
  Difficulty=3;
  Course=230;
@

qu.3.14.question=<div title="UW Statistics Bank/Discrete Probability Models/Other Models/Q$Q">Consider tossing two fair 6-sided dice. Define M to be the maximum of the throws on the dice and let m be the minimum.
<p>Then  P(M &le; $m) + P(m &le; $m) =</p>
<p>(3 decimals of accuracy.)</p>
</div>@
qu.3.14.answer.num=$Ans@
qu.3.14.answer.units=@
qu.3.14.showUnits=false@
qu.3.14.grading=toler_abs@
qu.3.14.err=.01@
qu.3.14.negStyle=minus@
qu.3.14.numStyle=thousands scientific dollars arithmetic@
qu.3.14.mode=Numeric@
qu.3.14.name=07. P(Max dice&le;m) + P(Min dice&le;m)@
qu.3.14.comment=<p>Note that the two events are independent. Also for any fair die: P(die &le; x) = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>x</mi><mrow><mn>6</mn></mrow></mfrac></mrow></mstyle></math>for <font size="3" face="Times New Roman"><em>x</em> = 1,..,6</font> .</p>
<p>Then P(M &le; $m) =&nbsp; P(both rolls &le; $m) = P(1st roll &le; $m)P(2nd roll &le; $m) = ($mOVER6ML)($mOVER6ML) = ($mOVER6ML)<sup>2</sup> = $mOVER362ML.</p>
<p>Now P(m &le; $m) = P((1st roll &le; $m) OR (2nd roll &le; $m)) = P(1st roll &le; $m) + P(2nd roll &le; $m) - P(both rolls &le; $m) = $mOVER6ML + $mOVER6ML - ($mOVER6ML)<sup>2</sup> = $MinLTmML</p>
<p>So P(M &le; $m) + P(m &le; $m) = $mOVER362ML+ $MinLTmML = $AnsML .</p>
<p align="center"><em><font size="2"><br />
</font></em></p>@
qu.3.14.editing=useHTML@
qu.3.14.solution=@
qu.3.14.algorithm=$Q=7;
$m=range(1,6,1);
$m2=$m^2;
$mOVER6ML=mathml("$m/6");
$mOVER362ML=mathml("$m2/36");
$MinLTmT=12*$m-$m^2;
$MinLTmML=mathml("$MinLTmT/36");
$Ans=decimal(4,2*$m/6);
$AnsT=2*$m;
$AnsML=mathml("$AnsT/6");@
qu.3.14.uid=4622d015-3f67-4c4f-98ac-43dfa241fc07@
qu.3.14.info=  Course=230;
  Type=numeric;
@

qu.3.15.mode=Multiple Choice@
qu.3.15.name=10. P(no x in 8-sided rolls)@
qu.3.15.comment=<div style="margin-top: 0px; margin-bottom: 2px;" class="shadedDiv descriptionSpan">Notice that for one roll, P(not $x) = 1 - P(is a $x) = 1 -&nbsp;$px = $npx<br />
So for four rolls:<br />
P(no $x in 4 rolls) = ($npx)<sup>4</sup> = $Ans</div>@
qu.3.15.editing=useHTML@
qu.3.15.solution=@
qu.3.15.algorithm=$Q="10";
$D7=switch(rint(2),0.05,0.10);
$D8=0.20-$D7;
$D1=switch(rint(3),0.05,0.1,0.2);
$D2=0.3-$D1;
$p=[$D1, $D2, 0.15, 0.2, 0.1, 0.05, $D7, $D8 ];
$x=range(1,8,1);
$px=switch($x-1,$p);
$npx=1-$px;
$Ans=decimal(4,$npx^4);
$Alt1=decimal(4,range(0.35,0.65,0.01)*$Ans);
$Alt2=decimal(4,$Ans+range(0.2,0.55,0.01)*(1-$Ans));
$Alt3=decimal(4,($Ans+switch(rint(2),$Alt1,$Alt2))/2);@
qu.3.15.uid=3139c417-de52-4e35-ae99-471f75e3548d@
qu.3.15.info=  Type=MC;
  Course=230;
@
qu.3.15.question=<div title="UW Statistics Bank/Discrete Probability Models/Other Models/Q$Q">An 8-sided die is loaded to give the following probabilities:<br />
<br />
<table cellspacing="2" cellpadding="2" bordercolor="#111111" border="1" style="border-collapse: collapse;">
    <tbody>
        <tr>
            <td>&nbsp;number</td>
            <td align="center">1</td>
            <td align="center">2</td>
            <td align="center">3</td>
            <td align="center">4</td>
            <td align="center">5</td>
            <td align="center">6</td>
            <td align="center">7</td>
            <td align="center">8</td>
        </tr>
        <tr>
            <td>probability</td>
            <td align="right">$D1</td>
            <td align="right">$D2</td>
            <td align="right">.15</td>
            <td align="right">.2</td>
            <td align="right">.1</td>
            <td align="right">.05</td>
            <td align="right">$D7</td>
            <td align="right">$D8</td>
        </tr>
    </tbody>
</table>
<br />
<div title="Stat230/Chapter6/Other Distributions/Q9">The die is rolled (independently) 4 times. Find the probability that&nbsp;$x does not occur.</div>
</div>@
qu.3.15.answer=1@
qu.3.15.choice.1=$Ans@
qu.3.15.choice.2=$Alt1@
qu.3.15.choice.3=$Alt2@
qu.3.15.choice.4=$Alt3@
qu.3.15.choice.5=None of the above@
qu.3.15.fixed=4@

qu.3.16.question=<div title="UW Statistics Bank/Discrete Probability Models/Other Models/Q$Q"><img hspace="4" align="$Align" src="__BASE_URI__DPM/OtherModels/Music$Which.gif" alt="Soothing music" title="Soothing music [IMG:Music$Which.gif]" />A psychologist studied the number of puzzles subjects were able to solve in a 5 minute period while listening to soothing music. Let X be the number of puzzles completed successfully by a subject. The psychologist found that X had the following probability distribution:</div>
<div align="center">
<table cellspacing="2" cellpadding="2" border="1">
    <tbody>
        <tr>
            <td>x</td>
            <td>1</td>
            <td>2</td>
            <td>3</td>
            <td>4</td>
        </tr>
        <tr>
            <td>P(X=x)</td>
            <td>$p1</td>
            <td>$p2</td>
            <td>$p3</td>
            <td>$p4</td>
        </tr>
    </tbody>
</table>
</div>
<p>Using the above data, what is the probability that a randomly chosen subject completes less than $n puzzles in the 5 minute period while listening to soothing music? (4 decimal accuracy)</p>@
qu.3.16.answer.num=$Ans@
qu.3.16.answer.units=@
qu.3.16.showUnits=false@
qu.3.16.grading=toler_abs@
qu.3.16.err=.001@
qu.3.16.negStyle=minus@
qu.3.16.numStyle=thousands scientific dollars arithmetic@
qu.3.16.mode=Numeric@
qu.3.16.name=15. P( < n successes)@
qu.3.16.comment=<div style="margin-top: 0px; margin-bottom: 2px;" class="shadedDiv descriptionSpan">P(x < $n) = P(x = 1)&nbsp; $Explain = $p1 $Explain2&nbsp; = $Ans</div>@
qu.3.16.editing=useHTML@
qu.3.16.solution=@
qu.3.16.algorithm=$Q=15;
$p1=decimal(2,range(0.05,0.35,0.05));
$p2=decimal(2,range(0.10,0.30,0.01));
$p3=decimal(2,range(0.1,0.3,0.1));
$p4=1-$p1-$p2-$p3;
$n=range(2,4,1);
$Ans=$p1+switch($n-2,0,$p2,$p2+$p3);
$Explain=switch($n-2,""," + P(x = 2)"," + P(x = 2) + P(x = 3)");
$Explain2=switch($n-2,""," + $p2"," + $p2 + $p3");
$Which=rint(4);
$Align=switch(rint(2),"Left","Right");@
qu.3.16.uid=7e4ead63-1b34-4c9c-a026-3480f543d78d@
qu.3.16.info=  Course=230;
  Type=numeric;
@

qu.3.17.mode=Multiple Choice@
qu.3.17.name=08. P(Two good then a bad set of brakes)@
qu.3.17.comment=<p>At first it may be tempting to use Hypergeometric since we have N = $t, r =&nbsp;$g successes (good brakes) and N - r =&nbsp;$f failures (bad brakes). We are selecting n = 3 objects without replacement. If X is the number of successes obtained, you may think we want P(X = 2). The problem is that the hypergeometric model does not address the issue of order.</p>
<p>Alternately, reason that the probability of drawing a good truck first is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi mathvariant='normal'>$g</mi><mrow><mi mathvariant='normal'>$t</mi></mrow></mfrac></mrow></mstyle></math>, then drawing another is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mi mathvariant='normal'>$g</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$t</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfrac></mrow></mstyle></math> and finally drawing a bad truck is then <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi mathvariant='normal'>$f</mi><mrow><mi mathvariant='normal'>$t</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn></mrow></mfrac></mrow></mstyle></math>. Multiply these together to get $ans and round up to $ANSWER.</p>@
qu.3.17.editing=useHTML@
qu.3.17.solution=@
qu.3.17.algorithm=$Q=3;
$t=range(10,35,1);
$rs=3;
$f=range(3,5,1);
$g=$t-$f;
$ans=($g/$t) * ($g-1)/($t-1) * $f/($t-2);
$ANSWER=decimal(2,$ans);
$Alt1=decimal(2,$g/$t);
$Alt2=decimal(2,($ANSWER+$Alt1)/2);
$Alt3=decimal(2, range(0.55,0.80,0.05)*$ANSWER);@
qu.3.17.uid=cfbb941d-e184-4477-85ec-bbb82b722010@
qu.3.17.info=  Course=230;
  Type=MC;
@
qu.3.17.question=<div title="UW Statistics Bank/Discrete Probability Models/Other Models/Q$Q">A delivery company has&nbsp;$t trucks,&nbsp;$f of which have faulty brakes. If three of the&nbsp;$t trucks are randomly selected, what is the approximate probability that the first two trucks have good brakes, and the third truck has faulty brakes?</div>@
qu.3.17.answer=1@
qu.3.17.choice.1=$ANSWER@
qu.3.17.choice.2=$Alt1@
qu.3.17.choice.3=$Alt2@
qu.3.17.choice.4=$Alt3@
qu.3.17.fixed=4@

qu.3.18.question=<div title="UW Statistics Bank/Discrete Probability Models/Other Models/Q$Q"><img hspace="4" align="$Align" src="__BASE_URI__DPM/OtherModels/Coin$Which.gif" alt="A coin" title="A coin [IMG:Coin$Which.gif]" />Suppose that 2 independent tosses of a coin having probability p = $pML of coming up heads are made. Then the probability of an even number of heads is (Please answer to 3 decimals of accuracy OR express your answer as a reduced fraction .)  NOTE: 0 is an even number!</div>@
qu.3.18.answer.num=$Ans@
qu.3.18.answer.units=@
qu.3.18.showUnits=false@
qu.3.18.grading=toler_abs@
qu.3.18.err=.01@
qu.3.18.negStyle=minus@
qu.3.18.numStyle=thousands scientific dollars arithmetic@
qu.3.18.mode=Numeric@
qu.3.18.name=09. P(even # heads)@
qu.3.18.comment=<p><em>p</em> is the probability of tossing a head. There are two outcomes that give us an even number of heads:</p>
<ol>
    <li>No heads - occurs with probability <em>(1-p)<sup>2</sup></em></li>
    <li>Two heads - occurs with probability <em>p<sup>2</sup></em></li>
</ol>
<p>So P(even number of heads) = <em>(1-p)<sup>2</sup> + p<sup>2</sup></em> = (1-$p)<sup>2</sup> +$p<sup>2</sup> = $Ans</p>@
qu.3.18.editing=useHTML@
qu.3.18.solution=@
qu.3.18.algorithm=$Q=9;
$a=range(2,6,1);
$pML=mathml("1/$a");
$p=decimal(2,1/$a);
$Ans=decimal(3,(1-$p)^2+$p^2);
$Which=rint(4);
$Align=switch(rint(2),"Left","Right");@
qu.3.18.uid=b57dcab1-b5db-4629-af06-5b63dfabf271@
qu.3.18.info=  Course=230;
  Type=numeric;
@

qu.3.19.question=<div title="UW Statistics Bank/Discrete Probability Models/Other Models/Q$Q"><img hspace="4" align="$Align" src="__BASE_URI__DPM/OtherModels/Music$Which.gif" alt="Soothing music" title="Soothing music [IMG:Music$Which.gif]" />A psychologist studied the number of puzzles subjects were able to solve in a 5 minute period while listening to soothing music. Let X be the number of puzzles completed successfully by a subject. The psychologist found that X had the following probability distribution:</div>
<div align="center">
<table cellspacing="2" cellpadding="2" border="1">
    <tbody>
        <tr>
            <td>x</td>
            <td>1</td>
            <td>2</td>
            <td>3</td>
            <td>4</td>
        </tr>
        <tr>
            <td>P(X=x)</td>
            <td>$p1</td>
            <td>$p2</td>
            <td>$p3</td>
            <td>$p4</td>
        </tr>
    </tbody>
</table>
</div>
<p>Using the above data, what is the probability that a randomly chosen subject completes at least $n puzzles in the 5 minute period while listening to soothing music? (4 decimal accuracy)</p>@
qu.3.19.answer.num=$Ans@
qu.3.19.answer.units=@
qu.3.19.showUnits=false@
qu.3.19.grading=toler_abs@
qu.3.19.err=.001@
qu.3.19.negStyle=minus@
qu.3.19.numStyle=thousands scientific dollars arithmetic@
qu.3.19.mode=Numeric@
qu.3.19.name=14. P( ≥ n successes)@
qu.3.19.comment=<div class="shadedDiv descriptionSpan" style="margin-top: 0px; margin-bottom: 2px;">P(x &ge; $n) = $Explain P(x = 4) = $Explain2 $p4 = $Ans</div>@
qu.3.19.editing=useHTML@
qu.3.19.solution=@
qu.3.19.algorithm=$Q=14;
$p1=decimal(2,range(0.05,0.35,0.05));
$p2=decimal(2,range(0.10,0.30,0.01));
$p3=decimal(2,range(0.1,0.3,0.1));
$p4=1-$p1-$p2-$p3;
$n=range(1,4,1);
$Ans=$p4+switch($n,0,$p1+$p2+$p3,$p2+$p3,$p3,0);
$Explain=switch($n,"Error","P(x = 1) + P(x = 2) + P(x = 3) + ","P(x = 2) + P(x = 3) + ","P(x = 3) + ","");
$Explain2=switch($n,"Error","$p1 + $p2 + $p3 + ","$p2 + $p3 + ","$p3 + ","");
$Which=rint(4);
$Align=switch(rint(2),"Left","Right");@
qu.3.19.uid=f1c7b010-ce39-4647-aed5-2fcfd85bd395@
qu.3.19.info=  Course=230;
  Type=numeric;
@

qu.3.20.mode=Multiple Choice@
qu.3.20.name=04. Find  value of c in probability function@
qu.3.20.comment=<div class="shadedDiv descriptionSpan" style="margin-top: 0px; margin-bottom: 2px;">To find <font size="3" face="Times New Roman"><em>c</em></font> use the fact that the probabilities must add up to 1:<br />
<br />
<font size="3" face="Times New Roman"><em>c</em></font>(<font size="3" face="Times New Roman">$k<sup>3</sup>&nbsp; + $kp1<sup>3</sup> + $kp2<sup>3</sup> + $kp3<sup>3</sup> + $kp4<sup>3</sup> = 1,</font>&nbsp; <font size="3" face="Times New Roman"><em>c</em>($Sum) = 1</font>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>c</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$Sum</mi></mrow></mfrac></mrow></mrow></mstyle></math></div>@
qu.3.20.editing=useHTML@
qu.3.20.solution=@
qu.3.20.algorithm=$Q=4;
$k=range(0,3,1);
$kp1=$k+1;
$kp2=$k+2;
$kp3=$k+3;
$kp4=$k+4;
$Sum=$k^3+$kp1^3+$kp2^3+$kp3^3+$kp4^3;
$AnsML=mathml("1/$Sum");
$Alt1Sum=$k^2+$kp1^2+$kp2^2+$kp3^2+$kp4^2;
$Alt1ML=mathml("1/$Alt1Sum");
$Alt2Sum=$k+$kp1+$kp2+$kp3+$kp4;
$Alt2ML=mathml("1/$Alt2Sum");
$Alt3Sum=int(range(1.2,1.5,0.05)*$Sum);
$Alt3ML=mathml("1/$Alt3Sum");
$Alt4ML=switch(rint(2),mathml("$Alt2Sum/$Sum"),mathml("$Alt2Sum/$Alt3Sum"));@
qu.3.20.uid=3cc8f84d-a343-4909-8071-04a5f459b358@
qu.3.20.info=  Course=230;
  Type=MC;
@
qu.3.20.question=<div title="UW Statistics Bank/Discrete Probability Models/Other Models/Q$Q">Let <em><font size="3" face="Times New Roman">X </font></em>represent a discrete random variable whose probability function is given by:<br />
<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>f</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mfrac linethickness='0'><mrow><msup><mi>cx</mi><mrow><mn>3</mn></mrow></msup></mrow><mrow><mn>0</mn></mrow></mfrac><mrow><mfrac linethickness='0'><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$k</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$kp1</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$kp2</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$kp3</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$kp4</mi></mrow><mrow><mi>otherwise</mi></mrow></mfrac></mrow></mrow></mrow></mstyle></math><br />
<br />
What is the value of <font size="3" face="Times New Roman"><em>c</em></font>?</div>@
qu.3.20.answer=1@
qu.3.20.choice.1=$AnsML@
qu.3.20.choice.2=$Alt1ML@
qu.3.20.choice.3=$Alt2ML@
qu.3.20.choice.4=$Alt3ML@
qu.3.20.choice.5=$Alt4ML@
qu.3.20.fixed=4@

qu.4.topic=Hypergeometric Distribution@

qu.4.1.question=<div title="UW Statistics Bank/Discrete Probability Models/Hypergeomteric/Q$Q">At a stage of a jury selection, six jurors are to be selected out of the remaining 15 prospective jurors, five of whom are women. Let X be the number of women finally selected. <br />
<br />
Find P(X = $n) . Answer to 4 decimal accuracy. Do NOT use any approximations here.</div>@
qu.4.1.answer.num=$Ans@
qu.4.1.answer.units=@
qu.4.1.showUnits=false@
qu.4.1.grading=toler_abs@
qu.4.1.err=.001@
qu.4.1.negStyle=minus@
qu.4.1.numStyle=thousands scientific dollars arithmetic@
qu.4.1.mode=Numeric@
qu.4.1.name=03. Jury selection - Find P(X = n)@
qu.4.1.comment=<p>There are n = 5 women and m = 10 men among the 15 prospective jurors, with r = 6 jurors to be chosen. We use combinations (the hypergeometric distribution) to derive the probability distribution of X = the number of women selected. The probability formula is:<br />
<br />
P(X = k) =<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>5</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi></mrow></mtd></mtr></mtable></mfenced></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>10</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>6</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>k</mi></mrow></mtd></mtr></mtable></mfenced></mrow></mfenced></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>15</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>6</mn></mrow></mtd></mtr></mtable></mfenced></mrow></mfenced></mrow></mfrac></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math> so P(X = $n) =&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>5</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi mathvariant='normal'>$n</mi></mrow></mtd></mtr></mtable></mfenced></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>10</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>6</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$n</mi></mrow></mtd></mtr></mtable></mfenced></mrow></mfenced></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>15</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>6</mn></mrow></mtd></mtr></mtable></mfenced></mrow></mfenced></mrow></mfrac></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math> = $Ans (rounded off).</p>@
qu.4.1.editing=useHTML@
qu.4.1.solution=@
qu.4.1.algorithm=$Q=3;
$n=range(0,5,1);
$PreAns=maple("with(Statistics);
X := RandomVariable(Hypergeometric(15, 5, 6));
ProbabilityFunction(X, $n)");
$Ans=decimal(4,$PreAns);@
qu.4.1.uid=807bdfe9-41ba-4b00-bd12-92c0b20d83d3@
qu.4.1.info=  Difficulty=3;
  Keyword=hypergeometric;
  Course=230;
  Type=numeric;
@

qu.4.2.question=<div title="UW Statistics Bank/Discrete Probability Models/Hypergeomteric/Q$Q"><img hspace="4" align="$Align" title="House [IMG:House$Which.gif]" alt="House" src="__BASE_URI__DPM/HyperGeometric/House$Which.gif" />$ch of the $N houses on my street have chimneys. If 3 houses on my street are chosen at random, what is the probability that two of them will have chimneys? (4 decimal accuracy)</div>@
qu.4.2.answer.num=$Ans@
qu.4.2.answer.units=@
qu.4.2.showUnits=false@
qu.4.2.grading=toler_abs@
qu.4.2.err=.001@
qu.4.2.negStyle=minus@
qu.4.2.numStyle=thousands scientific dollars arithmetic@
qu.4.2.mode=Numeric@
qu.4.2.name=02. Chimneys@
qu.4.2.comment=<div class="shadedDiv descriptionSpan" style="margin-top: 0px; margin-bottom: 2px;">This is an example of the <span style="font-weight: bold;">hypergeometric distribution<span style="font-weight: bold;">.</span></span> We have N =&nbsp;$N objects (the houses), call houses with chimneys type "S" (so r = $ch) and houses without chimneys type "F" (so N-r = $N-$ch).&nbsp; Pick 3 (n = 3) and we want the probability that x = 2: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mi>r</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>x</mi></mrow></mtd></mtr></mtable></mfenced></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mi>N</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>r</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>x</mi></mrow></mtd></mtr></mtable></mfenced></mrow></mfenced></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mi>N</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>n</mi></mrow></mtd></mtr></mtable></mfenced></mrow></mfenced></mrow></mfrac></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>&nbsp;&nbsp; so<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mn>2</mn></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mi mathvariant='normal'>$ch</mi></mrow></mtd></mtr><mtr><mtd><mrow><mn>2</mn></mrow></mtd></mtr></mtable></mfenced></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mi mathvariant='normal'>$N</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$ch</mi></mrow></mtd></mtr><mtr><mtd><mrow><mn>3</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn></mrow></mtd></mtr></mtable></mfenced></mrow></mfenced></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mi mathvariant='normal'>$N</mi></mrow></mtd></mtr><mtr><mtd><mrow><mn>3</mn></mrow></mtd></mtr></mtable></mfenced></mrow></mfenced></mrow></mfrac></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>= $PAns or&nbsp; $Ans.</div>@
qu.4.2.editing=useHTML@
qu.4.2.solution=@
qu.4.2.algorithm=$Q=2;
$N=range(15,40,1);
$Which=1+rint(5);
$Align=switch(rint(2),"Left","Right");
$ch=range(3,7,1);
$ans=maple("with(Statistics);
X := RandomVariable(Hypergeometric($N, $ch, 3));
ProbabilityFunction(X, 2)");
$PAns=mathml("$ans");
$Ans=decimal(3, $ans);@
qu.4.2.uid=b205523a-ce29-456e-8fbd-4722b098c8b6@
qu.4.2.info=  Difficulty=3;
  Keyword=hypergeometric;
  Course=230;
@

qu.4.3.mode=Inline@
qu.4.3.name=01. Picking chips@
qu.4.3.comment=<p>The sample space size is #Red + #$WhiteIs + #Blue chips = $r + $w + $b = $N&nbsp; (N).<br />
n = $n, the number selected.<br />
r = $r considering a red chip as a success.<br />
x = $x</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mi mathvariant='normal'>$r</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi mathvariant='normal'>$x</mi></mrow></mtd></mtr></mtable></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mi mathvariant='normal'>$N</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$r</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi mathvariant='normal'>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$x</mi></mrow></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mi mathvariant='normal'>$N</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi mathvariant='normal'>$n</mi></mrow></mtd></mtr></mtable></mrow></mfenced></mrow></mfrac></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$F1</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$F2</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'>$F3</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.4.3.editing=useHTML@
qu.4.3.solution=@
qu.4.3.algorithm=$Q = 1;
$r=range(4,7,1);
$w=range(5,9,1);
$b=range(3,6,1);
$WhiteIs=switch(rint(3),"white","yellow","green");
$N=$r+$w+$b;
$n=range(3,6);
$x=1;
$F1=fact($r)/(fact($x)*fact($r-$x));
$F2=fact($N-$r)/(fact($n-$x)*fact($N-$r-$n+$x));
$F3=fact($N)/(fact($n)*fact($N-$n));
$Ans=$F1*$F2/$F3;
$Which=rint(5);
$Align=switch(rint(2),"Left","Right");@
qu.4.3.uid=23463504-5291-48dc-bdc6-98e63b259dc5@
qu.4.3.info=  Course=230;
@
qu.4.3.weighting=1@
qu.4.3.numbering=alpha@
qu.4.3.part.1.name=sro_id_1@
qu.4.3.part.1.answer.units=@
qu.4.3.part.1.numStyle=thousands scientific  arithmetic@
qu.4.3.part.1.editing=useHTML@
qu.4.3.part.1.showUnits=false@
qu.4.3.part.1.err=0.02@
qu.4.3.part.1.question=(Unset)@
qu.4.3.part.1.mode=Numeric@
qu.4.3.part.1.grading=toler_abs@
qu.4.3.part.1.negStyle=minus@
qu.4.3.part.1.answer.num=$Ans@
qu.4.3.question=<div title="University of Waterloo Statistics Bank/Discrete Probability Models/Hypergeomteric/Q$Q"><img hspace="4" align="$Align" src="__BASE_URI__DPM/HyperGeometric/PokerChips$Which.gif" alt="Poker chips" title="Poker Chips [IMG:PokerChips$Which.gif]" />A bowl contains $N chips, of which $r are red, $w are $WhiteIs, and $b are blue. If $n chips are taken at random and without replacement, find the probability that there is only one red chip.<span> (please round to two decimal places) <span>&nbsp;</span><1><span>&nbsp;</span></span></div>@

qu.5.topic=Geometric Distribution@

qu.5.1.question=<div title="UW Statistics Bank/Discrete Probability Models/Geometrical Distributions/Q$Q"><img hspace="4" align="$Align" title="Paper bill(s) [IMG:Bill$Which.gif]" alt="Paper bill(s)" src="__BASE_URI__DPM/Geometric/Bill$Which.gif" />We inspect paper money  bills in order to find one whose serial number ends in a "$EndsIn". What is the probability that the  $n\\th billl we inspect is the first one whose serial number ends in "$EndsIn"? (Please answer to 4 decimals of accuracy.)</div>@
qu.5.1.answer.num=$Ans@
qu.5.1.answer.units=@
qu.5.1.showUnits=false@
qu.5.1.grading=toler_abs@
qu.5.1.err=.001@
qu.5.1.negStyle=minus@
qu.5.1.numStyle=thousands scientific dollars arithmetic@
qu.5.1.mode=Numeric@
qu.5.1.name=03. Dollar serial # ends in [0,9]@
qu.5.1.comment=<p>NOTE: You can use the <span style="font-style: italic;">geometric distribution</span> like I have here, or just use basic logic.<br />
<br />
We can assume that, on the average, one out of ten serial number ending digits is a "$EndsIn", so:</p>
<p>p = probability of success = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>10</mn></mrow></mfrac></mrow></mstyle></math></p>
<p>The probability of looking at&nbsp;$f bills before encountering a terminal "$EndsIn" is the probability of meeting the first "$EndsIn" on the $n\\th dollar. That probablity is P(X = $n) = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>p</mi><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>p</mi></mrow></mfenced><mrow><mi>r</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><mn>10</mn></mrow></mfrac><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mn>10</mn></mrow></mfrac></mrow></mfenced><mrow><mi mathvariant='normal'>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>
<hr style="width: 100%; height: 2px;" />
<p>Canadian bills include letters, although at the start of the serial number, not the end. Regardless, if you assumed there were 36 possible endings, then the analysis is similar:<br />
<br />
P(X = $n) = p(1 &minus; p)<sup>r&minus;1 </sup>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>36</mn></mrow></mfrac><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mn>36</mn></mrow></mfrac></mrow></mfenced><mrow><mi mathvariant='normal'>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><mn>36</mn></mrow></mfrac><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>35</mn><mrow><mn>36</mn></mrow></mfrac></mrow></mfenced><mrow><mi mathvariant='normal'>$nM1</mi></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$SAns</mi></mrow></mstyle></math></p>@
qu.5.1.editing=useHTML@
qu.5.1.solution=@
qu.5.1.algorithm=$Q=3;
$EndsIn=rint(9);
$n=range(10,25,1);
$nM1=$n-1;
$M=maple("with(Statistics):
X := RandomVariable(Geometric(0.1)):
X1:=ProbabilityFunction(X, $nM1):
Y := RandomVariable(Geometric(1/36)):
Y1:=ProbabilityFunction(Y, $nM1);
X1,Y1;
");
$Ans=decimal(4,switch(0,$M));
$SAns=decimal(4,switch(1,$M));
$Which=rint(4);
$Align=switch(rint(2),"Left","Right");@
qu.5.1.uid=db219ba1-25e8-4716-a72e-b2ceec22519d@
qu.5.1.info=  Course=230;
  Type=numeric;
@

qu.5.2.question=<div title="UW Statistics Bank/Discrete Probability Models/Geometrical Distributions/Q$Q"><img hspace="3" align="$Align" alt="Baseball Player" src="__BASE_URI__DPM/Geometric/Baseball$Which.gif" title="Baseball Player [IMG:Baseball$Which.gif]" />A baseball player has a $pp% chance of hitting the ball each time at bat, with successive times at bat being independent. Calculate the probability that he gets his first hit on his $n\\th time at bat. (Please answer to 3 decimals of accuracy.)</div>@
qu.5.2.answer.num=$Ans@
qu.5.2.answer.units=@
qu.5.2.showUnits=false@
qu.5.2.grading=toler_abs@
qu.5.2.err=.01@
qu.5.2.negStyle=minus@
qu.5.2.numStyle=thousands scientific dollars arithmetic@
qu.5.2.mode=Numeric@
qu.5.2.name=02. P(1st hit on $nth at bat)@
qu.5.2.comment=<div style="margin-top: 0px; margin-bottom: 2px;" class="shadedDiv descriptionSpan">This is an example of a <span style="font-weight: bold;">Geometric Distribution</span>, or the <span style="font-weight: bold;">Negative Binomial</span> if you want, or you can just use common sense. He needs to "not hit"&nbsp;$f times, then hit on the $n\\th. The probability of those&nbsp;$n events occurring in sequence is ($pnot)<sup>$f</sup>($p) = $Ans</div>@
qu.5.2.editing=useHTML@
qu.5.2.solution=@
qu.5.2.algorithm=$Q=2;
$Which=rint(4);
$Align=switch(rint(2),"Left","Right");
$p=decimal(1,range(0.2,0.6,0.1));
$pp=$p*100;
$pnot=1-$p;
$n=range(4,7,1);
$f=$n-1;
$ans=maple("with(Statistics);
X := RandomVariable(Geometric($p));
ProbabilityFunction(X, $f)");
$Ans=decimal(3,$ans);@
qu.5.2.uid=c947cd38-8d37-4a7a-bc6e-7ca2d15d3143@
qu.5.2.info=  Type=numeric;
  Course=230;
@

qu.5.3.question=<div title="UW Statistics Bank/Discrete Probability Models/Geometrical Distributions/Q$Q"><img hspace="4" align="$Align" title="A target [IMG:Target$Which.gif]" alt="A target" src="__BASE_URI__DPM/Geometric/Target$Which.gif" />If the probability of hitting a target is $p, find the probability it will be hit for the first time on the&nbsp;$n$Ending trial. Assume that when shooting at a target, each shot is an independent event.&nbsp; (Please answer to 3 decimals of accuracy.)</div>@
qu.5.3.answer.num=$Ans@
qu.5.3.answer.units=@
qu.5.3.showUnits=false@
qu.5.3.grading=toler_abs@
qu.5.3.err=.01@
qu.5.3.negStyle=minus@
qu.5.3.numStyle=thousands scientific dollars arithmetic@
qu.5.3.mode=Numeric@
qu.5.3.name=04. P(1st hit on nth trial)@
qu.5.3.comment=<p>You can do this by reason alone, or by using the <span style="font-style: italic;">geometric distribution<span style="font-style: italic;">.</span></span> If we let a success be a hit of the target, then p = probability of success on a given trial is $p. X is the number of the shot to record the first hit so the probability of the first hit on the third trial is P(X = $n) = p(1 &minus; p)<sup>r&minus;1</sup> = ($p)(1 &minus; $p)<sup>$n&minus;1</sup> =&nbsp; $Ans.<br />
<br />
Alternately reason that the probability of missing on your first&nbsp;$f shots is $pnot<sup>$f</sup> and then multiply that by the probability of hitting on your third shot ($p).</p>@
qu.5.3.editing=useHTML@
qu.5.3.solution=@
qu.5.3.algorithm=$Q=4;
$p=decimal(2,range(0.3,0.6,0.05));
$pnot=1-$p;
$n=range(3,9,1);
$Ending=if(eq($n,3),"rd","th");
$f=$n-1;
$ans=maple("with(Statistics);
X := RandomVariable(Geometric($p));
ProbabilityFunction(X, $f)");
$Ans=decimal(3, $ans);
$Which=rint(5);
$Align=switch(rint(2),"Left","Right");@
qu.5.3.uid=8bde45df-3545-4308-a262-6bfdcfae737c@
qu.5.3.info=  Course=230;
  Type=numeric;
@

qu.5.4.mode=Multiple Choice@
qu.5.4.name=06. P(1st on odd)/P(1st on even)@
qu.5.4.comment=<div>The physical setup is for a <span style="font-weight: bold;">geometric distribution</span>. Thus<font size="3" face="Times New Roman"> <em>f</em>(<em>x</em>) =&nbsp; (1 - <em>p</em>)<em><sup>x</sup>p</em></font>&nbsp; for <font size="3" face="Times New Roman"><em>x</em> = 0,1,2,....</font>&nbsp; For simplicity let <font size="3" face="Times New Roman"><em>m</em> = 1 - <em>p</em></font>.<br />
<font size="3" face="Times New Roman"><em>P</em>(<em>X</em> is odd) = <em>mp</em> + <em>m</em><sup>3</sup><em>p</em> + <em>m</em><sup>5</sup><em>p</em> + ... = <em>p</em>(<em>m</em> + <em>m</em><sup>3</sup> + <em>m</em><sup>5</sup> + ...) </font><br />
<p><font size="3" face="Times New Roman"><em>P</em>(<em>X</em> is even) = <em>p</em> + <em>m</em><sup>2</sup><em>p</em> + <em>m</em><sup>4</sup><em>p</em> + ... = <em>p</em>(1 + <em>m</em><sup>2</sup> + <em>m</em><sup>4</sup> + ...)</font></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>p</mi><mrow><mi>m</mi></mrow></mfrac><mfenced open='(' close=')' separators=','><mrow><mi>m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>m</mi><mrow><mn>3</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>m</mi><mrow><mn>5</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mfenced></mrow></mstyle></math></p>
<p>So <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>is</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>odd</mi></mrow></mfenced></mrow><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>is</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>even</mi></mrow></mfenced></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>m</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>p</mi></mrow></mstyle></math></p>
</div>@
qu.5.4.editing=useHTML@
qu.5.4.solution=@
qu.5.4.algorithm=$Q=6;@
qu.5.4.uid=82a3bb28-903c-464d-85d7-7771f78c7327@
qu.5.4.info=  Course=230;
  Type=MC;
  Algorithmic=no;
@
qu.5.4.question=<div title="UW Statistics Bank/Discrete Probability Models/Geometrical Distributions/Q$Q">An experiment is repeated independently until exactly one success is obtained with the probability of success on each trial being <em>p</em>. Suppose X is the number of failures before obtaining a single success.&nbsp;
<p>Then <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>is odd</mi></mrow></mfenced></mrow><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>is even</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>is:</p>
</div>@
qu.5.4.answer=1@
qu.5.4.choice.1=1 - <i>p</i>@
qu.5.4.choice.2=1@
qu.5.4.choice.3=<i>p</i>@
qu.5.4.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>p</mi></mrow></mfrac></mrow></mstyle></math>@
qu.5.4.choice.5=None of the above@
qu.5.4.fixed=4@

qu.5.5.question=<div title="UW Statistics Bank/Discrete Probability Models/Geometrical Distributions/Q$Q"><img hspace="4" align="$Align" title="A lizard [IMG:Lizard$Which.gif]" alt="A Lizard" src="__BASE_URI__DPM/Geometric/Lizard$Which.gif" />For a particular type of lizard, the probability that a given egg will hatch out a female lizard is <em>p</em> = $p If a nest of eggs is hatching sequentially, what is the probability that the first female to hatch will be egg number $n? Please answer to 4 decimals of accuracy.)</div>@
qu.5.5.answer.num=$Ans@
qu.5.5.answer.units=@
qu.5.5.showUnits=false@
qu.5.5.grading=toler_abs@
qu.5.5.err=.001@
qu.5.5.negStyle=minus@
qu.5.5.numStyle=thousands scientific dollars arithmetic@
qu.5.5.mode=Numeric@
qu.5.5.name=05. P(1st female lizard is #n)@
qu.5.5.comment=<p>This is just a <span style="font-weight: bold;">geometric distribution</span> - success is a female hatching, <span style="font-style: italic;">p</span> = P(female hatches) and <span style="font-style: italic;">k</span> is how many eggs hatch before the first female is seen. Notice that <em>k</em> is the number of the egg the first female hatches from, since she is not seen  until her egg has hatched.<br />
<br />
The probability that the first female will appear when the <span style="font-style: italic;">k</span>th eggs hatches then is :</p>
<p>(P(male))<em><sup>k-1</sup></em>P(female). <br />
= (1 - P(female))<sup><em>k-1</em></sup>P(female)<br />
=$q<sup>$n-1</sup>$p = $Ans</p>@
qu.5.5.editing=useHTML@
qu.5.5.solution=@
qu.5.5.algorithm=$Q=5;
$Which=rint(5);
$Align=switch(rint(2),"Left","Right");
$p=range(0.3000,0.7000,0.01);
$q=1-$p;
$n=range(1,7,1);
$Ans=decimal(4,$q^($n-1)*$p);@
qu.5.5.uid=868b28dd-b6fc-48d5-bcb1-99f047d200f0@
qu.5.5.info=  Course=230;
  Type=numeric;
@

qu.5.6.question=<div title="UW Statistics Bank/Discrete Probability Models/Geometrical Distributions/Q$Q"><img hspace="4" align="$Align" title="Blood donors [IMG:Blood$Which.gif]" alt="Blood donors" src="__BASE_URI__DPM/Geometric/Blood$Which.gif" />$pp% of people donating blood at a clinic have O<sup>+</sup> type blood. Find the probability that the first O<sup>+</sup> donor is the $Abbrev donor of the day. (Please answer to 4 decimals of accuracy )</div>@
qu.5.6.answer.num=$Ans@
qu.5.6.answer.units=@
qu.5.6.showUnits=false@
qu.5.6.grading=toler_abs@
qu.5.6.err=.001@
qu.5.6.negStyle=minus@
qu.5.6.numStyle=thousands scientific dollars arithmetic@
qu.5.6.mode=Numeric@
qu.5.6.name=01. Blood Donors@
qu.5.6.comment=<div class="shadedDiv descriptionSpan" style="margin-top: 0px; margin-bottom: 2px;">Define "success" here as a donor having O<sup>+</sup> type blood. Define a r.v. X to be the number of failures <u>before</u> the first success. Then X ~ G($p), that is X is Geometric with <span style="font-style: italic;">p</span> = $p.<br />
<br />
f (x) = $pnot<sup>x</sup> * $p so f($x) = $Ans</div>@
qu.5.6.editing=useHTML@
qu.5.6.solution=@
qu.5.6.algorithm=$Q=1;
$p=decimal(1,range(0.2,0.5,0.1));
$pp=$p*100;
$a=range(3,6,1);
$Abbrev=switch($a-3,"third","fourth","fifth","sixth");
$pnot=1-$p;
$x=$a-1;
$Ans=decimal(4,$pnot^($a-1)*$p);
$Which=rint(4);
$Align=switch(rint(2),"Left","Right");@
qu.5.6.uid=b158fd89-0fb0-4163-88ff-559aca0fbde7@
qu.5.6.info=  Course=230;
  Type=numeric;
@

qu.6.topic=Basics@

qu.6.1.mode=Non Permuting Multiple Selection@
qu.6.1.name=08. Which are prob. functions@
qu.6.1.comment=<p>Must check two things:</p>
<ol>
    <li><font size="3" face="Times New Roman">0 &le; <em>f</em>(<em>x</em>) &le; 1</font> for all <em><font size="3" face="Times New Roman">x</font></em>&nbsp; .&nbsp; This is so for all these functions except <font size="3" face="Times New Roman"><em>f</em>(<em>x</em>) =</font> $f3.</li>
    <li>Sum of all&nbsp; <font size="3" face="Times New Roman"><em>f</em>(<em>x</em>) = 1</font>&nbsp;&nbsp; .&nbsp; This is so&nbsp; for <font size="3" face="Times New Roman"><em>f</em>(<em>x</em>) =</font> $f2&nbsp; and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mfrac><mn>3</mn><mrow><mn>5</mn></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mfrac><mn>1</mn><mrow><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac></mrow></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.</li>
</ol>@
qu.6.1.editing=useHTML@
qu.6.1.solution=@
qu.6.1.algorithm=$Q=8;
$f1=switch(rint(2),mathml("x^2"),mathml("x^3"));
$r1="0.2, 0.3, 0.4";
$f2=mathml("x/3");
$r2="1, 2";
$Pick3=rint(2);
$f3=switch($Pick3,mathml("3/2-x"),mathml("2/3-x"));
$r3=switch($Pick3,"0, 1, 2,3","0, 1, 2");@
qu.6.1.uid=d8d728c2-4e2a-4a67-80a8-3e08f179a39e@
qu.6.1.info=  Type=MS;
  Course=230;
@
qu.6.1.question=<div title="UW Statistics Bank/Discrete Probability Models/Basics/Q$Q">Which of the following are probability functions?&nbsp; (Select all that are)</div>@
qu.6.1.answer=2, 4@
qu.6.1.choice.1=<font size="3" face="Times New Roman"><i>f</i>(<i>x</i>)</font> = $f1, <font size="3" face="Times New Roman"><i>x</i> = $r1</font>@
qu.6.1.choice.2=<font size="3" face="Times New Roman"><i>f</i>(<i>x</i>)</font> = $f2 , <font size="3" face="Times New Roman"><i>x</i> = $r2</font>@
qu.6.1.choice.3=<font size="3" face="Times New Roman"><i>f</i>(<i>x</i>)</font> = $f3 , <font size="3" face="Times New Roman"><i>x</i> = $r3</font>@
qu.6.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>3</mn><mrow><mn>5</mn></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mn>1</mn><mrow><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math> , <font size="3" face="Times New Roman"><i>x</i> = 0, 1, 2</font>@
qu.6.1.fixed=@

qu.6.2.mode=True False@
qu.6.2.name=03. If  a < b, P(X<=a) <= P(X <=b)@
qu.6.2.comment=<div class="shadedDiv descriptionSpan" style="margin-top: 0px; margin-bottom: 2px">This is True. If a < b then P(X &le; b) = P(X &le; a) + P(a < X &le; b) &ge; P(X &le; a) since probabilities are always &ge; 0.</div>@
qu.6.2.editing=useHTML@
qu.6.2.solution=@
qu.6.2.algorithm=@
qu.6.2.uid=088ab4d3-76b5-44c2-8fef-5a51aff8cdd7@
qu.6.2.info=  Difficulty=1;
  Course=230;
  Type=TF;
  Algorithmic=no;
@
qu.6.2.question=<div title="UW Statistics Bank/Discrete Probability Models/Basics/Q03">Let X be a random variable which takes on integer values, and suppose a and b are possible values of X. <br />
True or False: If  a < b then P(X &le; a) &le; P(X &le; b)</div>@
qu.6.2.answer=1@
qu.6.2.choice.1=True@
qu.6.2.choice.2=False@
qu.6.2.fixed=@

qu.6.3.mode=Multiple Choice@
qu.6.3.name=01+. Which is continuous?@
qu.6.3.comment=<p>Only "$Ans" involves a continuous measurement, the rest are discrete.</p>@
qu.6.3.editing=useHTML@
qu.6.3.solution=@
qu.6.3.algorithm=$Q="01+";
$Alt1=switch(rint(2),"the number of brothers a randomly chosen person has","the number of birds swimming in a pond at a given moment");
$Alt2=switch(rint(2),"the number of cars owned by a randomly chosen adult male","the number of children living in a randomly chosen house");
$Ans=switch(rint(3),"the time it takes for a randomly chosen woman to run 100 meters","the volume of water flowing over Niagara Falls during a randomly chosen minute","the time it takes a randomly chosen person to commute to work");
$Alt3="number of orders received by a mail order company in a randomly chosen week";@
qu.6.3.uid=f98d95d4-ac2c-48fb-8c13-dda04c88e6c4@
qu.6.3.info=  Type=MC;
@
qu.6.3.question=<div title="UW Statistics Bank/Continuous Distributions/Basics/Q$Q">Which of the following random variables would be considered continuous?</div>@
qu.6.3.answer=2@
qu.6.3.choice.1=$Alt1@
qu.6.3.choice.2=$Ans@
qu.6.3.choice.3=$Alt2@
qu.6.3.choice.4=$Alt3@
qu.6.3.fixed=@

qu.6.4.mode=Multiple Choice@
qu.6.4.name=09. What is a random variable?@
qu.6.4.comment=@
qu.6.4.editing=useHTML@
qu.6.4.solution=@
qu.6.4.algorithm=$Q="09";
$Alt1=switch(rint(2),"the particular sample obtained from simple random sampling","the dual space of a probability function.");
$Ans="a variable whose value is a numerical outcome of a random phenomenon";
$Alt2=switch(rint(2),"the particular variable selected by random sampling from an initial list of possible variables","any single-valued function defined on a sample space");
$Alt3=switch(rint(2),"any number that has an unknown and unpredictable value","a heuristically defined variable that mimics random behavior");@
qu.6.4.uid=0cda386d-979b-4027-af89-f352ab63af42@
qu.6.4.info=  Type=MC;
@
qu.6.4.question=<div title="UW Statistics Bank/Discrete Probability Models/Basics/Q$Q">
What is a random variable?</div>@
qu.6.4.answer=2@
qu.6.4.choice.1=$Alt1@
qu.6.4.choice.2=$Ans@
qu.6.4.choice.3=$Alt2@
qu.6.4.choice.4=$Alt3@
qu.6.4.fixed=@

qu.6.5.mode=Non Permuting Multiple Choice@
qu.6.5.name=11. Given CDF, what is P(X>n) ?@
qu.6.5.comment=<p>F($n) = P(X &le; $n), so P(X > $n) = 1 - P(X &le; $n) = 1 - F($n) .</p>@
qu.6.5.editing=useHTML@
qu.6.5.solution=@
qu.6.5.algorithm=$Q=11;
$n=range(5,25,1);
$n1=$n+1;@
qu.6.5.uid=20914fe7-0fc6-4b95-a42b-1738269f07ad@
qu.6.5.info=  Type=MC;
  Keyword=cdf;
@
qu.6.5.question=<div title="UW Statistics Bank/Discrete Probability Models/Basics/Q$Q">For a random variable with cumulative distribution function F(x), P(X>$n)=</div>@
qu.6.5.answer=2@
qu.6.5.choice.1=F($n)@
qu.6.5.choice.2=1-F($n)@
qu.6.5.choice.3=F($n1)@
qu.6.5.choice.4=1-F($n1)@
qu.6.5.choice.5=None of the above@
qu.6.5.fixed=@

qu.6.6.question=<div title="UW Statistics Bank/Discrete Probability Models/Basics/Q$Q">A number from the set {1,2,3} is chosen at random. A coin is tossed as many times as the number. X is a random variable defined to be the number of heads found in those tosses.
<p>Find P(X = $Pick) (4 decimal accuracy).</p>
</div>@
qu.6.6.answer.num=$Ans@
qu.6.6.answer.units=@
qu.6.6.showUnits=false@
qu.6.6.grading=toler_abs@
qu.6.6.err=.001@
qu.6.6.negStyle=minus@
qu.6.6.numStyle=thousands scientific dollars arithmetic@
qu.6.6.mode=Numeric@
qu.6.6.name=04. Probabilty of getting X number of heads@
qu.6.6.comment=<p>Notice that you could toss the coin up to 3 times, so the number of possible heads X takes on values from the set {0,1,2,3}.</p>
<table cellspacing="0" cellpadding="3" border="1" x:str="" style="border-collapse: collapse;">
    <tbody>
        <tr height="17" style="height: 12.75pt;">
            <td style="border: 0.5pt solid windowtext; padding: 0px; font-weight: bold; font-size: 10pt; vertical-align: bottom; color: windowtext; font-style: normal; font-family: Arial,sans-serif; white-space: nowrap; text-decoration: none;">Number<br />
            Picked</td>
            <td style="font-weight: bold;">Prob.</td>
            <td align="center" colspan="8" style="font-weight: bold;">Possible              Equally-Likely Outcomes</td>
            <td nowrap="" style="font-weight: bold;" colspan="4">
            <p align="center">P(n heads)</p>
            </td>
        </tr>
        <tr>
            <td align="center">&nbsp;</td>
            <td align="center">&nbsp;</td>
            <td>&nbsp;</td>
            <td>&nbsp;</td>
            <td>&nbsp;</td>
            <td>&nbsp;</td>
            <td>&nbsp;</td>
            <td>&nbsp;</td>
            <td>&nbsp;</td>
            <td>&nbsp;</td>
            <td align="right">$Col0Color 0$EndSpan</td>
            <td align="right">$Col0Color 1$EndSpan</td>
            <td align="right">$Col0Color 2$EndSpan</td>
            <td align="right">$Col0Color 3$EndSpan</td>
        </tr>
        <tr>
            <td align="center">1</td>
            <td align="center">1/3</td>
            <td>H</td>
            <td>T</td>
            <td>&nbsp;</td>
            <td>&nbsp;</td>
            <td>&nbsp;</td>
            <td>&nbsp;</td>
            <td>&nbsp;</td>
            <td>&nbsp;</td>
            <td align="right">$Col0Color 1/2$EndSpan</td>
            <td align="right">$Col1Color 1/2$EndSpan</td>
            <td align="right">$Col2Color -$EndSpan</td>
            <td align="right">$Col3Color -$EndSpan</td>
        </tr>
        <tr>
            <td align="center">2</td>
            <td align="center">1/3</td>
            <td>HH</td>
            <td>HT</td>
            <td>TH</td>
            <td>TT</td>
            <td>&nbsp;</td>
            <td>&nbsp;</td>
            <td>&nbsp;</td>
            <td>&nbsp;</td>
            <td align="right">$Col0Color 1/4$EndSpan</td>
            <td align="right">$Col1Color 1/2$EndSpan</td>
            <td align="right">$Col2Color 1/4$EndSpan</td>
            <td align="right">$Col3Color -$EndSpan</td>
        </tr>
        <tr>
            <td align="center">3</td>
            <td align="center">1/3</td>
            <td>HHH</td>
            <td>HHT</td>
            <td>HTH</td>
            <td>HTT</td>
            <td>THH</td>
            <td>THT</td>
            <td>TTH</td>
            <td>TTT</td>
            <td align="right">$Col0Color 1/8$EndSpan</td>
            <td align="right">$Col1Color 3/8$EndSpan</td>
            <td align="right">$Col2Color 3/8$EndSpan</td>
            <td align="right">$Col3Color 1/8$EndSpan</td>
        </tr>
        <tr>
            <td>&nbsp;</td>
            <td>&nbsp;</td>
            <td>&nbsp;</td>
            <td>&nbsp;</td>
            <td>&nbsp;</td>
            <td>&nbsp;</td>
            <td>&nbsp;</td>
            <td>&nbsp;</td>
            <td>&nbsp;</td>
            <td>&nbsp;</td>
            <td align="right">$Col0Color 7/24$EndSpan</td>
            <td align="right">$Col1Color 11/24$EndSpan</td>
            <td align="right">$Col2Color 5/24$EndSpan</td>
            <td align="right">$Col3Color 1/24$EndSpan</td>
        </tr>
    </tbody>
</table>
<p><br />
The table shows the possible tosses for each of the numbers 1,2, and 3. The probability for any X value then is found by adding up the fraction of times that number of Heads occurs. <br />
<br />
Then P(X = $a) = $AnsML</p>@
qu.6.6.editing=useHTML@
qu.6.6.solution=@
qu.6.6.algorithm=$Q=4;
$Pick=range(0,3,1);
$AnsT=switch($Pick,7,11,5,1);
$AnsML=mathml("$AnsT/24");
$Ans=$AnsT/24;
$Col0Color=if(eq($Pick,0),"<span style='color:green'>","");
$Col1Color=if(eq($Pick,1),"<span style='color:green'>","");
$Col2Color=if(eq($Pick,2),"<span style='color:green'>","");
$Col3Color=if(eq($Pick,3),"<span style='color:green'>","");
$EndSpan="</span>";@
qu.6.6.uid=df4aea09-ff2c-48ef-b30d-3ad0f58a4d80@
qu.6.6.info=  Difficulty=2;
  Course=230;
  Type=numeric;
@

qu.6.7.mode=Multiple Choice@
qu.6.7.name=10. Properties of a discrete pdf@
qu.6.7.comment=<p>First, add up the given probabilities :&nbsp; $P2 + $P3 + $P4 = $PartialSum so we know that P(X = 0) + P(X = 1) = 1 - $PartialSum = $Remnant.</p>
<p>Use this and the fact that probabilities are positive to test each of the alternatives:</p>
<ul>
    <li>P(X = 0) = P(X = $Pick1)&nbsp; . By chance this may be true, but it does not have to be as there is nothing in what we are told to force this equality.</li>
    <li>&nbsp;P(X = 0) < P(X = $Pick2) . As above.</li>
    <li>&nbsp;P(X = 0) + P(X = $Pick3) = $AltSum1&nbsp; No, we do not know what P(X = 0) is so we cannot determine this sum.</li>
    <li>&nbsp;P(X = 0) + P(X = 1) + P(X = 2) = $Sum . Yes. We know that P(X = 0) + P(X = 1) = $Remnant and are given P(X = 2) = $P2 so just add them.</li>
</ul>@
qu.6.7.editing=useHTML@
qu.6.7.solution=@
qu.6.7.algorithm=$Q="10";
$P2=range(0.05,0.35,0.1);
$P3=range(0.05,0.35,0.1);
$P4=range(0.05,0.75-$P2-$P3);
$PartialSum=$P2+$P3+$P4;
$Remnant=1-$PartialSum;
$Sum=1-$P3-$P4;
$OneMSum=1-$Sum;
$AltSum1=$P4+range(0.05,$OneMSum,0.05);
$Pick1=rint(1,5);
$Pick2=rint(1,5);
$Pick3=rint(2,5);@
qu.6.7.uid=4749953d-caad-4679-a634-e3e36acd2a62@
qu.6.7.info=  Type=MC;
  Course=230;
@
qu.6.7.question=<div title="UW Statistics Bank/Discrete Probability Models/Basics/Q$Q">Given a probability distribution in which the random variable X assumes only the values 0,1,2,3,4, suppose P(X=2) = $P2, P(X=3) = $P3, and P(X=4) = $P4, which of the following must be true?</div>@
qu.6.7.answer=3@
qu.6.7.choice.1=P(X=0) = P(X= $Pick1)@
qu.6.7.choice.2=P(X=0) < P(X=$Pick2)@
qu.6.7.choice.3=P(X=0) + P(X=1) + P(X=2) = $Sum@
qu.6.7.choice.4=P(X=0) + P(X=$Pick3) = $AltSum1@
qu.6.7.fixed=4@

qu.6.8.mode=True False@
qu.6.8.name=06. If P(X<=a) < P(X <=b),  a < b@
qu.6.8.comment=<p>This is True. If it wasn't, the only possibilities are:</p>
<ol>
    <li>that a = b . Not possible since P(X &le; a) < P(X &le; b)</li>
    <li>a > b. In this case then P(X &le; a) = P(X &le; b) + P(b < X &le; a) &ge; P(X &le; b) a contradiction.</li>
</ol>@
qu.6.8.editing=useHTML@
qu.6.8.solution=@
qu.6.8.algorithm=@
qu.6.8.uid=12baad37-35df-4254-a205-6a4e8deed2e8@
qu.6.8.info=  Difficulty=1;
  Keyword=cdf;
  Course=230;
  Type=TF;
  Algorithmic=no;
@
qu.6.8.question=<div title="UW Statistics Bank/Discrete Probability Models/Basics/Q06">Let X be a random variable which takes on integer values, and suppose a and b are possible values of X. <br />
<br />
True or False: If P(X &le; a) < P(X &le; b) then a < b</div>@
qu.6.8.answer=1@
qu.6.8.choice.1=True@
qu.6.8.choice.2=False@
qu.6.8.fixed=@

qu.6.9.question=<div title="UW Statistics Bank/Discrete Probability Models/Basics/Q$Q"><img hspace="4" align="$Align" src="__BASE_URI__DPM/Basics/Coin0.gif" alt="A coin" title="Imagine a coin here... [IMG:Coin$Which.gif]" />A fair coin is flipped 4 times. Let X be the number of heads in these 4 tosses. Find  P(X &le; $t) (3 decimal accuracy).</div>@
qu.6.9.answer.num=$Ans@
qu.6.9.answer.units=@
qu.6.9.showUnits=false@
qu.6.9.grading=toler_abs@
qu.6.9.err=.01@
qu.6.9.negStyle=minus@
qu.6.9.numStyle=thousands scientific dollars arithmetic@
qu.6.9.mode=Numeric@
qu.6.9.name=01. P(# heads <= n)@
qu.6.9.comment=<p>The sixteen possible outcomes are shown below. Those satisfying the criteria "Number of heads &le; $t"&nbsp; are shown in green. Count them up to conclude that P(# H &le; $t) = $AnsML.</p>
<div align="center"><center>
<table cellspacing="0" cellpadding="2" bordercolor="#111111" border="1" id="AutoNumber1" style="border-collapse: collapse;">
    <tbody>
        <tr>
            <td>$Row0Color TTTT  $EndSpan</td>
        </tr>
        <tr>
            <td>$Row1Color TTTH, TTHT, THTT, HTTT$EndSpan</td>
        </tr>
        <tr>
            <td>$Row2Color TTHH, THHT, HHTT, THTH, HTHT, HTTH  $EndSpan</td>
        </tr>
        <tr>
            <td>$Row3Color HHHT, HHTH, HTHH, THHH  $EndSpan</td>
        </tr>
        <tr>
            <td>$Row4Color HHHH  $EndSpan</td>
        </tr>
    </tbody>
</table>
</center></div>@
qu.6.9.editing=useHTML@
qu.6.9.solution=@
qu.6.9.algorithm=$Q="01";
$t=range(1,4,1);
$AnsT=switch($t-1,5,11,15,16);
$Ans=$AnsT/16;
$AnsML=mathml("$AnsT/16");
$Which=rint(4);
$Align=switch(rint(2),"Left","Right");
$Row0Color="<span style=color:green>";
$Row1Color="<span style=color:green>";
$Row2Color=if(gt($t,1),"<span style='color:green'>","");
$Row3Color=if(gt($t,2),"<span style='color:green'>","");
$Row4Color=if(eq($t,4),"<span style='color:green'>","");
$EndSpan="</span>";@
qu.6.9.uid=d6404620-25d0-428f-a495-2204c3f2d2a9@
qu.6.9.info=  Difficulty=2;
  Keyword=cdf;
  Course=230;
  Type=numeric;
@

qu.6.10.mode=Matching@
qu.6.10.name=07. Two Dice CDF@
qu.6.10.comment=<table cellpadding="3">
    <tbody>
        <tr>
            <td><img width="425" height="293" title="CDF for sum of two dice [IMG:TwoDiceCdf.gif]" alt="CDF for sum of two dice" src="__BASE_URI__DPM/Basics/TwoDiceCdf.gif" /></td>
            <td>
            <table width="192" cellspacing="0" cellpadding="0" border="1" style="border-collapse: collapse;">
                <tbody>
                    <tr>
                        <td>Dice Sum</td>
                        <td align="center">Probability<br />
                        /36</td>
                        <td align="center">Cumulative<br />
                        /36</td>
                    </tr>
                    <tr>
                        <td align="right">2</td>
                        <td align="right">1</td>
                        <td align="right">1</td>
                    </tr>
                    <tr>
                        <td align="right">3</td>
                        <td align="right">2</td>
                        <td align="right">3</td>
                    </tr>
                    <tr>
                        <td align="right">4</td>
                        <td align="right">3</td>
                        <td align="right">6</td>
                    </tr>
                    <tr>
                        <td align="right">5</td>
                        <td align="right">4</td>
                        <td align="right">10</td>
                    </tr>
                    <tr>
                        <td align="right">6</td>
                        <td align="right">5</td>
                        <td align="right">15</td>
                    </tr>
                    <tr>
                        <td align="right">7</td>
                        <td align="right">6</td>
                        <td align="right">21</td>
                    </tr>
                    <tr>
                        <td align="right">8</td>
                        <td align="right">5</td>
                        <td align="right">26</td>
                    </tr>
                    <tr>
                        <td align="right">9</td>
                        <td align="right">4</td>
                        <td align="right">30</td>
                    </tr>
                    <tr>
                        <td align="right">10</td>
                        <td align="right">3</td>
                        <td align="right">33</td>
                    </tr>
                    <tr>
                        <td align="right">11</td>
                        <td align="right">2</td>
                        <td align="right">35</td>
                    </tr>
                    <tr>
                        <td align="right">12</td>
                        <td align="right">1</td>
                        <td align="right">36</td>
                    </tr>
                </tbody>
            </table>
            </td>
        </tr>
    </tbody>
</table>
<p><em>Sorry, matching questions do not display well in the marking scheme.</em></p>@
qu.6.10.editing=useHTML@
qu.6.10.solution=@
qu.6.10.algorithm=$Q="07";
$S1=switch(rint(2),2,12);
$S2=switch(rint(2),3,11);
$S3=switch(rint(2),4,10);
$S4=switch(rint(2),5,9);
$S5=switch(rint(3),6,7,8);
$P=(0,1,3,6,10,15,21,26,30,33,35,36);
$P1=switch($S1-1,$P);
$P2=switch($S2-1,$P);
$P3=switch($S3-1,$P);
$P4=switch($S4-1,$P);
$P5=switch($S5-1,$P);
$Which=rint(4);
$Align=switch(rint(2),"Left","Right");@
qu.6.10.uid=bc15c640-e71d-4f66-88c3-b05907df00fd@
qu.6.10.info=  Difficulty=2;
  Keyword=cdf;
  Course=230;
  Type=Matching;
@
qu.6.10.format.columns=3@
qu.6.10.question=<div title="UW Statistics Bank/Discrete Probability Models/Basics/Q$Q"><img hspace="4" align="$Align" src="__BASE_URI__DPM/Basics/Dice$Which.gif" alt="Two dice" title="Two dice [IMG:TwoDice.gif]" />Two dice are rolled and a random variable X is defined as the sum of the two dice. This yields a <span style="font-style: italic;">cumulative distribution function</span> F(x)= P(X &le; x). For example F(12) = 1 since (of course) EVERY roll will yield a total of 12 or less. Match each possible value of X (the sum of the dice) with its cumulative probability:&nbsp;
<p>Match each of the numbered items in the list with the numbers in the drop-down menus:</p>
</div>@
qu.6.10.term.1=Sum is $S5@
qu.6.10.term.1.def.1=Cumulative Probability is $P5/36@
qu.6.10.term.2=Sum is $S4@
qu.6.10.term.2.def.1=Cumulative Probability is $P4/36@
qu.6.10.term.3=Sum is $S1@
qu.6.10.term.3.def.1=Cumulative Probability is $P1/36@
qu.6.10.term.4=Sum is $S2@
qu.6.10.term.4.def.1=Cumulative Probability is $P2/36@
qu.6.10.term.5=Sum is $S3@
qu.6.10.term.5.def.1=Cumulative Probability is $P3/36@

qu.6.11.mode=Matching@
qu.6.11.name=02. Two Dice Probability Distribution@
qu.6.11.comment=<p>There are 36 different outcomes for the sum of two dice. (We consider the dice to be distinguishable, so e.g.&nbsp;<img width="107" hspace="4" height="61" align="absMiddle" title="Dice are 2 and 3 [IMG:yellow2030.gif]" alt="Dice are 2 and 3." src="__BASE_URI__DPM/Basics/yellow2030.gif" /> is a different roll than (though the same sum as) <img width="97" hspace="4" height="60" align="absMiddle" title="Dice are 2 and 3 [IMG:yellow3121.gif]" alt="Dice are 3 and 2." src="__BASE_URI__DPM/Basics/yellow3121.gif" />&nbsp; ). Then the probability of any sum is just the number of possible rolls adding to that number divided by 36. The following table gives us the numerators:</p>
<p>
<table cellspacing="1" cellpadding="1" border="1" align="center">
    <tbody>
        <tr>
            <td align="center">Sum</td>
            <td align="center"># Rolls</td>
            <td align="center">Sum</td>
            <td align="center"># Rolls</td>
        </tr>
        <tr>
            <td align="right">2</td>
            <td align="right">1</td>
            <td align="right">8</td>
            <td align="right">5</td>
        </tr>
        <tr>
            <td align="right">3</td>
            <td align="right">2</td>
            <td align="right">9</td>
            <td align="right">4</td>
        </tr>
        <tr>
            <td align="right">4</td>
            <td align="right">3</td>
            <td align="right">10</td>
            <td align="right">3</td>
        </tr>
        <tr>
            <td align="right">5</td>
            <td align="right">4</td>
            <td align="right">11</td>
            <td align="right">2</td>
        </tr>
        <tr>
            <td align="right">6</td>
            <td align="right">5</td>
            <td align="right">12</td>
            <td align="right">1</td>
        </tr>
        <tr>
            <td align="right">7</td>
            <td align="right">6</td>
            <td>&nbsp;</td>
            <td>&nbsp;</td>
        </tr>
    </tbody>
</table>
</p>@
qu.6.11.editing=useHTML@
qu.6.11.solution=@
qu.6.11.algorithm=$Q="02";
$S1=switch(rint(2),2,12);
$S2=switch(rint(2),3,11);
$S3=switch(rint(2),4,10);
$S4=switch(rint(2),5,9);
$S5=switch(rint(3),6,7,8);
$P=(0,1,2,3,4,5,6,5,4,3,2,1);
$P1=1;
$P2=2;
$P3=3;
$P4=4;
$P5=switch($S5-1,$P);
$P1ML=mathml("$P1/36");
$P2ML=mathml("$P2/36");
$P3ML=mathml("$P3/36");
$P4ML=mathml("$P4/36");
$P5ML=mathml("$P5/36");@
qu.6.11.uid=525add76-c583-4ced-ba79-1ab4c2e04554@
qu.6.11.info=  Difficulty=2;
  Keyword=cdf;
  Course=230;
  Type=Matching;
@
qu.6.11.format.columns=3@
qu.6.11.question=<div title="STAT230/Chapter 5/Basic Questions/Q$Q">Two dice are rolled and the sum of the dice is recorded. This yields a probability distribution: each possible sum (2,3,...,12) has a probability associated with it. In this question you are to match the sums with their probabilities.
<p>&nbsp;</p>
<p>Match each of the numbered items in the list with the numbers in the drop-down menus:</p>
</div>@
qu.6.11.term.1=Sum is $S1@
qu.6.11.term.1.def.1=Probability is $P1ML@
qu.6.11.term.2=Sum is $S2@
qu.6.11.term.2.def.1=Probability is $P2ML@
qu.6.11.term.3=Sum is $S4@
qu.6.11.term.3.def.1=Probability is $P4ML@
qu.6.11.term.4=Sum is $S5@
qu.6.11.term.4.def.1=Probability is $P5ML@
qu.6.11.term.5=Sum is $S3@
qu.6.11.term.5.def.1=Probability is $P3ML@

qu.6.12.mode=True False@
qu.6.12.name=05. If P(X≤a) ≤ P(X ≤ b),  a < b@
qu.6.12.comment=<div style="margin-top: 0px; margin-bottom: 2px;" class="shadedDiv descriptionSpan">This is False, for the most obvious reason: if a = b then the hypothesis holds but the conclusion is obviously wrong.</div>@
qu.6.12.editing=useHTML@
qu.6.12.solution=@
qu.6.12.algorithm=@
qu.6.12.uid=e490864d-1be9-4be3-8673-fc43428562db@
qu.6.12.info=  Difficulty=1;
  Keyword=cdf;
  Course=230;
  Type=TF;
  Algorithmic=no;
@
qu.6.12.question=<div title="University of Waterloo Statistics Bank/Discrete Probability Models/Basics/Q05">Let X be a random variable which takes on integer values, and suppose a and b are possible values of X. <br />
<br />
True or False: If  P(X &le; a) &le; P(X &le; b) then a < b .</div>@
qu.6.12.answer=2@
qu.6.12.choice.1=True@
qu.6.12.choice.2=False@
qu.6.12.fixed=@

qu.7.topic=Negative Binomial Distribution@

qu.7.1.mode=Non Permuting Multiple Choice@
qu.7.1.name=03. Model door-door sales.@
qu.7.1.comment=<div style="margin-top: 0px; margin-bottom: 2px;" class="shadedDiv descriptionSpan">This is best modelled by the <span style="font-style: italic;">Negative Binomial</span>. Why? X ~ NB(k,p) when X is the number of failures before the kth success. Here <font size="3" face="Times New Roman"><em>k</em> = 5</font> (success is selling a bar) and the actual number of houses visited is <font size="3" face="Times New Roman"><em>X</em> + 5</font> so P(Last bar sold at house <font size="3" face="Times New Roman"><span style="font-style: italic;">n</span>) = P(X = <span style="font-style: italic;">n</span> - 5) = P(<span style="font-style: italic;">X - n</span> = 5)</font> and so with <font size="3" face="Times New Roman"><em>k</em> = <em>n </em>- 5;&nbsp; <em>r </em>= 5</font> this is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>NB</mi><mfenced open='(' close=')' separators=','><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>5</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>5</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>p</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><munder><mrow><mfenced open='(' close=')' separators=','><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>5</mn></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>5</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>5</mn></mrow></munder></mrow></mfenced><msup><mi>p</mi><mrow><mn>5</mn></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>p</mi></mrow></mfenced><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>5</mn></mrow></msup></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mn>5</mn></mrow></mstyle></math></div>@
qu.7.1.editing=useHTML@
qu.7.1.solution=@
qu.7.1.algorithm=$Q=3;
$p=decimal(1,range(0.1,0.7,0.1));
$q=1-$p;@
qu.7.1.uid=6ea2dfdd-d19b-4e4a-bab6-fcbe6a8a2e00@
qu.7.1.info=  Course=230;
  Type=MC;
@
qu.7.1.question=<div title="Stat230/Chapter 5/Negative Binomial Distribution/Q$Q">Pat is required to sell candy bars to raise money for the 6th grade field trip. There are thirty houses in the neighborhood, and Pat is not supposed to return home until five candy bars have been sold. So the child goes door to door, selling candy bars. At each house, there is a $p probability of selling one candy bar and a $q probability of selling nothing. We are interested in the probability that she sells the last (that is the 5th) candy bar at the nth house.
<p>Which probability distribution best models this physical situation?</p>
</div>@
qu.7.1.answer=3@
qu.7.1.choice.1=Uniform@
qu.7.1.choice.2=Hypergeometric@
qu.7.1.choice.3=Negative Binomial@
qu.7.1.choice.4=Binomial@
qu.7.1.choice.5=A combination of two of the above@
qu.7.1.fixed=@

qu.7.2.mode=Multiple Choice@
qu.7.2.name=05. f(finding # students)@
qu.7.2.comment=<p>This is a negative binomial setup (note that it is the <u>second</u> version discussed in your text, not the first). The number of successes is k = $MinNum, we want to calculate how many ways we can select x - $MinNum failures from x - 1 trials (x - 1 since the last trial is a success).&nbsp; This is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><munderover><mi mathcolor='#0000ff'></mi><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$MinNum</mi></mrow><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></munderover></mrow></mfenced></mrow></mstyle></math> . Using P(success) = $P and thus P(failure) = $OneMP; we have the result:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$MinNum</mi></mrow></mtd></mtr></mtable></mrow></mfenced><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$P</mi></mrow></mfenced><mrow><mi mathvariant='normal'>$MinNum</mi></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$OneMP</mi></mrow></mfenced><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$MinNum</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$MinNum</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$MinNumP1</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow></mrow></mtd></mtr></mtable></mrow></mstyle></math></p>
<p>&nbsp;</p>@
qu.7.2.editing=useHTML@
qu.7.2.solution=@
qu.7.2.algorithm=$Q="05";
$MinNum=range(20,35,5);
$P=range(0.15,0.45,0.05);
$OneMP=1-$P;
$Ppc=100*$P;
$MinNumP1=$MinNum+1;
$MinNumM1=$MinNum-1;@
qu.7.2.uid=ca8d0469-90e5-4112-b34e-7aa3ee1e79dd@
qu.7.2.info=  Course=230;
  Type=MC;
@
qu.7.2.question=<div title="UW Statistics Bank/Discrete Probability Models/Negative Binomial Distribution/Q$Q">An instructor will hold a review session if you find $MinNum students (besides yourself) who will agree to attend. Suppose each student you ask answers independently and has a $Ppc% chance of agreeing to go. Let X be the total number of students you must ask in order to find $MinNum to go. Which of the following is the probability function of X?</div>@
qu.7.2.answer=2@
qu.7.2.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><munderover><mi mathcolor='#0000ff'></mi><mrow><mn>1</mn></mrow><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$MinNum</mi></mrow></munderover></mrow></mfenced><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$P</mi></mrow></mfenced><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$MinNum</mi></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$OneMP</mi></mrow></mfenced><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$MinNum</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msup></mrow><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$MinNum</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$MinNumP1</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math>@
qu.7.2.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><munderover><mi mathcolor='#0000ff'></mi><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$MinNum</mi></mrow><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></munderover></mrow></mfenced><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$P</mi></mrow></mfenced><mrow><mi mathvariant='normal'>$MinNum</mi></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$OneMP</mi></mrow></mfenced><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$MinNum</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$MinNum</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$MinNumP1</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>@
qu.7.2.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><munderover><mi mathcolor='#0000ff'></mi><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$MinNumM1</mi></mrow><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$MinNum</mi></mrow></munderover></mrow></mfenced><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$P</mi></mrow></mfenced><mrow><mi></mi></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$OneMP</mi></mrow></mfenced><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$MinNum</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msup></mrow><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$MinNum</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$MinNumP1</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math>@
qu.7.2.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$P</mi></mrow></mfenced><mrow><mi mathvariant='normal'>$MinNum</mi></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$OneMP</mi></mrow></mfenced><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$MinNum</mi></mrow></msup></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$MinNum</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$MinNumP1</mi></mrow></mstyle></math>@
qu.7.2.choice.5=You cannot determine this with the information given.@
qu.7.2.fixed=4@

qu.7.3.question=<div title="UW Statistics Bank/Discrete Probability Models/Negative Binomial Distribution/Q$Q"><img hspace="4" align="$Align" src="__BASE_URI__DPM/NegativeBinomial/Baseball$Which.gif" alt="Baseball" title="Baseball [IMG:Baseball$Which.gif]" />A baseball player has a $pp% chance of hitting the ball each time at bat, with successive times at bat being independent. Calculate the probability that he needs more than 6 times at bat in order to get his third hit. (Please answer to 4 decimals of accuracy.)

</div>@
qu.7.3.answer.num=$Ans@
qu.7.3.answer.units=@
qu.7.3.showUnits=false@
qu.7.3.grading=toler_abs@
qu.7.3.err=.001@
qu.7.3.negStyle=minus@
qu.7.3.numStyle=thousands scientific dollars arithmetic@
qu.7.3.mode=Numeric@
qu.7.3.name=04. P(need > 6 ABs to get 3 hits)@
qu.7.3.comment=<div class="shadedDiv descriptionSpan" style="margin-top: 0px; margin-bottom: 2px;">
<p>This is an example of a <span style="font-weight: bold;">Negative Binomial</span> distribution.</p>
<p>Let X = number of at-bats without a hit needed before the third hit, Y be the number of at-bats needed to get his third hit <br />
Then <em>X</em> = <em>Y</em> - 3 and <em>X</em> ~ NB(<em>k</em> = 3, <em>p</em> = $p) and</p>
<p>P(<em>X</em> = <em>x</em>) = f(<em>x</em>) =<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>2</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>x</mi></mrow></mtd></mtr></mtable></mfenced></mrow></mfenced></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>p</mi></mrow></mfenced><mrow><mn>3</mn></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>p</mi></mrow></mfenced><mrow><mi>x</mi></mrow></msup></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math> .</p>
<p>Then we want P(Y > 6) = P(X > 3) = 1 - P(X &le; 3) = 1 - P(X=3) - P(X=2) - P(X=1) - P(X=0) <br />
= $Ans</p>
</div>@
qu.7.3.editing=useHTML@
qu.7.3.solution=@
qu.7.3.algorithm=$Q=4;
$p=decimal(1,range(0.1,0.6,0.1));
$pp=100*$p;
$ans=maple("with(Statistics);
X := RandomVariable(NegativeBinomial(3, $p));
1-ProbabilityFunction(X, 3)-ProbabilityFunction(X, 2)-ProbabilityFunction(X, 1)-ProbabilityFunction(X, 0)");
$Ans=decimal(4, $ans);
$Which=rint(4);
$Align=switch(rint(2),"Left","Right");@
qu.7.3.uid=7f571324-59c7-4603-8f34-b3ddd2b3d251@
qu.7.3.info=  Course=230;
  Type=numeric;
@

qu.7.4.question=<div title="UW Statistics Bank/Discrete Probability Models/Negative Binomial Distribution/Q$Q"><img hspace="4" align="$Align" title="Basketball player [IMG:Basketball$Which.gif]" alt="Basketball Player" src="__BASE_URI__DPM/NegativeBinomial/Basketball$Which.gif" />Bob is a high school basketball player.  He is a $pp% free throw shooter.   		    That means his probability of making a free throw is $p.  During the  		    season, what is the probability that Bob makes his $WNumSunk free throw on his $WY shot? (4 decimal accuracy)</div>@
qu.7.4.answer.num=$Ans@
qu.7.4.answer.units=@
qu.7.4.showUnits=false@
qu.7.4.grading=toler_abs@
qu.7.4.err=.001@
qu.7.4.negStyle=minus@
qu.7.4.numStyle=thousands scientific dollars arithmetic@
qu.7.4.mode=Numeric@
qu.7.4.name=06. Free throws@
qu.7.4.comment=<p>This is an example of a <span style="font-weight: bold;">Negative Binomial</span> distribution. We have Y = $Y as the number of throws needed to get his $WNumSunk basket. Let X = number of throws <u>without</u> a basket needed before the $WNumSunk&nbsp; basket.</p>
<p><br />
Then X = Y - $NumSunk and X ~ NB(k = $NumSunk, p = $p) and</p>
<p>&nbsp;</p>
<p>P(X = x) = f(x) =<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><munderover><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mi>x</mi></mrow><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$NumSunk</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></munderover></mrow></mfenced></mrow><mrow><msup><mi mathvariant='normal'>$p</mi><mrow><mn>3</mn></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$p</mi></mrow></mfenced><mrow><mi>x</mi></mrow></msup></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math> so f($X) = $Ans .</p>
<p><em>Another Solution:</em> Determine how many ways you can get $WNumSunk baskets in $WY tries, given that you stop when you sink the last basket needed. This means you just need to determine how many ways to sink $NumSunkM1 baskets in $YM1 trials, which is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><munderover><mrow><mo mathcolor='#0000ff' lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mi mathvariant='normal'>$NumSunkM1</mi></mrow><mrow><mi mathvariant='normal'>$YM1</mi></mrow></munderover></mrow></mfenced></mrow></mstyle></math> . The probabilty of sinking $NumSunk baskets is $p<sup>$NumSunk</sup> and the probability of missing the rest is (1-$p)<sup>$Y-$NumSunk</sup>.</p>
<p>Put it all together to find the probability is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><munderover><mrow><mo mathcolor='#0000ff' lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mi mathvariant='normal'>$NumSunkM1</mi></mrow><mrow><mi mathvariant='normal'>$YM1</mi></mrow></munderover></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>$p</mi><mrow><mi mathvariant='normal'>$NumSunk</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$p</mi></mrow></mfenced><mrow><mi mathvariant='normal'>$Y</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$NumSunk</mi></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>@
qu.7.4.editing=useHTML@
qu.7.4.solution=@
qu.7.4.algorithm=$Q=6;
$p=decimal(1,range(0.1,0.6,0.1));
$pp=100*$p;
$Y=range(5,10,1);
$YM1=$Y-1;
$NumSunk = range(2,$Y-2,1);
$NumSunkM1=$NumSunk-1;
$X=$Y-$NumSunk;
$WNumSunk=switch($NumSunk,0,"first","second","third","fourth","fifth","sixth","seventh","eigth","ninth","tenth");
$WY=switch($Y,0,"first","second","third","fourth","fifth","sixth","seventh","eigth","ninth","tenth");
$ans=maple("with(Statistics);
X := RandomVariable(NegativeBinomial($NumSunk, $p));
ProbabilityFunction(X, $Y-$NumSunk)");
$Ans=decimal(4, $ans);
$Which=rint(6);
$Align=switch(rint(2),"Left","Right");@
qu.7.4.uid=702bd5cd-c262-4168-8172-00b4afae414d@
qu.7.4.info=  Course=230;
  Type=numeric;
@

qu.7.5.question=<div title="UW Statistics Bank/Discrete Probability Models/Negative Binomial Distribution/Q$Q"><img hspace="4" align="$Align" title="Blood [IMG:Blood$Which.gif]" alt="Blood" src="__BASE_URI__DPM/NegativeBinomial/Blood$Which.gif" />$pp% of people donating blood at a clinic have O<sup>+</sup> type blood. Find the probability that the second O<sup>+</sup> donor is the fourth donor of the day. (4 decimal accuracy)</div>@
qu.7.5.answer.num=$Ans@
qu.7.5.answer.units=@
qu.7.5.showUnits=false@
qu.7.5.grading=toler_abs@
qu.7.5.err=.001@
qu.7.5.negStyle=minus@
qu.7.5.numStyle=thousands scientific dollars arithmetic@
qu.7.5.mode=Numeric@
qu.7.5.name=01. P(k-th O+ donor is n-th donor)@
qu.7.5.comment=<div style="margin-top: 0px; margin-bottom: 2px;" class="shadedDiv descriptionSpan">Define "success" here as a donor having O<sup>+</sup> type blood. Define a r.v. Y to be the number of failures before the <span style="text-decoration: underline;">second</span> success. Then Y ~ Nb(2,<em>$p</em>), that is Y is Negative Binomial.<br />
<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>x</mi></mrow></mtd></mtr></mtable></mfenced></mrow></mfenced><msup><mi>p</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>p</mi></mrow></mfenced><mrow><mi>x</mi></mrow></msup></mrow></mstyle></math>&nbsp;<br />
so<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mn>2</mn></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>2</mn></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>3</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>2</mn></mrow></mtd></mtr></mtable></mfenced></mrow></mfenced></mrow><mrow><msup><mi mathvariant='normal'>$p</mi><mrow><mn>2</mn></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$p</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math> = $Ans</div>@
qu.7.5.editing=useHTML@
qu.7.5.solution=@
qu.7.5.algorithm=$Q=1;
$p=decimal(1,range(0.1,0.5,0.1));
$pp=100*$p;
$PreAns=maple("with(Statistics): X := RandomVariable(NegativeBinomial(2,$p)):ProbabilityFunction(X, 2);
");
$Ans=decimal(4,$PreAns);
$Which=rint(5);
$Align=switch(rint(2),"Left","Right");@
qu.7.5.uid=022a62e0-bd24-4e33-93ab-880884ae6525@
qu.7.5.info=  Course=230;
  Type=numeric;
@

qu.7.6.question=<div title="UW Statistics Bank/Discrete Probability Models/Negative Binomial Distribution/Q$Q">
<img hspace="4" align="$Align" title="Baseball [IMG:Baseball$Which.gif]" alt="Baseball" src="__BASE_URI__DPM/NegativeBinomial/Baseball$Which.gif" />A baseball player has a $pp% chance of hitting the ball each time at bat, with successive times at bat being independent. Calculate the probability that he gets his third hit on his sixth time at bat. (Please answer to 4 decimals of accuracy.)</div>@
qu.7.6.answer.num=$Ans@
qu.7.6.answer.units=@
qu.7.6.showUnits=false@
qu.7.6.grading=toler_abs@
qu.7.6.err=.001@
qu.7.6.negStyle=minus@
qu.7.6.numStyle=thousands scientific dollars arithmetic@
qu.7.6.mode=Numeric@
qu.7.6.name=02. P(kth hit on nth at bat)@
qu.7.6.comment=<div class="shadedDiv descriptionSpan" style="margin-top: 0px; margin-bottom: 2px;">
<p>This is an example of a <span style="font-weight: bold;">Negative Binomial</span> distribution with p = $p, k = 3 and X = 3.&nbsp;</p>
<p>Then f(3) =<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mo mathcolor='#0000ff' lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>5</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>3</mn></mrow></mtd></mtr></mtable></mfenced></mrow></mfenced></mrow></mstyle></math><font size="3" face="Times New Roman">($p)<sup>3</sup>(1-$p)<sup>3</sup> = $Ans</font></p>
</div>@
qu.7.6.editing=useHTML@
qu.7.6.solution=@
qu.7.6.algorithm=$Q=2;
$Which=rint(4);
$Align=switch(rint(2),"Left","Right");
$p=decimal(1,range(0.1,0.5,0.1));
$pp=100*$p;
$PreAns=maple("with(Statistics): X := RandomVariable(NegativeBinomial(3,$p)):
ProbabilityFunction(X, 3);
");
$Ans=decimal(4,$PreAns);@
qu.7.6.uid=1944277b-e5fc-4985-8746-41f165dd321d@
qu.7.6.info=  Difficulty=3;
  Keyword=Negative Binomial;
  Course=230;
  Type=numeric;
@

qu.8.topic=Uniform Distributions@

qu.8.1.question=<div title="UW Statistics Bank/Discrete Probability Models/Uniform Distributions/Q$Q">
A uniform probability distribution <strong><em>on a set of consecutive integers</em></strong> is given by:
<p>&nbsp;</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>f</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mfrac linethickness='0'><mrow><mfrac><mn>1</mn><mrow><mi>P</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mn>0</mn></mrow></mfrac><mrow><mfrac linethickness='0'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Start</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$StartP1</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$End</mi></mrow><mrow><mi mathvariant='normal'>otherwise</mi></mrow></mfrac></mrow></mrow></mrow></mstyle></math></p>
<p>&nbsp;</p>
<p>What is P ?</p>@
qu.8.1.answer.num=$Gap@
qu.8.1.answer.units=@
qu.8.1.showUnits=false@
qu.8.1.grading=exact_value@
qu.8.1.negStyle=minus@
qu.8.1.numStyle=thousands scientific dollars arithmetic@
qu.8.1.mode=Numeric@
qu.8.1.name=05. Find the Probability@
qu.8.1.comment=<p>&nbsp;Since this is a Uniform distribution defined on $End - ($Start) + 1 = $Gap points, each point must have the same probability. That value is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$Gap</mi></mrow></mfrac></mrow></mstyle></math>so P = $Gap.</p>@
qu.8.1.editing=useHTML@
qu.8.1.solution=@
qu.8.1.algorithm=$Q=5;
$Start=range(-5,5,1);
$StartP1=$Start+1;
$End=range($Start+4,$Start+10,1);
$Gap=$End-$Start+1;@
qu.8.1.uid=f8c3f47e-3289-4341-b2e8-f38fc8f0f1cf@
qu.8.1.info=  Difficulty=0;
  Keyword=uniform;
  Type=numeric;
@

qu.8.2.mode=Multiple Choice@
qu.8.2.name=03. P(X > n)@
qu.8.2.comment=<p>Add up the probabilities associated with all values x such that x > $GT . Since this is a Uniform Distribution this is just:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac></mrow></mstyle></math>(# points x > $GT) = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$GT</mi></mrow></mfenced></mrow><mrow></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math> = $AnsML because the distribution is defined on 1,2,...,$n .</p>@
qu.8.2.editing=useHTML@
qu.8.2.solution=@
qu.8.2.algorithm=$Q=3;
$n=range(4,15,1);
$GT=range(1,$n-1);
$Ans=($n-$GT)/$n;
$AnsML=mathml("($n-$GT)/$n");
$Alt1ML=mathml("($GT+1)/$n");
$Alt2ML=mathml("($GT-1)/$n");
$Alt3ML=switch(rint(2),mathml("$GT/($n-1)"),mathml("($GT+1)/($GT+5)"));
$Alt4ML=switch(rint(2),mathml("($n-$GT)/($n+2)"),mathml("($GT+1)/($n+1)"));@
qu.8.2.uid=1d3b7248-cb79-4fdd-a891-880250492594@
qu.8.2.info=  Difficulty=0;
  Keyword=uniform;
  Type=MC;
@
qu.8.2.question=<div title="UW Statistics Bank/Discrete Probability Models/Uniform Distributions/Q$Q">In a uniform probability distribution for a discrete random variable X the probability function is given by:
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>2</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$n</mi></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mi mathvariant='normal'>otherwise</mi></mrow></mtd></mtr></mtable></mfenced><mo lspace='0.0em' rspace='0.0em'></mo></mrow></mstyle></math></p>
<p><br />
Then P(X > $GT) is:</p>
</div>@
qu.8.2.answer=1@
qu.8.2.choice.1=$AnsML@
qu.8.2.choice.2=$Alt1ML@
qu.8.2.choice.3=$Alt2ML@
qu.8.2.choice.4=$Alt3ML@
qu.8.2.choice.5=$Alt4ML@
qu.8.2.fixed=4@

qu.8.3.mode=Multiple Selection@
qu.8.3.name=06. Properties@
qu.8.3.comment=<p>A Uniform distribution is characterized by having the probability being the same for every point in its domain. If the domain has <font size="3" face="Times New Roman"><em>n</em></font> points, the the probability of each point is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mi>n</mi></mrow></mfrac></mrow></mstyle></math>.</p>
<ul>
    <li><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></mfenced></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>N</mi></mrow></mstyle></math> is true since the domain has <font size="3" face="Times New Roman"><em>N</em></font> points.</li>
    <li><font size="3" face="Times New Roman"><em>P</em>(<em>X</em> = 1) = $IsGood1</font>&nbsp; can be true if <font size="3" face="Times New Roman"><em>N</em> = $IsGood1T</font> .</li>
    <li><font size="3" face="Times New Roman"><em>P</em>(<em>X</em> = 1) = <em>P</em>(<em>X</em> = <em>N</em> - 1)</font> True, in fact the probability of all points is the same.</li>
    <li><font size="3" face="Times New Roman"><em>P</em>(<em>X</em> = 1) = $NoGood1</font><font size="3" face="Times New Roman"> </font>This cannot be true as $NoGood1 is not the reciprocal of any integer.</li>
    <li><font size="3" face="Times New Roman"><em>P</em>(<em>X</em> = 1) < <em>P</em>(<em>X</em> = <em>N</em> - 1) </font>Not true, the probability at every point is the same.</li>
    <li><font size="3" face="Times New Roman"><em>P</em>(<em>X</em> = <em>N</em>) =&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><msqrt><mrow><mi mathvariant='normal'>$NoGood2</mi></mrow></msqrt></mrow></mfrac></mrow></mstyle></math> </font>Not true as &nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><msqrt><mrow><mi mathvariant='normal'>$NoGood2</mi></mrow></msqrt></mrow></mfrac></mrow></mstyle></math> is not the reciprocal of any integer.</li>
</ul>@
qu.8.3.editing=useHTML@
qu.8.3.solution=@
qu.8.3.algorithm=$Q=6;
$NoGood1 = switch(rint(4),0.3,0.7,0.9,0.15);
$NoGood2=switch(rint(4),2,3,5,7);
$IsGood1=switch(rint(4),0.1,0.2,0.25,0.5);
$IsGood1T=1/$IsGood1;@
qu.8.3.uid=7463ac71-9e0c-4f2b-830b-8efcb5fb8f2d@
qu.8.3.info=  Type=MS;
  Difficulty=1;
  Keyword=uniform;
@
qu.8.3.question=<div title="UW Statistics Bank/Discrete Probability Models/Uniform Distributions/Q$Q">Let X be a uniformly distributed random variable defined on the set of integers 1, ..., N for some N > 1 .
<p>&nbsp;</p>
<p>Which of the following could be true (select all that are) ?</p>
</div>@
qu.8.3.answer=1, 3, 5@
qu.8.3.choice.1=P(X = 1) = $IsGood1@
qu.8.3.choice.2=P(X = 1) = $NoGood1@
qu.8.3.choice.3=P(X = 1) = P(X = N - 1)@
qu.8.3.choice.4=P(X = 1) < P(X = N - 1)@
qu.8.3.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>1</mn></mrow></mfenced></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>N</mi></mrow></mstyle></math>@
qu.8.3.choice.6=P(X = N) = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><msqrt><mrow><mi mathvariant='normal'>$NoGood2</mi></mrow></msqrt></mrow></mfrac></mrow></mstyle></math>@
qu.8.3.fixed=@

qu.8.4.mode=Multiple Choice@
qu.8.4.name=02. P(X <= n)@
qu.8.4.comment=<p>Add up the probabilities associated with all values x such that x < $LT . Since this is a Uniform Distribution this is just:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac></mrow></mstyle></math>(# points x < $LT) = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$LT</mi></mrow><mrow><mi></mi></mrow></mstyle></math> = $AnsML because the distribution is defined on 1,2,...,$n .</p>@
qu.8.4.editing=useHTML@
qu.8.4.solution=@
qu.8.4.algorithm=$Q=2;
$n=range(4,15,1);
$LT=range(1,$n-1);
$Ans=$LT/$n;
$AnsML=mathml("$LT/$n");
$Alt1ML=mathml("($LT+1)/$n");
$Alt2ML=mathml("($LT-1)/$n");
$Alt3ML=switch(rint(2),mathml("$LT/($n-1)"),mathml("($LT+1)/($LT+5)"));
$Alt4ML=switch(rint(2),mathml("($n-$LT)/($n+1)"),mathml("($LT+1)/($n+1)"));@
qu.8.4.uid=d2e54b5a-5681-4be0-83ab-a2858b2942a6@
qu.8.4.info=  Difficulty=0;
  Keyword=uniform;
  Type=MC;
@
qu.8.4.question=<div title="UW Statistics Bank/Discrete Probability Models/Uniform Distributions/Q$Q">In a uniform probability distribution for a discrete random variable X the probability function is given by:
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>2</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$n</mi></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mi mathvariant='normal'>otherwise</mi></mrow></mtd></mtr></mtable></mfenced><mo lspace='0.0em' rspace='0.0em'></mo></mrow></mstyle></math></p>
<p><br />
Then P(X&le;$LT) is:</p>
</div>@
qu.8.4.answer=1@
qu.8.4.choice.1=$AnsML@
qu.8.4.choice.2=$Alt1ML@
qu.8.4.choice.3=$Alt2ML@
qu.8.4.choice.4=$Alt3ML@
qu.8.4.choice.5=$Alt4ML@
qu.8.4.fixed=4@

qu.8.5.mode=Multiple Choice@
qu.8.5.name=01. P(X < 0)@
qu.8.5.comment=<p>The answer is 0 of course since f(x) is non-zero only on the integers 1,...,$n .</p>@
qu.8.5.editing=useHTML@
qu.8.5.solution=@
qu.8.5.algorithm=$Q=1;
$n=range(4,15,1);
$Ans=0;
$Alt1=mathml("1/$n");
$Alt2=switch(rint(3),1,mathml("($n-1)/$n"),mathml("$n/($n+1)"));
$Alt3=switch(rint(2),mathml("($n-2)/($n-1)"),mathml("4/($n+1)"));
$Alt4=switch(rint(2),mathml("1/$n^2"),mathml("($n-1)/$n^2"));@
qu.8.5.uid=8e526324-6f4a-469e-8356-3060b4a83f27@
qu.8.5.info=  Difficulty=0;
  Keyword=uniform;
  Type=MC;
@
qu.8.5.question=<div title="UW Statistics Bank/Discrete Probability Models/Uniform Distributions/Q$Q">In a uniform probability distribution for a discrete random variable X the probability function is given by:
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mfrac linethickness='0'><mrow><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac></mrow><mrow><mn>0</mn></mrow></mfrac><mfrac linethickness='0'><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$n</mi></mrow><mrow><mi>otherwise</mi></mrow></mfrac></mrow></mstyle></math><br />
<br />
Then P(X < 0) is:</p>
</div>@
qu.8.5.answer=1@
qu.8.5.choice.1=$Ans@
qu.8.5.choice.2=$Alt1@
qu.8.5.choice.3=$Alt2@
qu.8.5.choice.4=$Alt3@
qu.8.5.choice.5=$Alt4@
qu.8.5.fixed=4@

qu.8.6.question=<div title="UW Statistics Bank/Discrete Probability Models/Uniform Distributions/Q$Q">A uniform probability distribution <strong><em>on a set of consecutive integers</em></strong> is given by:&nbsp;
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>f</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mfrac linethickness='0'><mrow><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$Gap</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mn>0</mn></mrow></mfrac><mrow><mfrac linethickness='0'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Start</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$StartP1</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>N</mi></mrow><mrow><mi mathvariant='normal'>otherwise</mi></mrow></mfrac></mrow></mrow></mrow></mstyle></math></p>
<p>&nbsp;</p>
<p>What is N ?</p>
</div>@
qu.8.6.answer.num=$End@
qu.8.6.answer.units=@
qu.8.6.showUnits=false@
qu.8.6.grading=exact_value@
qu.8.6.negStyle=minus@
qu.8.6.numStyle=thousands scientific dollars arithmetic@
qu.8.6.mode=Numeric@
qu.8.6.name=04. Find the domain@
qu.8.6.comment=<p>The denominator is your clue. Since this is a Uniform distribution, each point must have the same probability. That value is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$Gap</mi></mrow></mfrac></mrow></mstyle></math>so there must be $Gap points. Since they are consecutive, N = $Start + $Gap - 1 = $End.</p>@
qu.8.6.editing=useHTML@
qu.8.6.solution=@
qu.8.6.algorithm=$Q=4;
$Start=range(-5,5,1);
$StartP1=$Start+1;
$End=range($Start+4,$Start+10,1);
$Gap=$End-$Start+1;@
qu.8.6.uid=aca14d8b-d862-418c-b545-eab024f19a78@
qu.8.6.info=  Difficulty=0;
  Keyword=uniform;
  Type=numeric;
@

qu.9.topic=Moment Generating Functions@

qu.9.1.question=<div title="University of Waterloo Statistics Bank/Discrete Probability Models/Moment Generating Functions/Q$Q">A random variable X has moment generating function <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>M</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi mathvariant='normal'>$p</mi><msup><mi>e</mi><mrow><mi>t</mi></mrow></msup></mrow><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$q</mi><msup><mi>e</mi><mrow><mi>t</mi></mrow></msup></mrow></mfrac></mrow><mrow></mrow></mrow></mstyle></math>. <br />
<p>Find E(X) . (Please answer to 4 decimals of accuracy.)</p>
</div>@
qu.9.1.answer.num=$Ans@
qu.9.1.answer.units=@
qu.9.1.showUnits=false@
qu.9.1.grading=toler_abs@
qu.9.1.err=0.001@
qu.9.1.negStyle=minus@
qu.9.1.numStyle=thousands scientific dollars arithmetic@
qu.9.1.mode=Numeric@
qu.9.1.name=02. Find E(X) from the m.g.f.@
qu.9.1.comment=<p>Recall that E(X) = M'(0)</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>M</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$q</mi><msup><mi>e</mi><mrow><mi>t</mi></mrow></msup></mrow></mfenced><mi mathvariant='normal'>$p</mi><msup><mi>e</mi><mrow><mi>t</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$p</mi><msup><mi>e</mi><mrow><mi>t</mi></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$q</mi><msup><mi>e</mi><mrow><mi>t</mi></mrow></msup></mrow></mfenced></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$q</mi><msup><mi>e</mi><mrow><mi>t</mi></mrow></msup></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>M</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mn>0</mn></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$q</mi></mrow></mfenced><mi mathvariant='normal'>$p</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$p</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$q</mi></mrow></mfenced></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$q</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$p</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>@
qu.9.1.editing=useHTML@
qu.9.1.solution=@
qu.9.1.algorithm=$Q=2;
$p=range(0.1,0.9,0.1);
$q=1-$p;
$Ans=decimal(2,1/$p);@
qu.9.1.uid=16e9a6e8-0986-439e-bf12-e161d45de97a@
qu.9.1.info=  Type=numeric;
  Course=230;
@

qu.9.2.mode=Multiple Choice@
qu.9.2.name=08. E(X<sup>3</sup>) from mgf@
qu.9.2.comment=<p>
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>E</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>X</mi><mrow><mn>3</mn></mrow></msup></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>M</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mn>0</mn></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$C1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>e</mi><mrow><mi>t</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>8</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$C2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>e</mi><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>t</mi></mrow></msup><msub><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Ans</mi></mrow></mstyle></math></p>@
qu.9.2.editing=useHTML@
qu.9.2.solution=@
qu.9.2.algorithm=$Q="08";
$C0=range(0.1,0.4,0.05);
$C1=range(0.1,0.7-$C0,0.05);
$C2=1-$C1-$C0;
$Ans=decimal(2,$C1 + 8*$C2);
$Alt1=decimal(2,range(0.2,0.8,0.01)*$Ans);
$Alt2=decimal(2,range(1.1,1.7,0.01)*$Ans);
$Alt3=decimal(2,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.9.2.uid=10137b35-e81c-4f91-8eeb-fc8d3a10bc0f@
qu.9.2.info=  Type=MC;
@
qu.9.2.question=<div title="University of Waterloo Statistics Bank/Discrete Probability Models/Moment Generating Functions/Q$Q">A random variable X has moment generating function<font size="3" face="Times New Roman"> <em>M</em>(<em>t</em>) = $C0 + $C1 e<em><sup>t</sup></em> + $C2 e<em><sup>2t</sup></em></font>. Which of the following is <font size="3" face="Times New Roman"><em>E</em>(<em>X</em><sup>3</sup>)</font> ?</div>@
qu.9.2.answer=1@
qu.9.2.choice.1=$Ans@
qu.9.2.choice.2=$Alt1@
qu.9.2.choice.3=$Alt2@
qu.9.2.choice.4=$Alt3@
qu.9.2.fixed=@

qu.9.3.question=<div title="University of Waterloo Statistics Bank/Discrete Probability Models/Moment Generating Functions/Q$Q">A random variable X has moment generating function<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>M</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$C0</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$C1</mi><msup><mi>e</mi><mrow><mi>t</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$C2</mi><msup><mi>e</mi><mrow><mn>2</mn><mi>t</mi></mrow></msup></mrow></mstyle></math>&nbsp;. Find E(X). (4 decimal accuracy)</div>@
qu.9.3.answer.num=$Ans@
qu.9.3.answer.units=@
qu.9.3.showUnits=false@
qu.9.3.grading=toler_abs@
qu.9.3.err=.001@
qu.9.3.negStyle=minus@
qu.9.3.numStyle=thousands scientific dollars arithmetic@
qu.9.3.mode=Numeric@
qu.9.3.name=04a. E(X) from mg@
qu.9.3.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>E</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>M</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mn>0</mn></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$C1</mi><msup><mi>e</mi><mrow><mi>t</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mn>2</mn></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$C2</mi></mrow></mfenced><msup><mi>e</mi><mrow><mn>2</mn><mi>t</mi></mrow></msup><msub><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.9.3.editing=useHTML@
qu.9.3.solution=@
qu.9.3.algorithm=$Q="04a";
$C0=range(0.1,0.4,0.05);
$C1=range(0.1,0.7-$C0,0.05);
$C2=1-$C1-$C0;
$Ans=$C1+2*$C2;@
qu.9.3.uid=bf60b21a-a15a-4320-b89f-67a8dd2b5ad1@
qu.9.3.info=  Course=230;
  Type=numeric;
@

qu.9.4.mode=Multiple Choice@
qu.9.4.name=04b. E(X) from mgf@
qu.9.4.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>E</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>M</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mn>0</mn></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$C1</mi><msup><mi>e</mi><mrow><mi>t</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mn>2</mn></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$C2</mi></mrow></mfenced><msup><mi>e</mi><mrow><mn>2</mn><mi>t</mi></mrow></msup><msub><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$EX</mi></mrow></mstyle></math></p>@
qu.9.4.editing=useHTML@
qu.9.4.solution=@
qu.9.4.algorithm=$Q="04b";
$C0=range(0.1,0.4,0.05);
$C1=range(0.1,0.7-$C0,0.05);
$C2=1-$C1-$C0;
$EX=$C1+2*$C2;
$Alt1=decimal(2,range(0.2,0.8,0.01)*$EX);
$Alt2=decimal(2,range(1.1,1.7,0.01)*$EX);
$Alt3=decimal(2,0.5*($EX+switch(rint(2),$Alt1,$Alt2)));@
qu.9.4.uid=64f908dc-2d2b-4458-b8fa-b1f7f9b7cb7e@
qu.9.4.info=  Course=230;
  Type=MC;
@
qu.9.4.question=<div title="University of Waterloo Statistics Bank/Discrete Probability Models/Moment Generating Functions/Q$Q">A random variable X has moment generating function<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>M</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$C0</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$C1</mi><msup><mi>e</mi><mrow><mi>t</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$C2</mi><msup><mi>e</mi><mrow><mn>2</mn><mi>t</mi></mrow></msup></mrow></mstyle></math>&nbsp;. Which of the following is E(X)?</div>@
qu.9.4.answer=1@
qu.9.4.choice.1=$EX@
qu.9.4.choice.2=$Alt1@
qu.9.4.choice.3=$Alt2@
qu.9.4.choice.4=$Alt3@
qu.9.4.fixed=@

qu.9.5.mode=True False@
qu.9.5.name=09. Does same mgf => same pdf?@
qu.9.5.comment=<p>TRUE, since the mgf UNIQUELY identifies the distribution.</p>@
qu.9.5.editing=useHTML@
qu.9.5.solution=@
qu.9.5.algorithm=@
qu.9.5.uid=d59f37b8-dd02-456a-ba28-1fe95b7722d2@
qu.9.5.info=  Algorithmic=no;
  Type=TF;
@
qu.9.5.question=<div title="University of Waterloo Statistics Bank/Discrete Probability Models/Moment Generating Functions/Q9">Two discrete random variables X and Y are known to have exactly the same moment generating function M(t) . This implies that they have exactly the same probability function.</div>@
qu.9.5.answer=1@
qu.9.5.choice.1=True@
qu.9.5.choice.2=False@
qu.9.5.fixed=@

qu.9.6.mode=Multiple Choice@
qu.9.6.name=07b. E(X<sup>2</sup>) from mgf@
qu.9.6.comment=<p><font size="3" face="Times New Roman"><em>E</em>(<em>X</em><sup>2</sup>) = <em>M</em>''(0) = $C1 e<em><sup>t</sup></em> + 4($C2) e<sup>2<em>t</em></sup>|<sub><em>t</em>=0</sub>&nbsp;= $C1 + 4($C2)&nbsp; = $Ans</font></p>@
qu.9.6.editing=useHTML@
qu.9.6.solution=@
qu.9.6.algorithm=$Q="07b";
$C0=range(0.1,0.4,0.05);
$C1=range(0.1,0.7-$C0,0.05);
$C2=1-$C1-$C0;
$Ans=decimal(2,$C1 + 4*$C2);
$Alt1=decimal(2,range(0.2,0.8,0.01)*$Ans);
$Alt2=decimal(2,range(1.1,1.7,0.01)*$Ans);
$Alt3=decimal(2,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.9.6.uid=1d6effcf-e684-4693-8df2-83735b6417bd@
qu.9.6.info=  Type=MC;
@
qu.9.6.question=<div title="University of Waterloo Statistics Bank/Discrete Probability Models/Moment Generating Functions/Q$Q">A random variable X has moment generating function <font size="3" face="Times New Roman"><em>M</em>(<em>t</em>) = $C0 + $C1 e<em><sup>t</sup></em> + $C2 e<sup>2<em>t</em></sup></font>. Which of the following is <font size="3" face="Times New Roman"><em>E</em>(<em>X</em><sup>2</sup>)</font> ?</div>@
qu.9.6.answer=1@
qu.9.6.choice.1=$Ans@
qu.9.6.choice.2=$Alt1@
qu.9.6.choice.3=$Alt2@
qu.9.6.choice.4=$Alt3@
qu.9.6.fixed=@

qu.9.7.mode=Non Permuting Multiple Selection@
qu.9.7.name=03. Which are not mgf?@
qu.9.7.comment=<p><br />
In no particular order:</p>
<ul>
    <li>e<sup>$c t</sup> is actually the mgf for the pdf&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfenced open='{' close='' separators=','><mrow><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>1</mn></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$c</mi></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mi>otherwise</mi></mrow></mtd></mtr></mtable></mfenced></mrow></mfenced></mrow></mrow></mstyle></math></li>
    <li><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>e</mi><mrow><mi mathvariant='normal'>$Lambda</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>e</mi><mrow><mi>t</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced></mrow></msup></mrow></mstyle></math>is the mgf for Poisson($Lambda).</li>
    <li>A Moment generating function must be non-decreasing and M(0)=1. Since M(t)=1-t is decreasing it is not a MGF.</li>
    <li>For M(t) = 1 + t + t<sup>2</sup>/2 we ask the question: is it possible to find a probability distribution for X such that E(e<sup>tX</sup>) = 1 + t + t<sup>2</sup>/2 ? If so: <br />
    <br />
    E(X) = M'(0) = 1 and <br />
    var(X) = M''(0) - [M'(0)]<sup>2</sup> = 1 -1 = 0<br />
    <br />
    Since the variance is 0 and the mean is 1, X must be the constant 1. But if so the mgf is actually M(t) = e<sup>t</sup> &ne; 1 + t + t<sup>2</sup>/2, a contradiction.</li>
</ul>@
qu.9.7.editing=useHTML@
qu.9.7.solution=@
qu.9.7.algorithm=$Q=3;
$Lambda=range(2,5,1);
$c=range(2,10,1);@
qu.9.7.uid=ac1a6954-c697-48b9-81c9-e2ad53fde271@
qu.9.7.info=  Type=MS;
  Course=230;
@
qu.9.7.question=<div title="University of Waterloo Statistics Bank/Discrete Probability Models/Moment Generating Functions/Q$Q">
Which of the following functions cannot be moment generating functions? (NOTE: Select ALL choices which are NOT moment generating functions, there may be more than 1.)</div>@
qu.9.7.answer=2, 4@
qu.9.7.choice.1=M(t) = e<sup>$c t</sup>@
qu.9.7.choice.2=M(t) = 1 - t@
qu.9.7.choice.3=M(t) = e<sup>$Lambda(e<sup>t</sup>-1)</sup>@
qu.9.7.choice.4=M(t) = 1 + t + t<sup>2</sup>/2@
qu.9.7.fixed=@

qu.9.8.question=<div title="University of Waterloo Statistics Bank/Discrete Probability Models/Moment Generating Functions/Q$Q">
A random variable X has moment generating function <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>M</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi mathvariant='normal'>$p</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>e</mi><mrow><mi>t</mi></mrow></msup></mrow><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>e</mi><mrow><mi>t</mi></mrow></msup></mrow></mfrac></mrow><mrow></mrow></mrow></mstyle></math>.&nbsp;
<p>By expanding this as a series expansion in powers of e<sup>t</sup> find P(X = $X) (Please answer to 3 decimals of accuracy.)</p>
</div>@
qu.9.8.answer.num=$Ans@
qu.9.8.answer.units=@
qu.9.8.showUnits=false@
qu.9.8.grading=toler_abs@
qu.9.8.err=0.01@
qu.9.8.negStyle=minus@
qu.9.8.numStyle=thousands scientific dollars arithmetic@
qu.9.8.mode=Numeric@
qu.9.8.name=01. Find P(X=n) from the m.g.f.@
qu.9.8.comment=<p>The trick here is to recall the sum of an infinite Geometric Series:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>s</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow><mrow><mi>&infin;</mi></mrow></munderover><msup><mi>ar</mi><mrow><mi>k</mi></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>a</mi><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>r</mi></mrow></mfrac></mrow></mrow></mstyle></math></p>
<p>In our question a = $p e<sup>t</sup> and 1 - r = 1 - $q e<sup>t</sup>&nbsp; (i.e. r=$q e<sup>t</sup>) so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>M</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi mathvariant='normal'>$p</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>e</mi><mrow><mi>t</mi></mrow></msup></mrow><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>e</mi><mrow><mi>t</mi></mrow></msup></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>ar</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>ar</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$p</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>e</mi><mrow><mi>t</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$p</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>e</mi><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>t</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$p</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>$q</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>e</mi><mrow><mn>3</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>t</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math></p>
<p>However we also know that if X takes on non-negative integer values, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>M</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>E</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>e</mi><mrow><mi>xt</mi></mrow></msup></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>j</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow><mrow><mi>&infin;</mi></mrow></munderover><msup><mi>e</mi><mrow><mi>jt</mi></mrow></msup><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>j</mi></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mfenced></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>e</mi><mrow><mi>t</mi></mrow></msup><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>e</mi><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>t</mi></mrow></msup><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>2</mn></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>e</mi><mrow><mn>3</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>t</mi></mrow></msup><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>3</mn></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math></p>
<p>Setting like powers of e<sup>t</sup> equal: P(X=$X) = $p*$Match = $Ans</p>@
qu.9.8.editing=useHTML@
qu.9.8.hint.1=Realize that the given expression is the sum of a geometric series.@
qu.9.8.solution=@
qu.9.8.algorithm=$Q=1;
$p=range(0.1,0.9,0.1);
$q=1-$p;
$X=range(1,3,1);
$Match=switch($X,1,1,$q,mathml("$q^2"));
$Ans=$p*$q^($X-1);@
qu.9.8.uid=330a8c50-3cbe-4ca8-bf2a-acdfa92128ec@
qu.9.8.info=  Type=numeric;
@

qu.9.9.question=<div title="University of Waterloo Statistics Bank/Discrete Probability Models/Moment Generating Functions/Q$Q">A random variable X has moment generating function <font size="3" face="Times New Roman"><em>M</em>(<em>t</em>) = $C0 + $C1 e<em><sup>t</sup></em> + $C2 e<sup>2<em>t</em></sup></font>. Find <font size="3" face="Times New Roman"><em>E</em>(<em>X</em><sup>2</sup>)</font> (4 decimal accuracy)</div>@
qu.9.9.answer.num=$Ans@
qu.9.9.answer.units=@
qu.9.9.showUnits=false@
qu.9.9.grading=toler_abs@
qu.9.9.err=.001@
qu.9.9.negStyle=minus@
qu.9.9.numStyle=thousands scientific dollars arithmetic@
qu.9.9.mode=Numeric@
qu.9.9.name=07a. E(X<sup>2</sup>) from mgf@
qu.9.9.comment=<p><font size="3" face="Times New Roman"><em>E</em>(<em>X</em><sup>2</sup>) = <em>M</em>''(0) = $C1 e<em><sup>t</sup></em> + 4($C2) e<sup>2<em>t</em></sup>|<sub><em>t</em>=0</sub>&nbsp;= $C1 + 4($C2)&nbsp; = $Ans</font></p>@
qu.9.9.editing=useHTML@
qu.9.9.solution=@
qu.9.9.algorithm=$Q="07a";
$C0=range(0.1,0.4,0.05);
$C1=range(0.1,0.7-$C0,0.05);
$C2=1-$C1-$C0;
$Ans=decimal(4,$C1 + 4*$C2);@
qu.9.9.uid=68a76c8d-e896-4f53-8b75-743ba8df659d@
qu.9.9.info=  Type=numeric;
@

qu.9.10.mode=True False@
qu.9.10.name=05. Same mgf's => Same Var(X)'s ?@
qu.9.10.comment=<p>TRUE, since <font size="3" face="Times New Roman">Var(<em>X</em>) = M''(0) - (M'(0))&sup2;</font> and they have the same mgf.</p>@
qu.9.10.editing=useHTML@
qu.9.10.solution=@
qu.9.10.algorithm=@
qu.9.10.uid=9aac3e31-d454-49de-bc34-0a65cc2af235@
qu.9.10.info=  Course=230;
  Algorithmic=no;
  Type=TF;
@
qu.9.10.question=<div title="University of Waterloo Statistics Bank/Discrete Probability Models/Moment Generating Functions/Q05">Two discrete random variables X and Y are known to have exactly the same moment generating function M(t) . This implies that Var(X) = Var(Y).</div>@
qu.9.10.answer=1@
qu.9.10.choice.1=True@
qu.9.10.choice.2=False@
qu.9.10.fixed=@

qu.9.11.mode=Multiple Choice@
qu.9.11.name=06. Var(X) from mgf@
qu.9.11.comment=<p>Var(X) = M''(0) - [M'(0)]<sup>2</sup> = $C1 e<sup>t</sup> + 4*$C2 e<sup>2t</sup>|<sub>t=0</sub> - &nbsp; [$C1 e<sup>t</sup> + 2*$C2 e<sup>2t</sup> |<sub>t=0</sub> ]<sup>2</sup></p>
<p>= $C1 + 4*$C2 - [$C1 + 2*$C2]<sup>2</sup> = $VarX</p>@
qu.9.11.editing=useHTML@
qu.9.11.solution=@
qu.9.11.algorithm=$Q="06";
$C0=range(0.1,0.4,0.05);
$C1=range(0.1,0.7-$C0,0.05);
$C2=1-$C1-$C0;
$VarX=decimal(2,$C1 + 4*$C2 - ($C1 + 2*$C2)^2);
$Alt1=decimal(2,range(0.2,0.8,0.01)*$VarX);
$Alt2=decimal(2,range(1.1,1.7,0.01)*$VarX);
$Alt3=decimal(2,0.5*($VarX+switch(rint(2),$Alt1,$Alt2)));@
qu.9.11.uid=bbf35fb5-7b11-4074-9243-1bcd57bbadf1@
qu.9.11.info=  Type=MC;
@
qu.9.11.question=<div title="University of Waterloo Statistics Bank/Discrete Probability Models/Moment Generating Functions/Q$Q">A random variable X has moment generating function <font size="3" face="Times New Roman"><em>M</em>(<em>t</em>) = $C0 + $C1 e<em><sup>t</sup></em> + $C2 e<sup>2<em>t</em></sup></font>. Which of the following is Var(X)?</div>@
qu.9.11.answer=1@
qu.9.11.choice.1=$VarX@
qu.9.11.choice.2=$Alt1@
qu.9.11.choice.3=$Alt2@
qu.9.11.choice.4=$Alt3@
qu.9.11.fixed=@

qu.9.12.mode=True False@
qu.9.12.name=10. Same mgfs but different E()?@
qu.9.12.comment=<p>FALSE, Since E(X) = M'(0) and they have the same mgf so E(Y) = M'(0) = E(X).</p>@
qu.9.12.editing=useHTML@
qu.9.12.solution=@
qu.9.12.algorithm=@
qu.9.12.uid=efd03938-2c85-4e75-807f-c7433a704be5@
qu.9.12.info=  Type=TF;
  Algorithmic=no;
@
qu.9.12.question=<div title="University of Waterloo Statistics Bank/Discrete Probability Models/Moment Generating Functions/Q10">
Two discrete random variables X and Y are known to have exactly the same moment generating function M(t) . Then it is still possible that E(X) and E(Y) are different.</div>@
qu.9.12.answer=2@
qu.9.12.choice.1=True@
qu.9.12.choice.2=False@
qu.9.12.fixed=@

