qu.1.topic=Discrete Random Variables@

qu.1.1.mode=Inline@
qu.1.1.name=Identify type of discrete random variable@
qu.1.1.comment=@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$k1=rint(17);
$k2=rint(17);
$k3=rint(17);
$k4=rint(17);
$k5=rint(17);
$z=maple("S := $k1,$k2,$k3,$k4,$k5:
floor( nops({S})/nops([S]) )");
condition: $z;
$a=("'The number of complaints a customer service representative receives in a given day.'");
$b=("'A count of the number of dandelion weeds in a square metre of yard space.'");
$c=("'The number of bacteria per millilitre in a test tube.'");
$d=("'The number of cracks in a 5 metre stretch of sidewalk.'");
$e=("'A count of correct multiple choice answers a student got in a test of 20 questions, if they were guessing randomly at each question and each question had the same number of answer options.'");
$f=("'The number of winning coffee cups in a randomly selected sample of 75 cups.'");
$g=("'The number of defective tires out of 50 that were randomly selected by the quality control inspector.'");
$h=("'A count of the number of allergic reactions to a vaccine administered to a nursery of 15 infants.'");
$i=("'The first male kitten in a litter will be the third one born.'");
$j=("'The fourth sample of fungi exposed to a fungicide will be the first one to react to it.'");
$k=("'An amateur baseball player will get his first home run on his second batting chance.'");
$l=("'The first Hat Trick to be scored in a hockey playoff series will happen in the fifth game.'");
$m=("'The number of students who are in the B.Sc program in a sample of 15 randomly selected students, when 10 of the 30 students in the lab are registered in the B.Sc program.'");
$n=("'The number of females in a random sample of 10 students, when it is known that half of the 50 students in the course are women.'");
$o=("'A count of chipped dinner plates in a randomly selected sample of 5 plates, when in total only 2 of the 20 plates are actually chipped.'");
$p=("'The number of trays of blight-infected tomato seedlings in a random sample of 25, when 10 of the 70 tomato seedling trays in the nursery are blight-infected.'");
$q=("'The number of rotten blueberries in a random sample of 8 blueberries from a box of 50, when exactly 10% of the box of 50 blueberries are rotten.'");
$Answers=["'Poisson'","'Poisson'","'Poisson'","'Poisson'","'Binomial'","'Binomial'","'Binomial'","'Binomial'","'Geometric'","'Geometric'","'Geometric'","'Geometric'","'Hypergeometric'","'Hypergeometric'","'Hypergeometric'","'Hypergeometric'","'Hypergeometric'"];
$Dist1=["'Binomial'","'Binomial'","'Binomial'","'Binomial'","'Geometric'","'Geometric'","'Geometric'","'Geometric'","'Hypergeometric'","'Hypergeometric'","'Hypergeometric'","'Hypergeometric'","'Poisson'","'Poisson'","'Poisson'","'Poisson'","'Poisson'"];
$Dist2=["'Geometric'","'Geometric'","'Geometric'","'Geometric'","'Hypergeometric'","'Hypergeometric'","'Hypergeometric'","'Hypergeometric'","'Poisson'","'Poisson'","'Poisson'","'Poisson'","'Binomial'","'Binomial'","'Binomial'","'Binomial'","'Binomial'"];
$Dist3=["'Hypergeometric'","'Hypergeometric'","'Hypergeometric'","'Hypergeometric'","'Poisson'","'Poisson'","'Poisson'","'Poisson'","'Binomial'","'Binomial'","'Binomial'","'Binomial'","'Geometric'","'Geometric'","'Geometric'","'Geometric'","'Geometric'"];
$Q1=switch($k1, $a,$b,$c,$d,$e,$f,$g,$h,$i,$j,$k,$l,$m,$n,$o,$p,$q);
$A1=switch($k1, $Answers);
$D11=switch($k1, $Dist1);
$D12=switch($k1, $Dist2);
$D13=switch($k1, $Dist3);
$Q2=switch($k2, $a,$b,$c,$d,$e,$f,$g,$h,$i,$j,$k,$l,$m,$n,$o,$p,$q);
$A2=switch($k2, $Answers);
$D21=switch($k2, $Dist1);
$D22=switch($k2, $Dist2);
$D23=switch($k2, $Dist3);
$Q3=switch($k3, $a,$b,$c,$d,$e,$f,$g,$h,$i,$j,$k,$l,$m,$n,$o,$p,$q);
$A3=switch($k3, $Answers);
$D31=switch($k3, $Dist1);
$D32=switch($k3, $Dist2);
$D33=switch($k3, $Dist3);
$Q4=switch($k4, $a,$b,$c,$d,$e,$f,$g,$h,$i,$j,$k,$l,$m,$n,$o,$p,$q);
$A4=switch($k4, $Answers);
$D41=switch($k4, $Dist1);
$D42=switch($k4, $Dist2);
$D43=switch($k4, $Dist3);
$Q5=switch($k5, $a,$b,$c,$d,$e,$f,$g,$h,$i,$j,$k,$l,$m,$n,$o,$p,$q);
$A5=switch($k5, $Answers);
$D51=switch($k5, $Dist1);
$D52=switch($k5, $Dist2);
$D53=switch($k5, $Dist3);@
qu.1.1.uid=3ce0fcfc-9c2b-40d6-84bd-881e3c0d3aa6@
qu.1.1.info=  Course=Introductory Statistics;
  Topic=Discrete Distributions;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.1.1.weighting=1,1,1,1,1@
qu.1.1.numbering=alpha@
qu.1.1.part.1.grader=exact@
qu.1.1.part.1.name=sro_id_1@
qu.1.1.part.1.editing=useHTML@
qu.1.1.part.1.display.permute=true@
qu.1.1.part.1.answer.4=$D13@
qu.1.1.part.1.answer.3=$D12@
qu.1.1.part.1.question=(Unset)@
qu.1.1.part.1.answer.2=$D11@
qu.1.1.part.1.answer.1=$A1@
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qu.1.1.part.2.editing=useHTML@
qu.1.1.part.2.display.permute=true@
qu.1.1.part.2.answer.4=$D23@
qu.1.1.part.2.answer.3=$D22@
qu.1.1.part.2.question=(Unset)@
qu.1.1.part.2.answer.2=$D21@
qu.1.1.part.2.answer.1=$A2@
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qu.1.1.part.3.editing=useHTML@
qu.1.1.part.3.display.permute=true@
qu.1.1.part.3.answer.4=$D33@
qu.1.1.part.3.answer.3=$D32@
qu.1.1.part.3.question=(Unset)@
qu.1.1.part.3.answer.2=$D31@
qu.1.1.part.3.answer.1=$A3@
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qu.1.1.part.4.name=sro_id_4@
qu.1.1.part.4.editing=useHTML@
qu.1.1.part.4.display.permute=true@
qu.1.1.part.4.answer.4=$D43@
qu.1.1.part.4.answer.3=$D42@
qu.1.1.part.4.question=(Unset)@
qu.1.1.part.4.answer.2=$D41@
qu.1.1.part.4.answer.1=$A4@
qu.1.1.part.4.mode=List@
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qu.1.1.part.5.display.permute=true@
qu.1.1.part.5.answer.4=$D53@
qu.1.1.part.5.answer.3=$D52@
qu.1.1.part.5.question=(Unset)@
qu.1.1.part.5.answer.2=$D51@
qu.1.1.part.5.answer.1=$A5@
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qu.1.1.question=<div style="margin: 0cm 0cm 10pt"><p>For each of the following, determine whether the random variable is a Binomial, Poisson, Geometric, or Hypergeometric random variable.</p><p>&nbsp;</p><p>(N.B.&nbsp;In some of these scenarios, the random variable may not follow any of the distributions perfectly&nbsp;- choose the distribution that would be the most appropriate.)</p><p>&nbsp;</p><p>a)&nbsp;<span>&nbsp;</span><1><span>&nbsp;</span> $Q1</p><p>&nbsp;</p><p>b)&nbsp;<span>&nbsp;</span><2><span>&nbsp;</span> $Q2</p><p>&nbsp;</p><p>c)&nbsp;<span>&nbsp;</span><3><span>&nbsp;</span> $Q3</p><p>&nbsp;</p><p>d)&nbsp;<span>&nbsp;</span><4><span>&nbsp;</span> $Q4</p><p>&nbsp;</p><p>e)&nbsp;<span>&nbsp;</span><5><span>&nbsp; $Q5</span></p></div>@

qu.2.topic=Discrete Probability Distribution@

qu.2.1.mode=Inline@
qu.2.1.name=Determine probability values, calculate standard deviation@
qu.2.1.comment=<p>a)&nbsp; To find <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='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2.5</mn></mrow></mfenced></mrow></mstyle></math>, you first need to find the value of <em>C</em>.&nbsp; Recall that the probabilities must sum to 1, so $a C + $b C + $c C + $d C = 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><mo lspace='0.0em' rspace='0.0em' stretchy='true' accent='true'>&rArr;</mo></mrow></mstyle></math>10<em>C&nbsp;</em>= 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><mo lspace='0.0em' rspace='0.0em' stretchy='true' accent='true'>&rArr;</mo></mrow></mstyle></math><em>C</em> = 0.1.</p>
<p>Thefore, <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='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2.5</mn></mrow></mfenced></mrow></mstyle></math>&nbsp;= $c * 0.1 = $p3.</p>
<p>&nbsp;</p>
<p>b)&nbsp; To find the standard deviation, you first must find E[<em>X</em>] using the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>E</mi><mfenced open='[' close=']' separators=','><mrow><mi>X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi mathcolor='#800080'>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mn>4</mn></mrow></munderover><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>p</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>4.5</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$p1</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>3.5</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$p2</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2.5</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$p3</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1.5</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$p4</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$mean</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p>Then the variance can be found, using the formula:</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>Var</mi><mfenced open='[' close=']' separators=','><mrow><mi>X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi mathcolor='#800080'>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mn>4</mn></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mrow><mi>&mu;</mi></mrow><mrow><mi>X</mi></mrow></msub></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>p</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><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><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><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><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><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>4.5</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$mean</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$p1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>3.5</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$mean</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$p2</mi><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><msup><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2.5</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$mean</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$p3</mi><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><msup><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1.5</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$mean</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$p4</mi><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><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><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><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><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><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$var</mi><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p>Finally, the standard deviation is simply the square root of the variance, so <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>SD</mi><mfenced open='[' close=']' separators=','><mrow><mi>X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mi>$var</mi></mrow></msqrt><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>$StdDev</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>@
qu.2.1.editing=useHTML@
qu.2.1.solution=@
qu.2.1.algorithm=$a=rint(1,8);
$b=rint(1,9-$a);
$c=rint(1,10-($a+$b));
$d=10-($a+$b+$c);
$p1=$a/10;
$p2=$b/10;
$p3=$c/10;
$p4=$d/10;
$mean=(-4.5*$p1)+(-3.5*$p2)+(-2.5*$p3)+(-1.5*$p4);
$var=(((-4.5)^2*$p1)+((-3.5)^2*$p2)+((-2.5)^2*$p3)+((-1.5)^2*$p4))-$mean^2;
$StdDev=sqrt($var);@
qu.2.1.uid=675ff96e-4360-45bf-a012-ebbcc43317fc@
qu.2.1.info=  Course=Introductory Statistics;
  Topic=Discrete Probability Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.2.1.weighting=1,1@
qu.2.1.numbering=alpha@
qu.2.1.part.1.name=sro_id_1@
qu.2.1.part.1.answer.units=@
qu.2.1.part.1.numStyle=   @
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qu.2.1.part.1.grading=exact_value@
qu.2.1.part.1.negStyle=both@
qu.2.1.part.1.answer.num=$p3@
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qu.2.1.part.2.negStyle=both@
qu.2.1.part.2.answer.num=$StdDev@
qu.2.1.question=<p>Consider the following discrete probability distribution:</p><p>&nbsp;</p><p><table border="1" cellspacing="1" cellpadding="1" width="200" align="center">    <tbody>        <tr>            <td><em>X</em></td>            <td>-4.5</td>            <td>-3.5</td>            <td>-2.5</td>            <td>-1.5</td>        </tr>        <tr>            <td><em>P(X)</em></td>            <td>$a C</td>            <td>$b C</td>            <td>$c C</td>            <td>$d C</td>        </tr>    </tbody></table></p><p>&nbsp;</p><p>&nbsp;</p><p>a)&nbsp; What 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>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><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2.5</mn></mrow></mfenced></mrow></mstyle></math>?&nbsp;</p><p>&nbsp;</p><p>Your answer should be a numerical value that does not include <em>C</em>.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b) What is the standard deviation of <em>X</em>?&nbsp; </span></p><p>&nbsp;</p><p><span>Your answer should be a numerical value that does not include C.</span></p><p><span>Round your response to&nbsp;at least 3 decimal places.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.2.2.mode=Inline@
qu.2.2.name=Determine missing probability, calculate mean, variance for X@
qu.2.2.comment=<p>a)&nbsp; To determine the missing value, remember that the sum of the probabilities must equal 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 mathcolor='#800080'>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>p</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p>Therefore, $p1 + $p2 + $p3 + $p4 + <em>C</em> = 1 <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' accent='true'>&rArr;</mo></mrow></mstyle></math>1 - ($p1 + $p2 + $p3 + $p4) = <em>C</em> = $p5</p>
<p>&nbsp;</p>
<p>b)&nbsp; To calculate E[<em>X</em>], 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>E</mi><mfenced open='[' close=']' separators=','><mrow><mi>X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi mathcolor='#800080'>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>p</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow></mstyle></math>.&nbsp; Since <em>n</em> = 5, we get <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><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi mathcolor='#800080'>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mn>5</mn></mrow></munderover><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>p</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2.25</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$p1</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$p2</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mn>0.25</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$p3</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mn>1.5</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$p4</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mn>2.5</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$p5</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$mean</mi><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>
<p>c)&nbsp; To calculate Var[X], 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>Var</mi><mfenced open='[' close=']' separators=','><mrow><mi>X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi mathcolor='#800080'>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msub><mrow><mi>&mu;</mi></mrow><mrow><mi>X</mi></mrow></msub></mrow></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>p</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.&nbsp; Here, <em>n</em> = 5, 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><msub><mrow><mi>&mu;</mi></mrow><mrow><mi>X</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>E</mi><mfenced open='[' close=']' separators=','><mrow><mi>X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$mean</mi></mrow></mstyle></math>, so we get <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>Var</mi><mfenced open='[' close=']' separators=','><mrow><mi>X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi mathcolor='#800080'>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mn>5</mn></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$mean</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>p</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2.25</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$mean</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$p1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mn>0</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$mean</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$p2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mn>0.25</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$mean</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$p3</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mn>1.5</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$mean</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$p4</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mn>2.5</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$mean</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$p5</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$var</mi><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math></p>@
qu.2.2.editing=useHTML@
qu.2.2.solution=@
qu.2.2.algorithm=$p1=rand(0.1, 0.2, 2);
$p2=rand(0.1, 0.2, 2);
$p3=rand(0.1, 0.3, 2);
$p4=rand(0.1, 0.2, 2);
$p5=1-($p1+$p2+$p3+$p4);
condition:gt($p5,0);
$mean=(-2.25*$p1)+(0*$p2)+(0.25*$p3)+(1.5*$p4)+(2.5*$p5);
$var=((-2.25^2*$p1)+(0^2*$p2)+(0.25^2*$p3)+(1.5^2*$p4)+(2.5^2*$p5))-$mean^2;@
qu.2.2.uid=da565180-f7f1-43a0-adec-4f57e6059e58@
qu.2.2.info=  Course=Introductory Statistics;
  Topic=Discrete Probability Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.2.2.weighting=1,1,1@
qu.2.2.numbering=alpha@
qu.2.2.part.1.name=sro_id_1@
qu.2.2.part.1.answer.units=@
qu.2.2.part.1.numStyle=   @
qu.2.2.part.1.editing=useHTML@
qu.2.2.part.1.showUnits=false@
qu.2.2.part.1.question=(Unset)@
qu.2.2.part.1.mode=Numeric@
qu.2.2.part.1.grading=exact_value@
qu.2.2.part.1.negStyle=both@
qu.2.2.part.1.answer.num=$p5@
qu.2.2.part.2.name=sro_id_2@
qu.2.2.part.2.answer.units=@
qu.2.2.part.2.numStyle=   @
qu.2.2.part.2.editing=useHTML@
qu.2.2.part.2.showUnits=false@
qu.2.2.part.2.err=0.01@
qu.2.2.part.2.question=(Unset)@
qu.2.2.part.2.mode=Numeric@
qu.2.2.part.2.grading=toler_abs@
qu.2.2.part.2.negStyle=both@
qu.2.2.part.2.answer.num=$mean@
qu.2.2.part.3.name=sro_id_3@
qu.2.2.part.3.answer.units=@
qu.2.2.part.3.numStyle=   @
qu.2.2.part.3.editing=useHTML@
qu.2.2.part.3.showUnits=false@
qu.2.2.part.3.err=0.01@
qu.2.2.part.3.question=(Unset)@
qu.2.2.part.3.mode=Numeric@
qu.2.2.part.3.grading=toler_abs@
qu.2.2.part.3.negStyle=both@
qu.2.2.part.3.answer.num=$var@
qu.2.2.question=<p>Consider the following discrete probability distribution:</p><p>&nbsp;</p><p><table border="1" cellspacing="1" cellpadding="1" width="200" align="center">    <tbody>        <tr>            <td><em>X</em></td>            <td>-2.25</td>            <td>0</td>            <td>0.25</td>            <td>1.5</td>            <td>2.5</td>        </tr>        <tr>            <td><em>P(X)</em></td>            <td>$p1</td>            <td>$p2</td>            <td>$p3</td>            <td>$p4</td>            <td><em>C</em></td>        </tr>    </tbody></table></p><p>&nbsp;</p><p>&nbsp;</p><p>a) What is the missing value, <em>C</em>?</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p>b)&nbsp; What is E[<em>X</em>]?&nbsp; Your answer should be a numeric value that does not include <em>C</em>.</p><p>&nbsp;</p><p>Round your answer to&nbsp;at least 3 decimal places.</p><p><span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>c)&nbsp; What is Var[<em>X</em>]?&nbsp; Your answer should be a numeric value that does not include <em>C</em>.</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>Round your answer to&nbsp;at least 3 decimal places.</span></p><p><span><span>&nbsp;</span><3><span>&nbsp;</span></span></p>@

qu.2.3.mode=Inline@
qu.2.3.name=Calculate P(X = x | X < y)@
qu.2.3.comment=<p>a)&nbsp; The <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='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>0</mn></mrow></mfenced></mrow></mstyle></math>&nbsp;can be read directly from the discrete probability distribution table, and is equal to $p2.</p>
<p>&nbsp;</p>
<p>b)&nbsp; To calculate <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>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>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><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'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>5</mn><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mstyle></math>, use the general 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>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><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><mfrac><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi></mrow></mfenced></mrow><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.&nbsp; Therefore, we are calculating <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><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><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'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>5</mn></mrow></mfenced></mrow><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'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>5</mn></mrow></mfenced></mrow></mfrac></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>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>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><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'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>5</mn></mrow></mfenced></mrow></mstyle></math>&nbsp;is the sum of the probabilities for <em>X</em> values that are equal to 0 <strong>and</strong> less than 5.&nbsp; In this case, <em>X</em> = 0 and <em>X</em> < 5 only intersect at the point <em>X</em> = 0, 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>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>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><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'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>5</mn></mrow></mfenced><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>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>0</mn></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$p2</mi></mrow></mstyle></math>.&nbsp; To determine <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='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>5</mn></mrow></mfenced></mrow></mstyle></math>, it is the sum of the probabilities for <em>X</em> values that are less than 5, 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>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'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>5</mn></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$p1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$p2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$p3</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ProbDenom</mi></mrow></mstyle></math>.</p>
<p>Therefore, <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>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>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><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'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>5</mn><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><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><mfrac><mi>$ProbNum</mi><mrow><mi>$ProbDenom</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$CondProb</mi></mrow></mstyle></math></p>@
qu.2.3.editing=useHTML@
qu.2.3.solution=@
qu.2.3.algorithm=$p1=rand(0.1, 0.2, 2);
$p2=rand(0.1, 0.2, 2);
$p3=rand(0.1, 0.25, 2);
$p4=rand(0.1, 0.25,2);
$p5=1-($p1+$p2+$p3+$p4);
condition:gt($p5,0);
$ProbNum=$p2;
$ProbDenom=$p1+$p2+$p3;
$CondProb=$ProbNum/$ProbDenom;@
qu.2.3.uid=a50e64ad-180a-462f-8323-5eb34d326d4b@
qu.2.3.info=  Course=Introductory Statistics;
  Topic=Discrete Probability Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.2.3.weighting=1,1@
qu.2.3.numbering=alpha@
qu.2.3.part.1.name=sro_id_1@
qu.2.3.part.1.answer.units=@
qu.2.3.part.1.numStyle=   @
qu.2.3.part.1.editing=useHTML@
qu.2.3.part.1.showUnits=false@
qu.2.3.part.1.question=(Unset)@
qu.2.3.part.1.mode=Numeric@
qu.2.3.part.1.grading=exact_value@
qu.2.3.part.1.negStyle=both@
qu.2.3.part.1.answer.num=$p2@
qu.2.3.part.2.name=sro_id_2@
qu.2.3.part.2.answer.units=@
qu.2.3.part.2.numStyle=   @
qu.2.3.part.2.editing=useHTML@
qu.2.3.part.2.showUnits=false@
qu.2.3.part.2.err=0.01@
qu.2.3.part.2.question=(Unset)@
qu.2.3.part.2.mode=Numeric@
qu.2.3.part.2.grading=toler_abs@
qu.2.3.part.2.negStyle=both@
qu.2.3.part.2.answer.num=$CondProb@
qu.2.3.question=<p>Consider the following discrete probability distribution:</p><p>&nbsp;</p><p><table border="1" cellspacing="1" cellpadding="1" width="200" align="center">    <tbody>        <tr>            <td><em>X</em></td>            <td>-1.3</td>            <td>0</td>            <td>2.1</td>            <td>5.4</td>            <td>6.2</td>        </tr>        <tr>            <td><em>P(X)</em></td>            <td>$p1</td>            <td>$p2</td>            <td>$p3</td>            <td>$p4</td>            <td>$p5</td>        </tr>    </tbody></table></p><p>&nbsp;</p><p>&nbsp;</p><p>a)&nbsp; What 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>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>0</mn></mrow></mfenced></mrow></mstyle></math>?</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; What 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>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><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>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><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'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>5</mn><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mstyle></math>?</span></p><p>&nbsp;</p><p><span>Round your answer to at least&nbsp;3 decimal places.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.2.4.mode=Inline@
qu.2.4.name=Calculate Mean, Variance of X@
qu.2.4.comment=<p>a)&nbsp; To calculate the expected value&nbsp;of <em>X</em>, E[<em>X</em>], use the formula:</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>E</mi><mfenced open='[' close=']' separators=','><mrow><mi>X</mi></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 mathcolor='#800080'>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>p</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p>Here, <em>n</em> = 4, since there are 4 terms in the discrete probability distribution, and the formula becomes:</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>E</mi><mfenced open='[' close=']' separators=','><mrow><mi>X</mi></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 mathcolor='#800080'>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mn>4</mn></mrow></munderover><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>p</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$x1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0.1</mn></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$x2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0.4</mn></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$x3</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0.3</mn></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$x4</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0.2</mn></mrow></mfenced></mrow></mstyle></math>=$mean</p>
<p>&nbsp;</p>
<p>b)&nbsp; To calculate the variance of <em>X</em>, Var[<em>X</em>], use the formula:</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>Var</mi><mfenced open='[' close=']' separators=','><mrow><mi>X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi mathcolor='#800080'>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mn>4</mn></mrow></munderover></mrow><msup><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mrow><mi>&mu;</mi></mrow><mrow><mi>X</mi></mrow></msub></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>p</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></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><msub><mrow><mi>&mu;</mi></mrow><mrow><mi>X</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>E</mi><mfenced open='[' close=']' separators=','><mrow><mi>X</mi></mrow></mfenced></mrow></mstyle></math></p>
<p>This becomes:</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>Var</mi><mfenced open='[' close=']' separators=','><mrow><mi>X</mi></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 mathcolor='#800080'>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mi></mi></munderover><msup><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>E</mi><mfenced open='[' close=']' separators=','><mrow><mi>X</mi></mrow></mfenced></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>p</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$x1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$mean</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0.1</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$x2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$mean</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0.4</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$x3</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$mean</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0.3</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$x4</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$mean</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0.2</mn></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p>= $var</p>@
qu.2.4.editing=useHTML@
qu.2.4.solution=@
qu.2.4.algorithm=$x1=rand(-1.0, -0.5, 2);
$x2=rand(-0.5, 0.5, 2);
$x3=rand(0.5, 1.0, 2);
$x4=rand(1.0, 1.5, 2);
$mean=(0.1*$x1)+(0.4*$x2)+(0.3*$x3)+(0.2*$x4);
$var=(($x1^2*0.1)+($x2^2*0.4)+($x3^2*0.3)+($x4^2*0.2))-$mean^2;@
qu.2.4.uid=7a37a2a0-c922-4b4c-8270-ec1fb302e5df@
qu.2.4.info=  Course=Introductory Statistics;
  Topic=Discrete Probability Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.2.4.weighting=1,1@
qu.2.4.numbering=alpha@
qu.2.4.part.1.name=sro_id_1@
qu.2.4.part.1.answer.units=@
qu.2.4.part.1.numStyle=   @
qu.2.4.part.1.editing=useHTML@
qu.2.4.part.1.showUnits=false@
qu.2.4.part.1.err=0.01@
qu.2.4.part.1.question=(Unset)@
qu.2.4.part.1.mode=Numeric@
qu.2.4.part.1.grading=toler_abs@
qu.2.4.part.1.negStyle=both@
qu.2.4.part.1.answer.num=$mean@
qu.2.4.part.2.name=sro_id_2@
qu.2.4.part.2.answer.units=@
qu.2.4.part.2.numStyle=   @
qu.2.4.part.2.editing=useHTML@
qu.2.4.part.2.showUnits=false@
qu.2.4.part.2.err=0.01@
qu.2.4.part.2.question=(Unset)@
qu.2.4.part.2.mode=Numeric@
qu.2.4.part.2.grading=toler_abs@
qu.2.4.part.2.negStyle=both@
qu.2.4.part.2.answer.num=$var@
qu.2.4.question=<p>Consider the following discrete probability distribution:</p><p>&nbsp;</p><p><table border="1" cellspacing="1" cellpadding="1" width="200" align="center">    <tbody>        <tr>            <td><em>X</em></td>            <td>$x1</td>            <td>$x2</td>            <td>$x3</td>            <td>$x4</td>        </tr>        <tr>            <td><em>P(X)</em></td>            <td>0.1</td>            <td>0.4</td>            <td>0.3</td>            <td>0.2</td>        </tr>    </tbody></table></p><p>&nbsp;</p><p>a)&nbsp; What is E[<em>X</em>]?</p><p>&nbsp;</p><p>Round your answer to at least&nbsp;3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; What is Var[<em>X</em>]?</span></p><p>&nbsp;</p><p><span>Round your answer to&nbsp;at least 3 decimal places.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.2.5.mode=Inline@
qu.2.5.name=Calculate P(X > x| X > y)@
qu.2.5.comment=<p>To determine the conditional probability, use the general 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>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><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><mfrac><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi></mrow></mfenced></mrow><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>This becomes <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>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$x1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2.5</mn><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$x1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><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'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2.5</mn></mrow></mfenced></mrow><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'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2.5</mn></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>, where&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='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$x1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><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'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2.5</mn></mrow></mfenced></mrow></mstyle></math>&nbsp;is the sum of all the probabilities for <em>X</em> values greater than $x1 <strong>and</strong> greater than 2.5, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2.5</mn></mrow></mfenced></mrow></mstyle></math>&nbsp;is the sum of all the probabilities for <em>X</em> values greater than 2.5.&nbsp; This results in <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$x1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><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'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2.5</mn></mrow></mfenced></mrow></mstyle></math>= $ProbNum, 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='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2.5</mn></mrow></mfenced></mrow></mstyle></math>&nbsp;= $ProbDenom.</p>
<p>&nbsp;</p>
<p>Therefore, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$x1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><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'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2.5</mn><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><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><mfrac><mi>$ProbNum</mi><mrow><mi>$ProbDenom</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$CondProb</mi></mrow></mstyle></math></p>@
qu.2.5.editing=useHTML@
qu.2.5.solution=@
qu.2.5.algorithm=$p1=rand(0.1, 0.3, 2);
$p2=rand(0.1, 0.2, 2);
$p3=rand(0.1, 0.3, 2);
$p4=1-($p1+$p2+$p3);
condition:gt($p4,0);
$k1=rint(2);
$ProbDenom=$p2+$p3+$p4;
$ProbNum1=$p3+$p4;
$ProbNum2=$p4;
$x1=switch($k1, 4.5, 6.5);
$ProbNum=switch($k1, $ProbNum1, $ProbNum2);
$CondProb=$ProbNum/$ProbDenom;@
qu.2.5.uid=7ce183ed-e593-4d02-81d5-98927e23a299@
qu.2.5.info=  Course=Introductory Statistics;
  Topic=Discrete Probability Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.2.5.weighting=1@
qu.2.5.numbering=alpha@
qu.2.5.part.1.name=sro_id_1@
qu.2.5.part.1.answer.units=@
qu.2.5.part.1.numStyle=   @
qu.2.5.part.1.editing=useHTML@
qu.2.5.part.1.showUnits=false@
qu.2.5.part.1.err=0.01@
qu.2.5.part.1.question=(Unset)@
qu.2.5.part.1.mode=Numeric@
qu.2.5.part.1.grading=toler_abs@
qu.2.5.part.1.negStyle=both@
qu.2.5.part.1.answer.num=$CondProb@
qu.2.5.question=<p>Consider the following discrete probability distribution:</p><p>&nbsp;</p><p><table border="1" cellspacing="1" cellpadding="1" width="200" align="center">    <tbody>        <tr>            <td><em>X</em></td>            <td>2</td>            <td>4</td>            <td>6</td>            <td>8</td>        </tr>        <tr>            <td><em>P(X)</em></td>            <td>$p1</td>            <td>$p2</td>            <td>$p3</td>            <td>$p4</td>        </tr>    </tbody></table></p><p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p><span>What 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>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$x1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><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'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2.5</mn><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mstyle></math>?</span></p><p>&nbsp;</p><p><span>Round your answer to&nbsp;at least 3 decimal places.</span></p><p><span><span>&nbsp;</span><1><span>&nbsp;</span></span></p>@

qu.3.topic=Binomial Distribution@

qu.3.1.mode=Inline@
qu.3.1.name=Calculate P(X = x), P(X > x), P(X <= x)@
qu.3.1.comment=<p>a)&nbsp; To calculate <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>$x</mi></mrow></mfenced></mrow></mstyle></math>, we can use the binomial formula such that:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msubsup><mi>C</mi><mrow><mi>$x</mi></mrow><mrow><mi>$n</mi></mrow></msubsup><msup><mi>$p</mi><mrow><mi>$x</mi></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><mi>$x</mi></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$x1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p>&nbsp;</p>
<p>b)&nbsp; To calculate <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'>&gt;</mo><mi>$x</mi></mrow></mfenced></mrow></mstyle></math>, we&nbsp;can use the binomial formula for values of <em>X</em> greater than $x <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$x</mi></mrow></mfenced><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><mi>$y</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n</mi></mrow></mfenced></mrow></mstyle></math>.&nbsp; Therefore, <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'>&gt;</mo><mi>$x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msubsup><mi>C</mi><mrow><mi>$y</mi></mrow><mrow><mi>$n</mi></mrow></msubsup><msup><mi>$p</mi><mrow><mi>$y</mi></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$p</mi></mrow></mfenced><mrow><mi>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$y</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msubsup><mi>C</mi><mrow><mi>$n</mi></mrow><mrow><mi>$n</mi></mrow></msubsup><msup><mi>$p</mi><mrow><mi>$n</mi></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$p</mi></mrow></mfenced><mrow><mi>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$n</mi></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$x2</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>
<p>c)&nbsp; To calculate <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'>&leq;</mo><mi>$x</mi></mrow></mfenced></mrow></mstyle></math>, we can use the binomial formula for values of <em>X</em> less than and equal to $x <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>$x</mi></mrow></mfenced><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><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><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$x</mi></mrow></mfenced></mrow></mstyle></math>.&nbsp; Alternately, we can use the fact that the complement of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>$x</mi></mrow></mfenced></mrow></mstyle></math>&nbsp;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>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$x</mi></mrow></mfenced></mrow></mstyle></math>, which was found in part (b).&nbsp; Therefore, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>$x</mi></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'>&gt;</mo><mi>$x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$x2</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$x3</mi></mrow></mstyle></math></p>@
qu.3.1.editing=useHTML@
qu.3.1.solution=@
qu.3.1.algorithm=$n=range(10,15);
$p=rand(0.8, 0.9, 2);
$x=$n-2;
$y=$n-1;
$m=maple("
X:=Statistics[RandomVariable](Binomial($n,$p)):
Px1:=Statistics[Probability](X=$x):
Px2:=Statistics[Probability](X>$x):
Px3:=Statistics[Probability](X<=$x):
Px1, Px2, Px3
");
$x1=switch(0,$m);
$x2=switch(1,$m);
$x3=switch(2,$m);@
qu.3.1.uid=c3c8d1ae-b857-48cb-b07d-78a88c6632bc@
qu.3.1.info=  Course=Introductory Statistics;
  Topic=Binomial Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.3.1.weighting=1,1,1@
qu.3.1.numbering=alpha@
qu.3.1.part.1.name=sro_id_1@
qu.3.1.part.1.answer.units=@
qu.3.1.part.1.numStyle=   @
qu.3.1.part.1.editing=useHTML@
qu.3.1.part.1.showUnits=false@
qu.3.1.part.1.err=0.0010@
qu.3.1.part.1.question=(Unset)@
qu.3.1.part.1.mode=Numeric@
qu.3.1.part.1.grading=toler_abs@
qu.3.1.part.1.negStyle=both@
qu.3.1.part.1.answer.num=$x1@
qu.3.1.part.2.name=sro_id_2@
qu.3.1.part.2.answer.units=@
qu.3.1.part.2.numStyle=   @
qu.3.1.part.2.editing=useHTML@
qu.3.1.part.2.showUnits=false@
qu.3.1.part.2.err=0.01@
qu.3.1.part.2.question=(Unset)@
qu.3.1.part.2.mode=Numeric@
qu.3.1.part.2.grading=toler_abs@
qu.3.1.part.2.negStyle=both@
qu.3.1.part.2.answer.num=$x2@
qu.3.1.part.3.name=sro_id_3@
qu.3.1.part.3.answer.units=@
qu.3.1.part.3.numStyle=   @
qu.3.1.part.3.editing=useHTML@
qu.3.1.part.3.showUnits=false@
qu.3.1.part.3.err=0.01@
qu.3.1.part.3.question=(Unset)@
qu.3.1.part.3.mode=Numeric@
qu.3.1.part.3.grading=toler_abs@
qu.3.1.part.3.negStyle=both@
qu.3.1.part.3.answer.num=$x3@
qu.3.1.question=<p>Let <em>X</em> be a random variable that follows a binomial distribution with <em>n</em> = $n, and probability of success <em>p</em> = $p.</p><p>&nbsp;</p><p>a)&nbsp; What is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$x</mi></mrow></mfenced></mrow></mstyle></math>?</p><p>&nbsp;</p><p>Round your response to&nbsp;at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b) &nbsp;What 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>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$x</mi></mrow></mfenced></mrow></mstyle></math>?</span></p><p>&nbsp;</p><p><span>Round your response to&nbsp;at least 3 decimal places.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span>c)&nbsp; What is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>$x</mi></mrow></mfenced></mrow></mstyle></math>?</span></span></p><p>&nbsp;</p><p><span><span>Round your response to&nbsp;at least 3 decimal places.</span></span></p><p><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></p>@

qu.3.2.mode=Multiple Selection@
qu.3.2.name=Definitions of Binomial 2@
qu.3.2.comment=@
qu.3.2.editing=useHTML@
qu.3.2.solution=@
qu.3.2.algorithm=@
qu.3.2.uid=1346d660-18d4-415f-b64e-31d41c157bac@
qu.3.2.info=  Course=Introductory Statistics;
  Topic=Binomial Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.3.2.question=<p>Which of the following statements are true?</p>
<p>&nbsp;</p>
<p>There may be more than&nbsp;one correct answer; select all that are true.</p>@
qu.3.2.answer=1, 2, 3@
qu.3.2.choice.1=In a binomial distribution, each trial can have one of two possible outcomes.@
qu.3.2.choice.2=There are two parameters in a binomial distribution.@
qu.3.2.choice.3=If X is a binomial random variable with parameters n and 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><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>n</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>p</mi><mrow><mi>n</mi></mrow></msup></mrow></mstyle></math>.@
qu.3.2.choice.4=The variance of a binomial random variable is greater than the number of trials.@
qu.3.2.choice.5=The largest value a binomial random variable can be is n + 1.@
qu.3.2.fixed=@

qu.3.3.mode=Multiple Selection@
qu.3.3.name=Definitions of Binomial 1@
qu.3.3.comment=@
qu.3.3.editing=useHTML@
qu.3.3.solution=@
qu.3.3.algorithm=@
qu.3.3.uid=3b88375e-ef10-4204-9602-e5ed8afab495@
qu.3.3.info=  Course=Introductory Statistics;
  Topic=Binomial Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.3.3.question=<p>Which of the following statements are true?</p>
<p>&nbsp;</p>
<p>There may be more than&nbsp;one correct answer; select all that are true.</p>@
qu.3.3.answer=1, 2@
qu.3.3.choice.1=In a binomial distribution, the random variable X is a count of the number of successes.@
qu.3.3.choice.2=If the probability of success is greater than zero, then the mean of a binomial random variable is greater than the variance.@
qu.3.3.choice.3=The probability of success, p, in a binomial distribution increases as X, the number of successes, increases.@
qu.3.3.choice.4=A binomial random variable X can take on a total of n possible values.@
qu.3.3.choice.5=The mean of a binomial random variable must be an integer, since a binomial random variable is discrete.@
qu.3.3.fixed=@

qu.3.4.mode=Inline@
qu.3.4.name=Calculate P(X > x)@
qu.3.4.comment=<p>To calculate <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='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$x</mi></mrow></mfenced></mrow></mstyle></math>, we can use the binomial formula for values of <em>X</em> greater than $x, and then sum the resulting probabilities.</p>
<p>Therefore, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$x</mi></mrow></mfenced><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><mi>$x1</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$x2</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n</mi></mrow></mfenced><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msubsup><mi>C</mi><mrow><mi>$x1</mi></mrow><mrow><mi>$n</mi></mrow></msubsup><msup><mi>$p</mi><mrow><mi>$x1</mi></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$p</mi></mrow></mfenced><mrow><mi>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$x1</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msubsup><mi>C</mi><mrow><mi>$x2</mi></mrow><mrow><mi>$n</mi></mrow></msubsup><msup><mi>$p</mi><mrow><mi>$x2</mi></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><mi>$x2</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msubsup><mi>C</mi><mrow><mi>$n</mi></mrow><mrow><mi>$n</mi></mrow></msubsup><msup><mi>$p</mi><mrow><mi>$n</mi></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$p</mi></mrow></mfenced><mrow><mi>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$n</mi></mrow></msup><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><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><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><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><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><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><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><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>$Prob</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>@
qu.3.4.editing=useHTML@
qu.3.4.solution=@
qu.3.4.algorithm=$n=range(15,20);
$p=rand(0.85, 0.95, 2);
$x=$n-3;
$Prob=maple("
X:=Statistics[RandomVariable](Binomial($n,$p)):
PX:=Statistics[Probability](X > $x):
PX
");
$x1=$n-2;
$x2=$n-1;@
qu.3.4.uid=645f4d04-bc45-4bff-a2c0-68d7bc7edcbb@
qu.3.4.info=  Course=Introductory Statistics;
  Topic=Binomial Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.3.4.weighting=1@
qu.3.4.numbering=alpha@
qu.3.4.part.1.name=sro_id_1@
qu.3.4.part.1.answer.units=@
qu.3.4.part.1.numStyle=   @
qu.3.4.part.1.editing=useHTML@
qu.3.4.part.1.showUnits=false@
qu.3.4.part.1.err=0.01@
qu.3.4.part.1.question=(Unset)@
qu.3.4.part.1.mode=Numeric@
qu.3.4.part.1.grading=toler_abs@
qu.3.4.part.1.negStyle=both@
qu.3.4.part.1.answer.num=$Prob@
qu.3.4.question=<p>Let <em>X</em> be a discrete random variable that follows a binomial distribution with <em>n</em> = $n and probability of success <em>p</em> = $p.</p><p>&nbsp;</p><p>What is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$x</mi></mrow></mfenced></mrow></mstyle></math>?</p><p>&nbsp;</p><p>Round your response to&nbsp;at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.3.5.mode=Inline@
qu.3.5.name=Definitions of Binomial 1&2: Random Selection of T/F@
qu.3.5.comment=@
qu.3.5.editing=useHTML@
qu.3.5.solution=@
qu.3.5.algorithm=$k1=rint(10);
$k2=rint(10);
$k3=rint(10);
$k4=rint(10);
$k5=rint(10);
$z=maple("S:=$k1,$k2,$k3,$k4,$k5:
floor(nops({S})/nops([S]))
");
condition:$z;
$a=("'In a binomial distribution, the random variable X is a count of the number of successes.'");
$b=("'If the probability of success is greater than zero, then the mean of a binomial random variable is greater than the variance.'");
$c=("'In a binomial distribution, each trial can have one of two possible outcomes.'");
$d=("'There are two parameters in a binomial distribution.'");
$e=maple("convert(cat(`If X is a binomial random variable with parameters n and p, then `,MathML[ExportPresentation](P(X=n)=p^n),`.`),string)");
$f=("'The probability of success, p, in a binomial distribution increases as X, the number of successes, increases.'");
$g=("'A binomial random variable, X, can take on a total of n possible values.'");
$h=("'The mean of a binomial random variable must be an integer, since a binomial random variable is discrete.'");
$i=("'The variance of a binomial random variable is greater than the number of trials.'");
$j=("'The largest value a binomial random variable can be is n + 1.'");
$Answers=["'True'","'True'","'True'","'True'","'True'","'False'","'False'","'False'","'False'","'False'"];
$Distractors=["'False'","'False'","'False'","'False'","'False'","'True'","'True'","'True'","'True'","'True'"];
$Q1=switch($k1, $a,$b,$c,$d,$e,$f,$g,$h,$i,$j);
$A1=switch($k1, $Answers);
$D1=switch($k1, $Distractors);
$Q2=switch($k2, $a,$b,$c,$d,$e,$f,$g,$h,$i,$j);
$A2=switch($k2, $Answers);
$D2=switch($k2, $Distractors);
$Q3=switch($k3, $a,$b,$c,$d,$e,$f,$g,$h,$i,$j);
$A3=switch($k3, $Answers);
$D3=switch($k3, $Distractors);
$Q4=switch($k4, $a,$b,$c,$d,$e,$f,$g,$h,$i,$j);
$A4=switch($k4, $Answers);
$D4=switch($k4, $Distractors);
$Q5=switch($k5, $a,$b,$c,$d,$e,$f,$g,$h,$i,$j);
$A5=switch($k5, $Answers);
$D5=switch($k5, $Distractors);@
qu.3.5.uid=d07d5351-ecb6-4136-9137-c2b7f4ac83b5@
qu.3.5.info=  Course=Introductory Statistics;
  Topic=Binomial Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.3.5.weighting=1,1,1,1,1@
qu.3.5.numbering=alpha@
qu.3.5.part.1.grader=exact@
qu.3.5.part.1.name=sro_id_1@
qu.3.5.part.1.editing=useHTML@
qu.3.5.part.1.display.permute=true@
qu.3.5.part.1.question=(Unset)@
qu.3.5.part.1.answer.2=$D1@
qu.3.5.part.1.answer.1=$A1@
qu.3.5.part.1.mode=List@
qu.3.5.part.1.display=menu@
qu.3.5.part.1.credit.2=0.0@
qu.3.5.part.1.credit.1=1.0@
qu.3.5.part.2.grader=exact@
qu.3.5.part.2.name=sro_id_2@
qu.3.5.part.2.editing=useHTML@
qu.3.5.part.2.display.permute=true@
qu.3.5.part.2.question=(Unset)@
qu.3.5.part.2.answer.2=$D2@
qu.3.5.part.2.answer.1=$A2@
qu.3.5.part.2.mode=List@
qu.3.5.part.2.display=menu@
qu.3.5.part.2.credit.2=0.0@
qu.3.5.part.2.credit.1=1.0@
qu.3.5.part.3.grader=exact@
qu.3.5.part.3.name=sro_id_3@
qu.3.5.part.3.editing=useHTML@
qu.3.5.part.3.display.permute=true@
qu.3.5.part.3.question=(Unset)@
qu.3.5.part.3.answer.2=$D3@
qu.3.5.part.3.answer.1=$A3@
qu.3.5.part.3.mode=List@
qu.3.5.part.3.display=menu@
qu.3.5.part.3.credit.2=0.0@
qu.3.5.part.3.credit.1=1.0@
qu.3.5.part.4.grader=exact@
qu.3.5.part.4.name=sro_id_4@
qu.3.5.part.4.editing=useHTML@
qu.3.5.part.4.display.permute=true@
qu.3.5.part.4.question=(Unset)@
qu.3.5.part.4.answer.2=$D4@
qu.3.5.part.4.answer.1=$A4@
qu.3.5.part.4.mode=List@
qu.3.5.part.4.display=menu@
qu.3.5.part.4.credit.2=0.0@
qu.3.5.part.4.credit.1=1.0@
qu.3.5.part.5.grader=exact@
qu.3.5.part.5.name=sro_id_5@
qu.3.5.part.5.editing=useHTML@
qu.3.5.part.5.display.permute=true@
qu.3.5.part.5.question=(Unset)@
qu.3.5.part.5.answer.2=$D5@
qu.3.5.part.5.answer.1=$A5@
qu.3.5.part.5.mode=List@
qu.3.5.part.5.display=menu@
qu.3.5.part.5.credit.2=0.0@
qu.3.5.part.5.credit.1=1.0@
qu.3.5.question=<p>Identify each of the following statements as either TRUE or FALSE.</p><p>&nbsp;</p><p>a)&nbsp;<span>&nbsp;</span><1><span>&nbsp;</span> $Q1</p><p>&nbsp;</p><p>b)&nbsp;<span>&nbsp;</span><2><span>&nbsp;</span> $Q2</p><p>&nbsp;</p><p>c)&nbsp;<span>&nbsp;</span><3><span>&nbsp;</span> $Q3</p><p>&nbsp;</p><p>d)&nbsp;<span>&nbsp;</span><4><span>&nbsp;</span> $Q4</p><p>&nbsp;</p><p>e)&nbsp;<span>&nbsp;</span><5><span>&nbsp;</span> $Q5</p>@

qu.3.6.mode=Inline@
qu.3.6.name=Japan Population Rate; Calculate mean, standard deviation@
qu.3.6.comment=<p>a)&nbsp; The random variable <em>X</em> follows a binomial distribution, with mean <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><mi>np</mi></mrow></mstyle></math>, where <em>n</em> is the sample size, and <em>p</em> is the probability of success.&nbsp; Therefore, the mean of <em>X</em> 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>&mu;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>50</mn><mfenced open='(' close=')' separators=','><mrow><mi>$Rate</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$mean</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The variance of a binomial distribution is given by <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>&sigma;</mi><mrow><mn>2</mn></mrow></msup><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></mrow></mstyle></math>, with the standard deviation then being equal to <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><msup><mi>&sigma;</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><msqrt><mrow><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></mrow></msqrt></mrow></mrow></mstyle></math>.&nbsp; Therefore, the standard deviation of <em>X</em> 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><msqrt><mrow><mn>50</mn><mfenced open='(' close=')' separators=','><mrow><mi>$Rate</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$Rate</mi></mrow></mfenced></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$StdDev</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>@
qu.3.6.editing=useHTML@
qu.3.6.solution=@
qu.3.6.algorithm=$OverSixty=rand(26, 27, 3);
$Rate=$OverSixty/100;
$mean=50*$Rate;
$StdDev=sqrt(50*$Rate*(1-$Rate));@
qu.3.6.uid=6c360b47-5998-45b6-bd35-57d82a9d4eb7@
qu.3.6.info=  Course=Introductory Statistics;
  Topic=Binomial Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Application;
@
qu.3.6.weighting=1,1@
qu.3.6.numbering=alpha@
qu.3.6.part.1.name=sro_id_1@
qu.3.6.part.1.answer.units=@
qu.3.6.part.1.numStyle=   @
qu.3.6.part.1.editing=useHTML@
qu.3.6.part.1.showUnits=false@
qu.3.6.part.1.err=0.0010@
qu.3.6.part.1.question=(Unset)@
qu.3.6.part.1.mode=Numeric@
qu.3.6.part.1.grading=toler_abs@
qu.3.6.part.1.negStyle=both@
qu.3.6.part.1.answer.num=$mean@
qu.3.6.part.2.name=sro_id_2@
qu.3.6.part.2.answer.units=@
qu.3.6.part.2.numStyle=   @
qu.3.6.part.2.editing=useHTML@
qu.3.6.part.2.showUnits=false@
qu.3.6.part.2.err=0.0010@
qu.3.6.part.2.question=(Unset)@
qu.3.6.part.2.mode=Numeric@
qu.3.6.part.2.grading=toler_abs@
qu.3.6.part.2.negStyle=both@
qu.3.6.part.2.answer.num=$StdDev@
qu.3.6.question=<p>Japan is reported to have among the highest percentage of citizens aged 60 years and over,&nbsp;at approximately $OverSixty % (i.e. $OverSixty % of its population falls within this age category).&nbsp; If 50 individuals are randomly selected from the population, and <em>X</em> is the number of individuals out of 50 that are 60 or more years old, what is:</p><p>&nbsp;</p><p>a)&nbsp; The mean of <em>X</em>?</p><p>&nbsp;</p><p>Round your answer to&nbsp;at least&nbsp;3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; The standard deviation of <em>X</em>?</span></p><p>&nbsp;</p><p><span>Round your answer to&nbsp;at least 3 decimal places.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.3.7.mode=Inline@
qu.3.7.name=Calculate P(X > x | X <= y)@
qu.3.7.comment=<p>To calculate <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='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$x1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$x2</mi></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>, we can start by using the general conditional probability 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>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</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>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi></mrow></mfenced></mrow><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.&nbsp;&nbsp; Here, this becomes <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>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$x1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>$x2</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><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><mfrac><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'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$x1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>$x2</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><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>$x2</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.&nbsp; In the numerator, <em>X</em> is greater than $x1 and less than or equal to $x2 at the values <em>X =&nbsp;</em>$x3 and <em>X</em> = $x2.&nbsp; Therefore, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$x1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>$x2</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$x3</mi></mrow></mfenced><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><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$x2</mi></mrow></mfenced></mrow></mstyle></math>, which we can&nbsp;now solve by using the binomial formula:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$x1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>$x2</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msubsup><mi>C</mi><mrow><mi>$x3</mi></mrow><mrow><mi>$n</mi></mrow></msubsup><msup><mi>$p</mi><mrow><mi>$x3</mi></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$p</mi></mrow></mfenced><mrow><mi>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$x3</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msubsup><mi>C</mi><mrow><mi>$x2</mi></mrow><mrow><mi>$n</mi></mrow></msubsup><msup><mi>$p</mi><mrow><mi>$x2</mi></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><mi>$x2</mi></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ProbNum</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p>In the&nbsp;denominator, <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'>&leq;</mo><mi>$x2</mi></mrow></mfenced></mrow></mstyle></math>&nbsp;can be found by using the binomial formula for values of <em>X</em> less than or equal to $x2, such that: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>$x2</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><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>0</mn></mrow></mfenced><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><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='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><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><mi>$x2</mi></mrow></mfenced></mrow></mstyle></math>.&nbsp; Alternately, it can be found by using the complement, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>$x2</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$x2</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>$x2</mi></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'>&gt;</mo><mi>$x2</mi></mrow></mfenced></mrow></mstyle></math>, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$x2</mi></mrow></mfenced><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><mi>$x4</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n</mi></mrow></mfenced></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Complement</mi></mrow></mstyle></math>.&nbsp; Thus, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>$x2</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$Complement</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ProbDenom</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.</p>
<p>Thefore, <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>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$x1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>$x2</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><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><mfrac><mi>$ProbNum</mi><mrow><mi>$ProbDenom</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$CondProb</mi></mrow></mstyle></math></p>@
qu.3.7.editing=useHTML@
qu.3.7.solution=@
qu.3.7.algorithm=$n=range(10,14);
$p=rand(0.7, 0.8, 2);
$x1=$n-4;
$x2=$n-2;
$x3=$n-3;
$x4=$n-1;
$Prob=maple("
X:=Statistics[RandomVariable](Binomial($n,$p)):
Px1:=Statistics[Probability](X = $x2):
Px2:=Statistics[Probability](X = $x3):
Px3:=Statistics[Probability](X <= $x2):
Px1, Px2, Px3
");
$Prob1=switch(0, $Prob);
$Prob2=switch(1, $Prob);
$ProbDenom=switch(2, $Prob);
$Complement=1-$ProbDenom;
$ProbNum=$Prob1 + $Prob2;
$CondProb=$ProbNum/$ProbDenom;
condition:lt($CondProb,1.0);@
qu.3.7.uid=e831ecd3-2f6e-4911-ae30-a041cfb0846a@
qu.3.7.info=  Course=Introductory Statistics;
  Topic=Binomial Distribution;
  Author=Lorna Deeth;
  Difficulty=Hard;
  Features=None;
  Type=Calculation;
@
qu.3.7.weighting=1@
qu.3.7.numbering=alpha@
qu.3.7.part.1.name=sro_id_1@
qu.3.7.part.1.answer.units=@
qu.3.7.part.1.numStyle=   @
qu.3.7.part.1.editing=useHTML@
qu.3.7.part.1.showUnits=false@
qu.3.7.part.1.err=0.01@
qu.3.7.part.1.question=(Unset)@
qu.3.7.part.1.mode=Numeric@
qu.3.7.part.1.grading=toler_abs@
qu.3.7.part.1.negStyle=both@
qu.3.7.part.1.answer.num=$CondProb@
qu.3.7.question=<p>Let <em>X</em> be a discrete random variable that follows a binomial distribution with <em>n</em> = $n and probability of success <em>p</em> = $p.</p><p>&nbsp;</p><p>What is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$x1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><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'>&leq;</mo><mi>$x2</mi></mrow></mfenced></mrow></mstyle></math>?</p><p>&nbsp;</p><p>Round your response to at least&nbsp;3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.3.8.mode=Inline@
qu.3.8.name=Canadian Exports; Calculate probability of at least one@
qu.3.8.comment=<p>If the random variable <em>X</em> is the number of exports destined for the European Union, where each of the 15 exports is either&nbsp;destined the EU or not with a constant&nbsp;probability of $ExportProb, then <em>X </em>follows a binomial distribution with <em>n</em> = 15 and probability of success <em>p</em> = $ExportProb.</p>
<p>To determine the probability that at least one shipment is destined for the European Union, we need to calculate <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='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&ge;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>1</mn></mrow></mfenced></mrow></mstyle></math>.&nbsp; This can be done using the binomial 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>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><mi>x</mi></mrow></mfenced><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><msubsup><mi>C</mi><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msubsup><msup><mi>p</mi><mrow><mi>x</mi></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><mi>x</mi></mrow></msup></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>, for <em>x</em> = 1, 2, ..., 15, and then adding up the resulting probabilities so that</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='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&ge;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><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><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><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>2</mn></mrow></mfenced><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><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><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><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>15</mn></mrow></mfenced></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>An alternate way to calculate the same value is to use the complement of <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></mrow><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&ge;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>1</mn><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mstyle></math>, 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><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>0</mn></mrow></mfenced></mrow></mstyle></math>.&nbsp; Therefore, <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='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&ge;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>1</mn></mrow></mfenced><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><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>0</mn></mrow></mfenced><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='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><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'>&ge;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>1</mn></mrow></mfenced><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><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>0</mn></mrow></mfenced></mrow></mstyle></math>.&nbsp; To find <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='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>0</mn></mrow></mfenced></mrow></mstyle></math>, we can use the binomial formula:</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='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>0</mn></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msubsup><mi>C</mi><mrow><mn>0</mn></mrow><mrow><mn>15</mn></mrow></msubsup><msup><mi>$ExportProb</mi><mrow><mn>0</mn></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$ExportProb</mi></mrow></mfenced><mrow><mn>15</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0</mn></mrow></msup></mrow></mstyle></math>= $Prob0</p>
<p>Therefore, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&ge;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$Prob0</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Prob1</mi></mrow></mstyle></math></p>@
qu.3.8.editing=useHTML@
qu.3.8.solution=@
qu.3.8.algorithm=$Export=rand(5,6,2);
$ExportProb=$Export/100;
$Prob0=(1-$ExportProb)^15;
$Prob1=1-$Prob0;@
qu.3.8.uid=44e866b5-4584-41a9-b737-c790eb6c8813@
qu.3.8.info=  Course=Introductory Statistics;
  Topic=Binomial Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Application;
@
qu.3.8.weighting=1@
qu.3.8.numbering=alpha@
qu.3.8.part.1.name=sro_id_1@
qu.3.8.part.1.answer.units=@
qu.3.8.part.1.numStyle=   @
qu.3.8.part.1.editing=useHTML@
qu.3.8.part.1.showUnits=false@
qu.3.8.part.1.err=0.0010@
qu.3.8.part.1.question=(Unset)@
qu.3.8.part.1.mode=Numeric@
qu.3.8.part.1.grading=toler_abs@
qu.3.8.part.1.negStyle=both@
qu.3.8.part.1.answer.num=$Prob1@
qu.3.8.question=<p>An economist estimates that&nbsp;the percentage of Canadian exports that are destined for the European Union is approximately $Export %.&nbsp; If 15 shipments for export are randomly selected, what is the probability that at least one of those shipments is destined for the European Union?</p><p>&nbsp;</p><p>Round your response to&nbsp;at least 3&nbsp;decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span>&nbsp;&nbsp;</p>@

qu.3.9.mode=Inline@
qu.3.9.name=Unemployment Rate; Calculate exact probability@
qu.3.9.comment=<p>To calculate the probability of exactly $x1 members being unemployed, we can use the&nbsp;Binomial formula to get:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$x1</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msubsup><mi>C</mi><mrow><mi>$x1</mi></mrow><mrow><mn>25</mn></mrow></msubsup><msup><mi>$Prob</mi><mrow><mi>$x1</mi></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$Prob</mi></mrow></mfenced><mrow><mn>25</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$x1</mi></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$m</mi></mrow></mstyle></math></p>@
qu.3.9.editing=useHTML@
qu.3.9.solution=@
qu.3.9.algorithm=$Unemploy=rand(7.0, 8.0, 2);
$Prob=$Unemploy/100;
$x1=range(2,5);
$m=maple("
X:=Statistics[RandomVariable](Binomial(25, $Prob)):
Px1:=Statistics[Probability](X=$x1):
Px1
");@
qu.3.9.uid=6eabc5e1-803e-4f7e-b3c5-71c64ede815b@
qu.3.9.info=  Course=Introductory Statistics;
  Topic=Binomial Distribution;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Application;
@
qu.3.9.weighting=1@
qu.3.9.numbering=alpha@
qu.3.9.part.1.name=sro_id_1@
qu.3.9.part.1.answer.units=@
qu.3.9.part.1.numStyle=   @
qu.3.9.part.1.editing=useHTML@
qu.3.9.part.1.showUnits=false@
qu.3.9.part.1.err=0.0010@
qu.3.9.part.1.question=(Unset)@
qu.3.9.part.1.mode=Numeric@
qu.3.9.part.1.grading=toler_abs@
qu.3.9.part.1.negStyle=both@
qu.3.9.part.1.answer.num=$m@
qu.3.9.question=<p>According to the <em>Statistics Canada</em> website, the unemployment rate in Canada is approximately $Unemploy % of the eligible workforce.&nbsp; If 25 members of the eligible workforce are randomly selected, what is the probability that exactly $x1 of them&nbsp;are unemployed?</p><p>&nbsp;</p><p>Round your response to&nbsp;3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span>&nbsp;</p>@

qu.3.10.mode=Inline@
qu.3.10.name=Calculate P(X <= x)@
qu.3.10.comment=<p>To calculate <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='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mn>2</mn></mrow></mfenced></mrow></mstyle></math>, we can use the binomial formula for values of <em>X</em>&nbsp;less than and equal to 2, and then sum the resulting probabilities.</p>
<p>Therefore, <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='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2</mn></mrow></mfenced><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>2</mn></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><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><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><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><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><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><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><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><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><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><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><msubsup><mi>C</mi><mrow><mn>2</mn></mrow><mrow><mi>$n</mi></mrow></msubsup><msup><mi>$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>$p</mi></mrow></mfenced><mrow><mi>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msubsup><mi>C</mi><mrow><mn>1</mn></mrow><mrow><mi>$n</mi></mrow></msubsup><msup><mi>$p</mi><mrow><mn>1</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>1</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msubsup><mi>C</mi><mrow><mn>0</mn></mrow><mrow><mi>$n</mi></mrow></msubsup><msup><mi>$p</mi><mrow><mn>0</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>0</mn></mrow></msup><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><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><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><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><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><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><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><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>$Prob</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p>&nbsp;</p>@
qu.3.10.editing=useHTML@
qu.3.10.solution=@
qu.3.10.algorithm=$n=range(15,20);
$p=rand(0.1, 0.2, 2);
$Prob=maple("
X:=Statistics[RandomVariable](Binomial($n,$p)):
PX:=Statistics[Probability](X <= 2):
PX
");@
qu.3.10.uid=0feb5c97-c965-4435-929a-b3a3ac84b564@
qu.3.10.info=  Course=Introductory Statistics;
  Topic=Binomial Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.3.10.weighting=1@
qu.3.10.numbering=alpha@
qu.3.10.part.1.name=sro_id_1@
qu.3.10.part.1.answer.units=@
qu.3.10.part.1.numStyle=   @
qu.3.10.part.1.editing=useHTML@
qu.3.10.part.1.showUnits=false@
qu.3.10.part.1.err=0.01@
qu.3.10.part.1.question=(Unset)@
qu.3.10.part.1.mode=Numeric@
qu.3.10.part.1.grading=toler_abs@
qu.3.10.part.1.negStyle=both@
qu.3.10.part.1.answer.num=$Prob@
qu.3.10.question=<p>Let <em>X</em> be a discrete random variable that follows a binomial distribution with <em>n</em> = $n and probability of success <em>p</em> = $p.</p><p>&nbsp;</p><p>What 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>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mn>2</mn></mrow></mfenced></mrow></mstyle></math>?</p><p>&nbsp;</p><p>Round your response to&nbsp;at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.3.11.mode=Inline@
qu.3.11.name=Calculate P(X = x)@
qu.3.11.comment=<p>To calculate <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>$x</mi></mrow></mfenced></mrow></mstyle></math>, we can use the binomial formula as follows:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msubsup><mi>C</mi><mrow><mi>$x</mi></mrow><mrow><mi>$n</mi></mrow></msubsup><msup><mi>$p</mi><mrow><mi>$x</mi></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><mi>$x</mi></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Prob</mi></mrow></mstyle></math></p>@
qu.3.11.editing=useHTML@
qu.3.11.solution=@
qu.3.11.algorithm=$n=range(10,12);
$p=rand(0.25, 0.35, 2);
$x=range(2,4);
$Prob=maple("
X:=Statistics[RandomVariable](Binomial($n,$p)):
PX:=Statistics[Probability](X = $x):
PX
");@
qu.3.11.uid=8d9f48e6-12dc-4113-8623-0bac4d75b00f@
qu.3.11.info=  Course=Introductory Statistics;
  Topic=Binomial Distribution;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
qu.3.11.weighting=1@
qu.3.11.numbering=alpha@
qu.3.11.part.1.name=sro_id_1@
qu.3.11.part.1.answer.units=@
qu.3.11.part.1.numStyle=   @
qu.3.11.part.1.editing=useHTML@
qu.3.11.part.1.showUnits=false@
qu.3.11.part.1.err=0.0010@
qu.3.11.part.1.question=(Unset)@
qu.3.11.part.1.mode=Numeric@
qu.3.11.part.1.grading=toler_abs@
qu.3.11.part.1.negStyle=both@
qu.3.11.part.1.answer.num=$Prob@
qu.3.11.question=<p>Let <em>X</em> be a discrete random variable that follows a binomial distribution with <em>n</em> = $n and probability of success <em>p</em> = $p.</p><p>&nbsp;</p><p>What is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$x</mi></mrow></mfenced></mrow></mstyle></math>?</p><p>&nbsp;</p><p>Round your response to&nbsp;at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.3.12.mode=Inline@
qu.3.12.name=Sweden Population Rate; Calculate expected value, probability of less than x@
qu.3.12.comment=<p>a)&nbsp; The random variable <em>X</em> follows a binomial distribution, with&nbsp;an&nbsp;expected value of&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>E</mi><mfenced open='[' close=']' separators=','><mrow><mi>X</mi></mrow></mfenced><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>&mu;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>np</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>, where <em>n</em> is the sample size, and <em>p</em> is the probability of success.&nbsp; Therefore, the&nbsp;expected value&nbsp;of <em>X</em> 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>&mu;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>20</mn><mfenced open='(' close=')' separators=','><mrow><mi>$Rate</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$mean</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The probability of less than 2 citizens being 80 years of age or older can be calculated using the binomial formula for <em>X </em>= 0 and <em>X</em> = 1.&nbsp; Therefore:</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'>&lt;</mo><mn>2</mn></mrow></mfenced><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><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><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><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><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><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><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><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><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><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><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msubsup><mi>C</mi><mrow><mn>0</mn></mrow><mrow><mn>20</mn></mrow></msubsup><msup><mi>$Rate</mi><mrow><mn>0</mn></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$Rate</mi></mrow></mfenced><mrow><mn>20</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msubsup><mi>C</mi><mrow><mn>1</mn></mrow><mrow><mn>20</mn></mrow></msubsup><mi>$Rate</mi><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$Rate</mi></mrow></mfenced><mrow><mn>20</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><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><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><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><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><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><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><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Prob</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>@
qu.3.12.editing=useHTML@
qu.3.12.solution=@
qu.3.12.algorithm=$OverEighty=rand(5, 6, 2);
$Rate=$OverEighty/100;
$mean=20*$Rate;
$Prob=maple("
X:=Statistics[RandomVariable](Binomial(20,$Rate)):
PX:=Statistics[Probability](X<2):
PX
");@
qu.3.12.uid=49ee573a-9461-48c7-9866-7a12cf2ae7f0@
qu.3.12.info=  Course=Introductory Statistics;
  Topic=Binomial Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Application;
@
qu.3.12.weighting=1,1@
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=   @
qu.3.12.part.1.editing=useHTML@
qu.3.12.part.1.showUnits=false@
qu.3.12.part.1.err=0.0010@
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=both@
qu.3.12.part.1.answer.num=$mean@
qu.3.12.part.2.name=sro_id_2@
qu.3.12.part.2.answer.units=@
qu.3.12.part.2.numStyle=   @
qu.3.12.part.2.editing=useHTML@
qu.3.12.part.2.showUnits=false@
qu.3.12.part.2.err=0.0010@
qu.3.12.part.2.question=(Unset)@
qu.3.12.part.2.mode=Numeric@
qu.3.12.part.2.grading=toler_abs@
qu.3.12.part.2.negStyle=both@
qu.3.12.part.2.answer.num=$Prob@
qu.3.12.question=<p>Sweden&nbsp;is reported to have among the highest percentage of citizens aged 80 years and over,&nbsp;at approximately $OverEighty % (i.e. $OverEighty % of it's population falls within this age category).&nbsp; Suppose 20 individuals are randomly selected from the population, and <em>X</em> is the number of individuals out of 20 that are 80 or more years old.</p><p>&nbsp;</p><p>a)&nbsp; On average, how many of the 20 individuals would be 80 years old or older?</p><p>&nbsp;</p><p>Round your answer to&nbsp;at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; What is the probability that less than 2 of the 20 individuals are 80 years old or older?</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>Round your answer to&nbsp;at least 3 decimal places.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.3.13.mode=Inline@
qu.3.13.name=Unemployment Rate; Calculate condional probability@
qu.3.13.comment=<p>To calculate the probability of exactly $x1 members being unemployed, given that 2 members of the labour force are unemployed, we need to calculate the conditional probability <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$x1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mn>2</mn><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mstyle></math>.&nbsp; This can be done using the general 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>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</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>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi></mrow></mfenced></mrow><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>, which in this case becomes <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><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$x1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mn>2</mn></mrow></mfenced></mrow><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mn>2</mn></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.&nbsp;</p>
<p>In the numerator,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$x1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><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'>&GreaterEqual;</mo><mn>2</mn></mrow></mfenced></mrow></mstyle></math>&nbsp;simplifies to <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='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$x1</mi></mrow></mfenced></mrow></mstyle></math>, as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$x1</mi></mrow></mstyle></math>&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>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2</mn></mrow></mstyle></math>intersect only at the point <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><mi>$x1</mi></mrow></mstyle></math>.&nbsp; To calculate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$x1</mi></mrow></mfenced></mrow></mstyle></math>, we can use the&nbsp;binomial formula, such that:</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>$x1</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msubsup><mi>C</mi><mrow><mi>$x1</mi></mrow><mrow><mn>25</mn></mrow></msubsup><msup><mi>$Prob</mi><mrow><mi>$x1</mi></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$Prob</mi></mrow></mfenced><mrow><mn>25</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$x1</mi></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ProbNum</mi></mrow></mstyle></math></p>
<p>In the denominator, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mn>2</mn></mrow></mfenced></mrow></mstyle></math>&nbsp;can be calculated by using the binomial formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msubsup><mi>C</mi><mrow><mi>x</mi></mrow><mrow><mn>25</mn></mrow></msubsup><msup><mi>$Prob</mi><mrow><mi>x</mi></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$Prob</mi></mrow></mfenced><mrow><mn>25</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>x</mi></mrow></msup></mrow></mstyle></math>&nbsp;&nbsp;&nbsp;for <em>x</em> = 2, 3, 4...25, and summing together the resulting probabilities.&nbsp; Alternately, we can use the binomial formula on the complement of <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'>&GreaterEqual;</mo><mn>2</mn></mrow></mfenced></mrow></mstyle></math>, 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><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mn>1</mn></mrow></mfenced></mrow></mstyle></math>.&nbsp; Since <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'>&GreaterEqual;</mo><mn>2</mn></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mn>2</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'>&leq;</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><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='0.2222222em' rspace='0.2222222em'>&plus;</mo><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></mfenced></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ProbDenom</mi></mrow></mstyle></math></p>
<p>&nbsp;Therefore, <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>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$x1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mn>2</mn><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>$ProbNum</mi><mrow><mi>$ProbDenom</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$CondProb</mi></mrow></mstyle></math></p>@
qu.3.13.editing=useHTML@
qu.3.13.solution=@
qu.3.13.algorithm=$Unemploy=rand(7.0, 8.0, 2);
$Prob=$Unemploy/100;
$x1=range(3,5);
$m=maple("
X:=Statistics[RandomVariable](Binomial(25, $Prob)):
Px1:=Statistics[Probability](X=$x1):
Px2:=Statistics[Probability](X>=2):
Px1, Px2
");
$ProbNum=switch(0,$m);
$ProbDenom=switch(1, $m);
$CondProb=$ProbNum/$ProbDenom;@
qu.3.13.uid=bb330166-03cb-42e3-b456-fda9e9cb1f90@
qu.3.13.info=  Course=Introductory Statistics;
  Topic=Binomial Distribution;
  Author=Lorna Deeth;
  Difficulty=Hard;
  Features=None;
  Type=Application;
@
qu.3.13.weighting=1@
qu.3.13.numbering=alpha@
qu.3.13.part.1.name=sro_id_1@
qu.3.13.part.1.answer.units=@
qu.3.13.part.1.numStyle=   @
qu.3.13.part.1.editing=useHTML@
qu.3.13.part.1.showUnits=false@
qu.3.13.part.1.err=0.01@
qu.3.13.part.1.question=(Unset)@
qu.3.13.part.1.mode=Numeric@
qu.3.13.part.1.grading=toler_abs@
qu.3.13.part.1.negStyle=both@
qu.3.13.part.1.answer.num=$CondProb@
qu.3.13.question=<p>According to the <em>Statistics Canada</em> website, the unemployment rate in Canada is approximately $Unemploy % of the eligible workforce.&nbsp; If 25 members of the eligible workforce are randomly selected, and it is&nbsp;determined that at least 2 of the 25&nbsp;are unemployed, what is the probability that exactly $x1 of the 25&nbsp;members&nbsp;are unemployed?</p><p>&nbsp;</p><p>&nbsp;</p><p>Round your response to&nbsp;at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span>&nbsp;</p>@

qu.4.topic=Geometric Distribution@

qu.4.1.mode=Inline@
qu.4.1.name=Definitions of Geometric 1&2: Random Selection of T/F@
qu.4.1.comment=@
qu.4.1.editing=useHTML@
qu.4.1.solution=@
qu.4.1.algorithm=$k1=rint(10);
$k2=rint(10);
$k3=rint(10);
$k4=rint(10);
$k5=rint(10);
$z=maple("S:=$k1,$k2,$k3,$k4,$k5:
floor(nops({S})/nops([S]))
");
condition:$z;
$a=("'In a geometric distribution, the random variable X can take on a countably infinite number of values.'");
$b=("'If X is a geometric random variable, then X  is the trial on which the first success occurs.'");
$c=("'The probability of success in a geometric distribution remains constant between each trial.'");
$d=("'If the probability of success in a geometric distribution is equal to 0.5, then the mean and the variance are the same.'");
$e=("'The mean of X, a geometric random variable, is 1/p, where p is the probability of success on a single trial.'");
$f=("'There are a finite number of independent trials in a geometric distribution.'");
$g=("'If the probability of success in a geometric distribution is greater than 0.5, then the variance is greater than the mean.'");
$h=("'If X is a geometric random variable, then the smallest value X can take on is 0.'");
$i=("'The geometric distribution is dependent upon two parameters.'");
$j=("'In a geometric distribution, if the first success occurs on the Xth trial, then on the first X � 1 trials there must have been only a small number of successes.'");
$Answers=["'True'","'True'","'True'","'True'","'True'","'False'","'False'","'False'","'False'","'False'"];
$Distractors=["'False'","'False'","'False'","'False'","'False'","'True'","'True'","'True'","'True'","'True'"];
$Q1=switch($k1, $a,$b,$c,$d,$e,$f,$g,$h,$i,$j);
$A1=switch($k1, $Answers);
$D1=switch($k1, $Distractors);
$Q2=switch($k2, $a,$b,$c,$d,$e,$f,$g,$h,$i,$j);
$A2=switch($k2, $Answers);
$D2=switch($k2, $Distractors);
$Q3=switch($k3, $a,$b,$c,$d,$e,$f,$g,$h,$i,$j);
$A3=switch($k3, $Answers);
$D3=switch($k3, $Distractors);
$Q4=switch($k4, $a,$b,$c,$d,$e,$f,$g,$h,$i,$j);
$A4=switch($k4, $Answers);
$D4=switch($k4, $Distractors);
$Q5=switch($k5, $a,$b,$c,$d,$e,$f,$g,$h,$i,$j);
$A5=switch($k5, $Answers);
$D5=switch($k5, $Distractors);@
qu.4.1.uid=98a302c5-5dce-42d7-8317-dc204fa396fb@
qu.4.1.info=  Course=Introductory Statistics;
  Topic=Geometric Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.4.1.weighting=1,1,1,1,1@
qu.4.1.numbering=alpha@
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qu.4.1.part.1.display.permute=true@
qu.4.1.part.1.question=(Unset)@
qu.4.1.part.1.answer.2=$D1@
qu.4.1.part.1.answer.1=$A1@
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qu.4.1.part.2.grader=exact@
qu.4.1.part.2.name=sro_id_2@
qu.4.1.part.2.editing=useHTML@
qu.4.1.part.2.display.permute=true@
qu.4.1.part.2.question=(Unset)@
qu.4.1.part.2.answer.2=$D2@
qu.4.1.part.2.answer.1=$A2@
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qu.4.1.part.3.display.permute=true@
qu.4.1.part.3.question=(Unset)@
qu.4.1.part.3.answer.2=$D3@
qu.4.1.part.3.answer.1=$A3@
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qu.4.1.part.4.name=sro_id_4@
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qu.4.1.part.4.display.permute=true@
qu.4.1.part.4.question=(Unset)@
qu.4.1.part.4.answer.2=$D4@
qu.4.1.part.4.answer.1=$A4@
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qu.4.1.part.5.grader=exact@
qu.4.1.part.5.name=sro_id_5@
qu.4.1.part.5.editing=useHTML@
qu.4.1.part.5.display.permute=true@
qu.4.1.part.5.question=(Unset)@
qu.4.1.part.5.answer.2=$D5@
qu.4.1.part.5.answer.1=$A5@
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qu.4.1.question=<p>Identify each of the following statements as either TRUE or FALSE.</p><p>&nbsp;</p><p>a)&nbsp;<span>&nbsp;</span><1><span>&nbsp;</span> $Q1</p><p>&nbsp;</p><p>b)&nbsp;<span>&nbsp;</span><2><span>&nbsp;</span> $Q2</p><p>&nbsp;</p><p>c)&nbsp;<span>&nbsp;</span><3><span>&nbsp;</span> $Q3</p><p>&nbsp;</p><p>d)&nbsp;<span>&nbsp;</span><4><span>&nbsp;</span> $Q4</p><p>&nbsp;</p><p>e)&nbsp;<span>&nbsp;</span><5><span>&nbsp;</span> $Q5</p>@

qu.4.2.mode=Multiple Selection@
qu.4.2.name=Definitions of Geometric 1@
qu.4.2.comment=@
qu.4.2.editing=useHTML@
qu.4.2.solution=@
qu.4.2.algorithm=@
qu.4.2.uid=a6cb6124-0130-4c9d-ad5a-88e53691c79a@
qu.4.2.info=  Course=Introductory Statistics;
  Topic=Geometric Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.4.2.question=<p>Which of the following statements are true?</p>
<p>&nbsp;</p>
<p>There may be more than&nbsp;one correct answer; select all that are true.</p>@
qu.4.2.answer=1, 2@
qu.4.2.choice.1=In a geometric distribution, the random variable X can take on a countably infinite number of values.@
qu.4.2.choice.2=If X is a geometric random variable, then X  is the trial on which the first "success" occurs.@
qu.4.2.choice.3=There are a finite number of independent trials in a geometric distribution.@
qu.4.2.choice.4=If the probability of success in a geometric distribution is greater than 0.5, then the variance is greater than the mean.@
qu.4.2.choice.5=If X is a geometric random variable, then the smallest value X can take on is 0.@
qu.4.2.fixed=@

qu.4.3.mode=Inline@
qu.4.3.name=Dentist Study: Calculate probability of first gingivitis.@
qu.4.3.comment=<p>If the random variable <em>X</em> is the trial on which the first success occurs, and 1 out of every&nbsp;$p1 people has gingivitis, then <em>X</em>&nbsp; follows a geometric distribution with probability of success 1/$p1 = $p.</p>
<p>To find the probability&nbsp;that the first success occurs on the $script trial,&nbsp;we can use the geometric formula: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>p</mi><mrow><mi></mi></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>p</mi></mrow></mfenced><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.&nbsp;&nbsp;Here, this becomes&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>$p</mi><mrow><mi></mi></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$p</mi></mrow></mfenced><mrow><mi>$x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Prob</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>&nbsp;</p>@
qu.4.3.editing=useHTML@
qu.4.3.solution=@
qu.4.3.algorithm=$k1=rint(2);
$k2=rint(2);
$x=switch($k1, 2,3);
$p1=switch($k2, 2,3);
$p=1/$p1;
$Prob=$p*((1-$p)^($x-1));
$script=switch($k1, "second", "third");@
qu.4.3.uid=7a797fb5-2744-4534-976e-013b3d874f50@
qu.4.3.info=  Course=Introductory Statistics;
  Topic=Geometric Distribution;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Application;
@
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=   @
qu.4.3.part.1.editing=useHTML@
qu.4.3.part.1.showUnits=false@
qu.4.3.part.1.err=0.0010@
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=both@
qu.4.3.part.1.answer.num=$Prob@
qu.4.3.question=<p>A dentist believes that 1 in $p1 people have gingivitis.&nbsp; Supposing that the dentist's belief is correct, that&nbsp;1 in $p1 people actually do have gingivitis, then on any given day in his office, what is the probability that the first case of gingivitis he sees will be on his $script patient?</p><p>&nbsp;</p><p>(We will assume for the purposes of this question that his patients are all independent of each other).</p><p>&nbsp;</p><p>Round your answer to&nbsp;at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.4.4.mode=Inline@
qu.4.4.name=Bowling Question: Calculate probability of no more than 3 attempts@
qu.4.4.comment=<p>If the random variable <em>X</em> is the trial on which Richard gets his first strike, with a $Shots % chance of success, then <em>X</em>&nbsp; follows a geometric distribution with <em>p</em> = $p.</p>
<p>To find the probability that it takes&nbsp;no more than&nbsp;3 attempts before Richard gets his first strike, we can use the geometric formula to find <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'>&leq;</mo><mn>3</mn></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.&nbsp; Using the geometric formula, 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>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><msup><mi>p</mi><mrow><mi></mi></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><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>, we get <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'>&leq;</mo><mn>3</mn></mrow></mfenced><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>1</mn></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><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><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><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><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><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><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><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><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><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>$p</mi><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$p</mi></mrow></mfenced><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>$p</mi><mrow><mi mathvariant='normal'></mi></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$p</mi></mrow></mfenced><mrow><mn>2</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$p</mi><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$p</mi></mrow></mfenced><mrow><mn>3</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Prob</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>@
qu.4.4.editing=useHTML@
qu.4.4.solution=@
qu.4.4.algorithm=$Shots=switch(rint(3), 60, 65, 70);
$p=$Shots/100;
$Prob=maple("
X:=Statistics[RandomVariable](Geometric($p)):
PX:=Statistics[Probability](X <= 2.0):
PX
");@
qu.4.4.uid=cc6263ed-d9b0-49e3-9850-6447e61ccc3c@
qu.4.4.info=  Course=Introductory Statistics;
  Topic=Geometric Distribution;
  Author=Lorna Deeth;
  Difficulty=Hard;
  Features=None;
  Type=Application;
@
qu.4.4.weighting=1@
qu.4.4.numbering=alpha@
qu.4.4.part.1.name=sro_id_1@
qu.4.4.part.1.answer.units=@
qu.4.4.part.1.numStyle= scientific  @
qu.4.4.part.1.editing=useHTML@
qu.4.4.part.1.showUnits=false@
qu.4.4.part.1.err=0.01@
qu.4.4.part.1.question=(Unset)@
qu.4.4.part.1.mode=Numeric@
qu.4.4.part.1.grading=toler_abs@
qu.4.4.part.1.negStyle=both@
qu.4.4.part.1.answer.num=$Prob@
qu.4.4.question=<p>Richard is an avid bowler, and can throw a strike $Shots % of the time.&nbsp; Assuming his shots are independent of each other, what is the probability that it will take Richard&nbsp;no more than&nbsp;3 shots before he gets his first strike?</p><p>&nbsp;</p><p>Round your answer to&nbsp;at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.4.5.mode=Multiple Selection@
qu.4.5.name=Definitions of Geometric 2@
qu.4.5.comment=@
qu.4.5.editing=useHTML@
qu.4.5.solution=@
qu.4.5.algorithm=@
qu.4.5.uid=08888d4e-2f9f-4265-a6c6-169a02dd07d5@
qu.4.5.info=  Course=Introductory Statistics;
  Topic=Geometric Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.4.5.question=<p>Which of the following statements are true?</p>
<p>&nbsp;</p>
<p>There may be more than&nbsp;one correct answer; select all that are true.</p>@
qu.4.5.answer=1, 2, 3@
qu.4.5.choice.1=The probability of success in a geometric distribution remains constant between each trial.@
qu.4.5.choice.2=If the probability of success in a geometric distribution is equal to 0.5, then the mean and the variance are the same.@
qu.4.5.choice.3=The mean of X, a geometric random variable, is 1/p, where p is the probability of success on a single trial.@
qu.4.5.choice.4=The geometric distribution is dependent upon two parameters.@
qu.4.5.choice.5=In a geometric distribution, if the first success occurs on the Xth trial, then on the first X - 1 trials there must have been only a small number of successes.@
qu.4.5.fixed=@

qu.4.6.mode=Inline@
qu.4.6.name=Calculate P(X > x)@
qu.4.6.comment=<p>To find <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'>&gt;</mo><mi>$x</mi></mrow></mfenced></mrow></mstyle></math>, it is necessary to work with the complement, which is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>$x</mi></mrow></mfenced><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>1</mn></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$x</mi></mrow></mfenced></mrow></mstyle></math>.&nbsp; Using the geometric formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>p</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>p</mi></mrow></mfenced><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math>, we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>$x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$p</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$p</mi></mrow></mfenced><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><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><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>$x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ProbComp</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p>Therefore, <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'>&gt;</mo><mi>$x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>$x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$ProbComp</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Prob</mi></mrow></mstyle></math></p>@
qu.4.6.editing=useHTML@
qu.4.6.solution=@
qu.4.6.algorithm=$p=rand(0.1, 0.3, 2);
$x=range(3,4);
$x1=$x-1;
$ProbComp=maple("
X:=Statistics[RandomVariable](Geometric($p)):
PX:=Statistics[Probability](X <= $x1):
PX
");
$Prob=1-$ProbComp;@
qu.4.6.uid=51cb45a5-4ced-4063-a9d6-42cff320839b@
qu.4.6.info=  Course=Introductory Statistics;
  Topic=Geometric Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.4.6.weighting=1@
qu.4.6.numbering=alpha@
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qu.4.6.part.1.answer.units=@
qu.4.6.part.1.numStyle=   @
qu.4.6.part.1.editing=useHTML@
qu.4.6.part.1.showUnits=false@
qu.4.6.part.1.err=0.01@
qu.4.6.part.1.question=(Unset)@
qu.4.6.part.1.mode=Numeric@
qu.4.6.part.1.grading=toler_abs@
qu.4.6.part.1.negStyle=both@
qu.4.6.part.1.answer.num=$Prob@
qu.4.6.question=<p>Let <em>X</em> be a discrete random variable that follows a geometric distribution with <em>p = </em>$p.</p><p>&nbsp;</p><p>What is P(X&nbsp;> $x)?</p><p>&nbsp;</p><p>Round your answer to&nbsp;at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.4.7.mode=Inline@
qu.4.7.name=Scratch 'N Win Question: Calculate probability of at least x trials@
qu.4.7.comment=<p>If the random variable <em>X</em> is the trial on which the first success occurs, then <em>X</em> follows a geometric distribution with a probability of success <em>p</em> = $p.&nbsp; To find the probability that it takes at least $x <em>Scratch 'N Win</em> tickets before you win your first prize, we need to determine <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$x</mi></mrow></mfenced></mrow></mstyle></math>.&nbsp; Here, it is necessary to work with the complement, 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><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>$x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xminus1</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p>Using the geometric formula, we get <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'>&lt;</mo><mi>x</mi></mrow></mfenced><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><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</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><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</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>$xminus1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ProbComp</mi></mrow></mstyle></math>.&nbsp; Therefore, <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'>&GreaterEqual;</mo><mi>$x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$ProbComp</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Prob</mi></mrow></mstyle></math></p>@
qu.4.7.editing=useHTML@
qu.4.7.solution=@
qu.4.7.algorithm=$p=rand(0.1, 0.15, 2);
$x=switch(rint(3), 3,4,5);
$xminus1=$x-1;
$Prob=maple("
X:=Statistics[RandomVariable](Geometric($p)):
PX:=Statistics[Probability](X >= $xminus1):
PX
");
$ProbComp=1-$Prob;@
qu.4.7.uid=3f9c745a-1bfe-4ac5-a9b1-1ced7d1a03d1@
qu.4.7.info=  Course=Introductory Statistics;
  Topic=Geometric Distribution;
  Author=Lorna Deeth;
  Difficulty=Hard;
  Features=None;
  Type=Application;
@
qu.4.7.weighting=1@
qu.4.7.numbering=alpha@
qu.4.7.part.1.name=sro_id_1@
qu.4.7.part.1.answer.units=@
qu.4.7.part.1.numStyle=   @
qu.4.7.part.1.editing=useHTML@
qu.4.7.part.1.showUnits=false@
qu.4.7.part.1.err=0.01@
qu.4.7.part.1.question=(Unset)@
qu.4.7.part.1.mode=Numeric@
qu.4.7.part.1.grading=toler_abs@
qu.4.7.part.1.negStyle=both@
qu.4.7.part.1.answer.num=$Prob@
qu.4.7.question=<p>The probability of winning a prize on a <em>Scratch 'N Win</em> ticket is $p.&nbsp; Assuming that the tickets are all independent of each other, what is the probability that it will take you at least $x tickets before you win your first prize?</p><p>&nbsp;</p><p>Round your answer to&nbsp;at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.4.8.mode=Inline@
qu.4.8.name=Calculate P(X <= x)@
qu.4.8.comment=<p>To find <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'>&leq;</mo><mi>$x</mi></mrow></mfenced></mrow></mstyle></math>,&nbsp;we can use the&nbsp;geometric formula to calculate&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>$x</mi></mrow></mfenced><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>1</mn></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$x</mi></mrow></mfenced></mrow></mstyle></math>.&nbsp; Here, this becomes <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>$x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</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><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</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>$x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Prob</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>@
qu.4.8.editing=useHTML@
qu.4.8.solution=@
qu.4.8.algorithm=$p=rand(0.4, 0.6, 2);
$x=range(2,4);
$x1=$x-1;
$Prob=maple("
X:=Statistics[RandomVariable](Geometric($p)):
PX:=Statistics[Probability](X <= $x1):
PX
");@
qu.4.8.uid=ec1e21ea-2f4b-439e-a61d-d623e29d174f@
qu.4.8.info=  Course=Introductory Statistics;
  Topic=Geometric Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.4.8.weighting=1@
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qu.4.8.part.1.name=sro_id_1@
qu.4.8.part.1.answer.units=@
qu.4.8.part.1.numStyle=   @
qu.4.8.part.1.editing=useHTML@
qu.4.8.part.1.showUnits=false@
qu.4.8.part.1.err=0.01@
qu.4.8.part.1.question=(Unset)@
qu.4.8.part.1.mode=Numeric@
qu.4.8.part.1.grading=toler_abs@
qu.4.8.part.1.negStyle=both@
qu.4.8.part.1.answer.num=$Prob@
qu.4.8.question=<p>Let <em>X</em> be a discrete random variable that follows a geometric distribution with <em>p = </em>$p.</p><p>&nbsp;</p><p>What is&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>$x</mi></mrow></mfenced></mrow></mstyle></math> ?</p><p>&nbsp;</p><p>Round your answer to&nbsp;at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.4.9.mode=Inline@
qu.4.9.name=Border Crossing: Probability of being searched.@
qu.4.9.comment=<p>If the random variable <em>X</em> is the trial on which the first success occurs, with a probability of success <em>p</em>, then <em>X</em> follows a geometric distribution.</p>
<p>Here, we want to determine the probability that your car is the first one that is searched when there are $x cars in front of you.&nbsp; This means there had to be $x failures before the first success occurred on trial $x1 (your car).&nbsp; We can then use the geometric formula to solve <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$x1</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$p</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$p</mi></mrow></mfenced><mrow><mi>$x1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Prob</mi></mrow></mstyle></math></p>@
qu.4.9.editing=useHTML@
qu.4.9.solution=@
qu.4.9.algorithm=$Border=switch(rint(2), 40, 50);
$p=1/$Border;
$x=range(30,35);
$x1=$x+1;
$Prob=$p*(1-$p)^($x);@
qu.4.9.uid=63094509-9844-4ca8-87e0-ba8fc7ccd699@
qu.4.9.info=  Course=Introductory Statistics;
  Topic=Geometric Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Application;
@
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qu.4.9.part.1.answer.num=$Prob@
qu.4.9.question=<p>When crossing into the United States at the Fort Erie Peace Bridge, approximately 1 in $Border cars are randomly selected for a search.&nbsp; If there are $x cars ahead of you in line, what is the probability that your car will be the first one selected for a random search?</p><p>&nbsp;</p><p>Round your answer to&nbsp;at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.4.10.mode=Inline@
qu.4.10.name=Calculate P(X = x)@
qu.4.10.comment=<p>Using the geometric 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>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><mi>p</mi><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><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math>, we get <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>$x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$p</mi><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><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Prob</mi></mrow></mstyle></math></p>@
qu.4.10.editing=useHTML@
qu.4.10.solution=@
qu.4.10.algorithm=$p=rand(0.1, 0.3, 2);
$x=range(5,9);
$Prob=$p*(1-$p)^($x-1);@
qu.4.10.uid=60f903f4-bd64-46ba-97c3-8eb66fcf4d35@
qu.4.10.info=  Course=Introductory Statistics;
  Topic=Geometric Distribution;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
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qu.4.10.part.1.answer.num=$Prob@
qu.4.10.question=<p>Let <em>X</em> be a discrete random variable that follows a geometric distribution with <em>p = </em>$p.</p><p>&nbsp;</p><p>What is P(X = $x)?</p><p>&nbsp;</p><p>Round your answer to at least&nbsp;3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.4.11.mode=Inline@
qu.4.11.name=Calculate P(X >= x | X < y)@
qu.4.11.comment=<p>To find <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'>&GreaterEqual;</mo><mi>$x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$y</mi></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>,&nbsp;we start with the basic conditional probability 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>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</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>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi></mrow></mfenced></mrow><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.&nbsp; Here, this becomes <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>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$y</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>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mi>$x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$y</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><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$y</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.&nbsp;</p>
<p>In the numerator, <em>X</em>&nbsp; is greater than or equal to $x and less than $y only at the values $x and $z.&nbsp; We&nbsp;can then&nbsp;use the&nbsp;geometric formula to calculate&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$x</mi><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$y</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$z</mi></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$p</mi><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$p</mi></mrow></mfenced><mrow><mi>$x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</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>$z</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$ProbNum</mi></mrow></mstyle></math>&nbsp;</p>
<p>In the denominator, we can calculate <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='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$y</mi></mrow></mfenced></mrow></mstyle></math>&nbsp;using the geometric formula for&nbsp;&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>$y</mi></mrow></mfenced><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>1</mn></mrow></mfenced><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><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='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><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><mi>$z</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ProbDenom</mi></mrow></mstyle></math></p>
<p>Therefore, <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>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$y</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>$ProbNum</mi><mrow><mi>$ProbDenom</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$CondProb</mi></mrow></mstyle></math></p>@
qu.4.11.editing=useHTML@
qu.4.11.solution=@
qu.4.11.algorithm=$p=rand(0.6, 0.8, 2);
$x=switch(rint(2), 2,3);
$x1=$x-1;
$y=$x+2;
$y1=$y-1;
$z=$x+1;
$z1=$z-1;
$Prob=maple("
X:=Statistics[RandomVariable](Geometric($p)):
PX:=Statistics[Probability](X < $y1):
PNum1:=Statistics[Probability](X = $x1):
PNum2:=Statistics[Probability](X = $z1):
PX, PNum1, PNum2
");
$ProbDenom=switch(0, $Prob);
$ProbNum=switch(1, $Prob) + switch(2, $Prob);
$CondProb=$ProbNum/$ProbDenom;@
qu.4.11.uid=5bea7b20-c618-490a-984b-d595c87e38f8@
qu.4.11.info=  Course=Introductory Statistics;
  Topic=Geometric Distribution;
  Author=Lorna Deeth;
  Difficulty=Hard;
  Features=None;
  Type=Calculation;
@
qu.4.11.weighting=1@
qu.4.11.numbering=alpha@
qu.4.11.part.1.name=sro_id_1@
qu.4.11.part.1.answer.units=@
qu.4.11.part.1.numStyle=   @
qu.4.11.part.1.editing=useHTML@
qu.4.11.part.1.showUnits=false@
qu.4.11.part.1.err=0.01@
qu.4.11.part.1.question=(Unset)@
qu.4.11.part.1.mode=Numeric@
qu.4.11.part.1.grading=toler_abs@
qu.4.11.part.1.negStyle=both@
qu.4.11.part.1.answer.num=$CondProb@
qu.4.11.question=<p>Let <em>X</em> be a discrete random variable that follows a geometric distribution with <em>p = </em>$p.</p><p>&nbsp;</p><p>What is&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mi>$x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><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'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$y</mi></mrow></mfenced></mrow></mstyle></math> ?</p><p>&nbsp;</p><p>Round your answer to at&nbsp;least&nbsp;3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.5.topic=Hypergeometric Distribution@

qu.5.1.mode=Inline@
qu.5.1.name=Men/Women Question: Calculate probability of x@
qu.5.1.comment=<p>If the random variable <em>X</em> is the number of successes out of <em>n</em> trials, when we are sampling without replacement (i.e. the probability of success changes from trial to trial), then <em>X</em> follows a hypergeometric distribution with a population size of $Pop, $NumMen successes in the population, and a sample size of $SampleSize.</p>
<p>To calculate the probability of getting $SampleMen men in a randomly selected sample of size $SampleSize, we can use the hypergeometric formula as:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SampleMen</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mi>$SampleMen</mi></mrow><mrow><mi>$NumMen</mi></mrow></msubsup></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mi>$SampleSize</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$SampleMen</mi></mrow><mrow><mi>$Pop</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$NumMen</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><msubsup><mi>C</mi><mrow><mi>$SampleSize</mi></mrow><mrow><mi>$Pop</mi></mrow></msubsup></mrow></mfrac></mrow></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Prob</mi></mrow></mstyle></math></p>@
qu.5.1.editing=useHTML@
qu.5.1.solution=@
qu.5.1.algorithm=$NumWomen=switch(rint(2), 30, 35);
$NumMen=switch(rint(2), 50, 55);
$Pop=$NumWomen+$NumMen;
$SampleSize=range(6,8);
$SampleMen=range(3, $SampleSize-2);
$Prob=maple("
X:=Statistics[RandomVariable](Hypergeometric($Pop, $NumMen, $SampleSize)):
PX:=Statistics[Probability](X=$SampleMen):
PXDecimal:=evalf(PX):
PXDecimal
");@
qu.5.1.uid=8b0f36eb-8313-41c9-8fa2-5418229d4fe6@
qu.5.1.info=  Course=Introductory Statistics;
  Topic=Hypergeometric Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Application;
@
qu.5.1.weighting=1@
qu.5.1.numbering=alpha@
qu.5.1.part.1.name=sro_id_1@
qu.5.1.part.1.answer.units=@
qu.5.1.part.1.numStyle=   @
qu.5.1.part.1.editing=useHTML@
qu.5.1.part.1.showUnits=false@
qu.5.1.part.1.err=0.01@
qu.5.1.part.1.question=(Unset)@
qu.5.1.part.1.mode=Numeric@
qu.5.1.part.1.grading=toler_abs@
qu.5.1.part.1.negStyle=both@
qu.5.1.part.1.answer.num=$Prob@
qu.5.1.question=<p>In a population of $Pop people, there are $NumWomen women and $NumMen men.&nbsp; If $SampleSize people are randomly selected without replacement, what is the probability that exactly $SampleMen of the people in the sample are male?</p><p>&nbsp;</p><p>Round your answer to at least&nbsp;3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.5.2.mode=Inline@
qu.5.2.name=Calculate P(X = x)@
qu.5.2.comment=<p>Using the hypergeometric formula of <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><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mi>x</mi></mrow><mrow><mi>M</mi></mrow></msubsup></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>x</mi></mrow><mrow><mi>N</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>M</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mi>n</mi></mrow><mrow><mi>N</mi></mrow></msubsup></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>, where <em>N</em> is the population size, <em>M</em> is the number of successes in the population, and <em>n</em> is the sample size, we 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><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mi>$x</mi></mrow><mrow><mi>$Success</mi></mrow></msubsup></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mi>$SampleSize</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$x</mi></mrow><mrow><mi>$Pop</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$Success</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mi>$SampleSize</mi></mrow><mrow><mi>$Pop</mi></mrow></msubsup></mrow></mfenced></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Prob</mi></mrow></mstyle></math>.</p>@
qu.5.2.editing=useHTML@
qu.5.2.solution=@
qu.5.2.algorithm=$Pop=range(20,24);
$Success=range(4,6);
$SampleSize=range(10,12);
$x=range(1, $Success-2);
$Prob=maple("
X:=Statistics[RandomVariable](Hypergeometric($Pop, $Success, $SampleSize)):
PX:=Statistics[Probability](X = $x):
PXDecimal:=evalf(PX):
PXDecimal
");@
qu.5.2.uid=20e10404-31d2-40fe-a426-345f8dc1569b@
qu.5.2.info=  Course=Introductory Statistics;
  Topic=Hypergeometric Distribution;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
qu.5.2.weighting=1@
qu.5.2.numbering=alpha@
qu.5.2.part.1.name=sro_id_1@
qu.5.2.part.1.answer.units=@
qu.5.2.part.1.numStyle=   @
qu.5.2.part.1.editing=useHTML@
qu.5.2.part.1.showUnits=false@
qu.5.2.part.1.err=0.0010@
qu.5.2.part.1.question=(Unset)@
qu.5.2.part.1.mode=Numeric@
qu.5.2.part.1.grading=toler_abs@
qu.5.2.part.1.negStyle=both@
qu.5.2.part.1.answer.num=$Prob@
qu.5.2.question=<p>Let <em>X</em> be a discrete random variable that follows a hypergeometric distribution with a population size of <em>N</em> = $Pop,&nbsp;the number of successes in the population&nbsp;is <em>M</em> = $Success, and a random sample of size <em>n</em> = $SampleSize is taken without replacement.</p><p>&nbsp;</p><p>What is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$x</mi></mrow></mfenced></mrow></mstyle></math>?</p><p>&nbsp;</p><p>Round your answer to&nbsp;at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.5.3.mode=Inline@
qu.5.3.name=Calculate P(X >= x)@
qu.5.3.comment=<p>Using the hypergeometric formula of <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><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mi>x</mi></mrow><mrow><mi>M</mi></mrow></msubsup></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>x</mi></mrow><mrow><mi>N</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>M</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mi>n</mi></mrow><mrow><mi>N</mi></mrow></msubsup></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>, where <em>N</em> is the population size, <em>M</em> is the number of successes in the population, and <em>n</em> is the sample size, we 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><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mi>$x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SampleSize</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mi>$x</mi></mrow><mrow><mi>$Success</mi></mrow></msubsup></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mi>$SampleSize</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$x</mi></mrow><mrow><mi>$Pop</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$Success</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mi>$SampleSize</mi></mrow><mrow><mi>$Pop</mi></mrow></msubsup></mrow></mfenced></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow></mrow><mrow><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mi>$SampleSize</mi></mrow><mrow><mi>$Success</mi></mrow></msubsup></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mi>$SampleSize</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$SampleSize</mi></mrow><mrow><mi>$Pop</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$Success</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mi>$SampleSize</mi></mrow><mrow><mi>$Pop</mi></mrow></msubsup></mrow></mfenced></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Prob</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>@
qu.5.3.editing=useHTML@
qu.5.3.solution=@
qu.5.3.algorithm=$Pop=range(16,18);
$Success=switch(rint(2), 8,9);
$SampleSize=switch(rint(2), 4,5);
$x=$SampleSize-1;
$Prob=maple("
X:=Statistics[RandomVariable](Hypergeometric($Pop, $Success, $SampleSize)):
PX:=Statistics[Probability](X >= $x):
PXDecimal:=evalf(PX):
PXDecimal
");@
qu.5.3.uid=2ce14f23-1de7-4a98-a731-25f9a5b1a931@
qu.5.3.info=  Course=Introductory Statistics;
  Topic=Hypergeometric Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.5.3.weighting=1@
qu.5.3.numbering=alpha@
qu.5.3.part.1.name=sro_id_1@
qu.5.3.part.1.answer.units=@
qu.5.3.part.1.numStyle=   @
qu.5.3.part.1.editing=useHTML@
qu.5.3.part.1.showUnits=false@
qu.5.3.part.1.err=0.01@
qu.5.3.part.1.question=(Unset)@
qu.5.3.part.1.mode=Numeric@
qu.5.3.part.1.grading=toler_abs@
qu.5.3.part.1.negStyle=both@
qu.5.3.part.1.answer.num=$Prob@
qu.5.3.question=<p>Let <em>X</em> be a discrete random variable that follows a hypergeometric distribution with a population size of <em>N</em> = $Pop,&nbsp;the number of successes in the population&nbsp;is <em>M</em> = $Success, and a random sample of size <em>n</em> = $SampleSize that is taken without replacement.</p><p>&nbsp;</p><p>What is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mi>$x</mi></mrow></mfenced></mrow></mstyle></math>?</p><p>&nbsp;</p><p>Round your answer to at least&nbsp;3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.5.4.mode=Multiple Selection@
qu.5.4.name=Definitions of Hypergeometric 1@
qu.5.4.comment=@
qu.5.4.editing=useHTML@
qu.5.4.solution=@
qu.5.4.algorithm=@
qu.5.4.uid=7d73f683-add6-411b-a7e3-9c8dcb95845e@
qu.5.4.info=  Course=Introductory Statistics;
  Topic=Hypergeometric Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.5.4.question=<p>Which of the following statements are true?</p>
<p>&nbsp;</p>
<p>There may be more than&nbsp;one correct answer; select all that are true.</p>@
qu.5.4.answer=2, 3@
qu.5.4.choice.1=The hypergeometric distribution is not appropriate to use whenever the probability of success changes depending on the outcome of previous trials.@
qu.5.4.choice.2=The hypergeometric distribution depends on the population size, the sample size, and number of successes within the population.@
qu.5.4.choice.3=If X is a hypergeometric random variable, then X represents a count of the number of successes in a set number of trials.@
qu.5.4.choice.4=In a hypergeometric distribution, the trials are independent of each other.@
qu.5.4.choice.5=If X is a hypergeometric random variable, then X can take on n different values.@
qu.5.4.fixed=@

qu.5.5.mode=Inline@
qu.5.5.name=Definitions of Hypergeometric 1&2: Random Selection of T/F@
qu.5.5.comment=@
qu.5.5.editing=useHTML@
qu.5.5.solution=@
qu.5.5.algorithm=$k1=rint(10);
$k2=rint(10);
$k3=rint(10);
$k4=rint(10);
$k5=rint(10);
$z=maple("S:=$k1,$k2,$k3,$k4,$k5:
floor(nops({S})/nops([S]))
");
condition:$z;
$a=("'The hypergeometric distribution is not appropriate to use whenever the probability of success changes depending on the outcome of previous trials.'");
$b=("'The hypergeometric distribution depends on the population size, the sample size, and number of successes within the population.'");
$c=("'If X is a hypergeometric random variable, then X represents a count of the number of successes in a set number of trials.'");
$d=("'In a hypergeometric distribution, the random variable X must be greater than or equal to 0.'");
$e=("'In a hypergeometric distribution, if x is the number of successes in a sample size of n, then n - x is the number of failures in the sample.'");
$f=("'In a hypergeometric distribution, the trials are independent of each other.'");
$g=("'If X is a hypergeometric random variable, then X can take on n different values.'");
$h=("'The maximum value the random variable X can take on in a hypergeometric distribution is M, the number of successes in the population.'");
$i=("'In a hypergeometric distribution, if M is the number of successes in the population, and N is the population size, then there are N+M failures in the population.'");
$j=("'In a hypergeometric distribution, if M is the number of success in the population, and x is the number of successes in the sample, then x must be strictly less than M.'");
$Answers=["'False'","'True'","'True'","'True'","'True'","'False'","'False'","'False'","'False'","'False'"];
$Distractors=["'True'","'False'","'False'","'False'","'False'","'True'","'True'","'True'","'True'","'True'"];
$Q1=switch($k1, $a,$b,$c,$d,$e,$f,$g,$h,$i,$j);
$A1=switch($k1, $Answers);
$D1=switch($k1, $Distractors);
$Q2=switch($k2, $a,$b,$c,$d,$e,$f,$g,$h,$i,$j);
$A2=switch($k2, $Answers);
$D2=switch($k2, $Distractors);
$Q3=switch($k3, $a,$b,$c,$d,$e,$f,$g,$h,$i,$j);
$A3=switch($k3, $Answers);
$D3=switch($k3, $Distractors);
$Q4=switch($k4, $a,$b,$c,$d,$e,$f,$g,$h,$i,$j);
$A4=switch($k4, $Answers);
$D4=switch($k4, $Distractors);
$Q5=switch($k5, $a,$b,$c,$d,$e,$f,$g,$h,$i,$j);
$A5=switch($k5, $Answers);
$D5=switch($k5, $Distractors);@
qu.5.5.uid=b7381146-343d-4b61-91ca-5999f22d650a@
qu.5.5.info=  Course=Introductory Statistics;
  Topic=Hypergeometric Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.5.5.weighting=1,1,1,1,1@
qu.5.5.numbering=alpha@
qu.5.5.part.1.grader=exact@
qu.5.5.part.1.name=sro_id_1@
qu.5.5.part.1.editing=useHTML@
qu.5.5.part.1.display.permute=true@
qu.5.5.part.1.question=(Unset)@
qu.5.5.part.1.answer.2=$D1@
qu.5.5.part.1.answer.1=$A1@
qu.5.5.part.1.mode=List@
qu.5.5.part.1.display=menu@
qu.5.5.part.1.credit.2=0.0@
qu.5.5.part.1.credit.1=1.0@
qu.5.5.part.2.grader=exact@
qu.5.5.part.2.name=sro_id_2@
qu.5.5.part.2.editing=useHTML@
qu.5.5.part.2.display.permute=true@
qu.5.5.part.2.question=(Unset)@
qu.5.5.part.2.answer.2=$D2@
qu.5.5.part.2.answer.1=$A2@
qu.5.5.part.2.mode=List@
qu.5.5.part.2.display=menu@
qu.5.5.part.2.credit.2=0.0@
qu.5.5.part.2.credit.1=1.0@
qu.5.5.part.3.grader=exact@
qu.5.5.part.3.name=sro_id_3@
qu.5.5.part.3.editing=useHTML@
qu.5.5.part.3.display.permute=true@
qu.5.5.part.3.question=(Unset)@
qu.5.5.part.3.answer.2=$D3@
qu.5.5.part.3.answer.1=$A3@
qu.5.5.part.3.mode=List@
qu.5.5.part.3.display=menu@
qu.5.5.part.3.credit.2=0.0@
qu.5.5.part.3.credit.1=1.0@
qu.5.5.part.4.grader=exact@
qu.5.5.part.4.name=sro_id_4@
qu.5.5.part.4.editing=useHTML@
qu.5.5.part.4.display.permute=true@
qu.5.5.part.4.question=(Unset)@
qu.5.5.part.4.answer.2=$D4@
qu.5.5.part.4.answer.1=$A4@
qu.5.5.part.4.mode=List@
qu.5.5.part.4.display=menu@
qu.5.5.part.4.credit.2=0.0@
qu.5.5.part.4.credit.1=1.0@
qu.5.5.part.5.grader=exact@
qu.5.5.part.5.name=sro_id_5@
qu.5.5.part.5.editing=useHTML@
qu.5.5.part.5.display.permute=true@
qu.5.5.part.5.question=(Unset)@
qu.5.5.part.5.answer.2=$D5@
qu.5.5.part.5.answer.1=$A5@
qu.5.5.part.5.mode=List@
qu.5.5.part.5.display=menu@
qu.5.5.part.5.credit.2=0.0@
qu.5.5.part.5.credit.1=1.0@
qu.5.5.question=<p>Identify each of the following statements as either TRUE or FALSE.</p><p>&nbsp;</p><p>a)&nbsp;<span>&nbsp;</span><1><span>&nbsp;</span> $Q1</p><p>&nbsp;</p><p>b)&nbsp;<span>&nbsp;</span><2><span>&nbsp;</span> $Q2</p><p>&nbsp;</p><p>c)&nbsp;<span>&nbsp;</span><3><span>&nbsp;</span> $Q3</p><p>&nbsp;</p><p>d)&nbsp;<span>&nbsp;</span><4><span>&nbsp;</span> $Q4</p><p>&nbsp;</p><p>e)&nbsp;<span>&nbsp;</span><5><span>&nbsp;</span> $Q5</p>@

qu.5.6.mode=Multiple Selection@
qu.5.6.name=Definitions of Hypergeometric 2@
qu.5.6.comment=@
qu.5.6.editing=useHTML@
qu.5.6.solution=@
qu.5.6.algorithm=@
qu.5.6.uid=6880b9ed-8c9d-4c8a-8faa-3706182048eb@
qu.5.6.info=  Course=Introductory Statistics;
  Topic=Hypergeometric Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.5.6.question=<p>Which of the following statements are true?</p>
<p>&nbsp;</p>
<p>There may be more than&nbsp;one correct answer; select all that are true.</p>@
qu.5.6.answer=1, 2@
qu.5.6.choice.1=In a hypergeometric distribution, the random variable X must be greater than or equal to 0.@
qu.5.6.choice.2=In a hypergeometric distribution, if X is the number of successes in a sample size of n, then n - x is the number of failures in the sample.@
qu.5.6.choice.3=The maximum value the random variable X can take on in a hypergeometric distribution is M, the number of successes in the population@
qu.5.6.choice.4=In a hypergeometric distribution, if M is the number of successes in the population, and N is the population size, then there are N + M failures in the population.@
qu.5.6.choice.5=In a hypergeometric distribution, if M is the number of success in the population, and X is the number of successes in the sample, then X must be strictly less than M@
qu.5.6.fixed=@

qu.5.7.mode=Inline@
qu.5.7.name=Single Parent Families: Calculate probability of all or none.@
qu.5.7.comment=<p>a)&nbsp; If the random variable <em>X</em> is the number of successes out of <em>n</em> trials, when sampling without replacement, then <em>X</em> follows a hypergeometric distribution with a population size of <em>N</em> = $N, the number of successes in the population is <em>M</em> = $M, and a randomly selected sample size of <em>n</em> = $n.</p>
<p>To determine the probability that none of the randomly selected children come from a single-parent family, we can use the hypergeometric formula to calculate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mn>0</mn></mrow><mrow><mi>$M</mi></mrow></msubsup></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mi>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0</mn></mrow><mrow><mi>$N</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$M</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mi>$n</mi></mrow><mrow><mi>$N</mi></mrow></msubsup></mrow></mfenced></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ProbA</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>
<p>b)&nbsp; Now, to calculate the probability that all of the randomly selected children come from a single-parent family, we can again use the hypergeometric formula to calculate <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>$n</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mi>$n</mi></mrow><mrow><mi>$M</mi></mrow></msubsup></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mi>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$n</mi></mrow><mrow><mi>$N</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$M</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mi>$n</mi></mrow><mrow><mi>$N</mi></mrow></msubsup></mrow></mfenced></mrow></mfrac></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ProbB</mi></mrow></mstyle></math></p>@
qu.5.7.editing=useHTML@
qu.5.7.solution=@
qu.5.7.algorithm=$N=switch(rint(2), 20,21);
$M=switch(rint(2), 8,9);
$n=switch(rint(2), 4,5);
$m=maple("
X:=Statistics[RandomVariable](Hypergeometric($N, $M, $n)):
PX1:=Statistics[Probability](X=0):
PX2:=Statistics[Probability](X=$n):
PX1Decimal:=evalf(PX1):
PX2Decimal:=evalf(PX2):
PX1Decimal, PX2Decimal
");
$ProbA=switch(0,$m);
$ProbB=switch(1,$m);@
qu.5.7.uid=efd8ce7e-9875-45db-8b85-6f76290f2f7a@
qu.5.7.info=  Course=Introductory Statistics;
  Topic=Hypergeometric Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Application;
@
qu.5.7.weighting=1,1@
qu.5.7.numbering=alpha@
qu.5.7.part.1.name=sro_id_1@
qu.5.7.part.1.answer.units=@
qu.5.7.part.1.numStyle=   @
qu.5.7.part.1.editing=useHTML@
qu.5.7.part.1.showUnits=false@
qu.5.7.part.1.err=0.0010@
qu.5.7.part.1.question=(Unset)@
qu.5.7.part.1.mode=Numeric@
qu.5.7.part.1.grading=toler_abs@
qu.5.7.part.1.negStyle=both@
qu.5.7.part.1.answer.num=$ProbA@
qu.5.7.part.2.name=sro_id_2@
qu.5.7.part.2.answer.units=@
qu.5.7.part.2.numStyle=   @
qu.5.7.part.2.editing=useHTML@
qu.5.7.part.2.showUnits=false@
qu.5.7.part.2.err=0.0010@
qu.5.7.part.2.question=(Unset)@
qu.5.7.part.2.mode=Numeric@
qu.5.7.part.2.grading=toler_abs@
qu.5.7.part.2.negStyle=both@
qu.5.7.part.2.answer.num=$ProbB@
qu.5.7.question=<p>In a classroom of $N children, $M of them are from single-parent families.&nbsp; If a random sample of $n students is selected without replacement, and assuming that the children are all independent of each other, what is:</p><p>&nbsp;</p><p>a)&nbsp; The probability that none of the randomly selected students are from a single-parent family?</p><p>&nbsp;</p><p>Round your answer to at least&nbsp;3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; The probability that all of the randomly selected students are from a single-parent family?</span></p><p>&nbsp;</p><p><span>Round your answer to at least&nbsp;3 decimal places.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.5.8.mode=Inline@
qu.5.8.name=Calculate P(X < x)@
qu.5.8.comment=<p>Using the hypergeometric formula of <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><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mi>x</mi></mrow><mrow><mi>M</mi></mrow></msubsup></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>x</mi></mrow><mrow><mi>N</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>M</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mi>n</mi></mrow><mrow><mi>N</mi></mrow></msubsup></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>, where <em>N</em> is the population size, <em>M</em> is the number of successes in the population, and <em>n</em> is the sample size, we 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><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>2</mn></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</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='0.2222222em' rspace='0.2222222em'>&plus;</mo><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><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mn>0</mn></mrow><mrow><mi>$Success</mi></mrow></msubsup></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mi>$SampleSize</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0</mn></mrow><mrow><mi>$Pop</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$Success</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mi>$SampleSize</mi></mrow><mrow><mi>$Pop</mi></mrow></msubsup></mrow></mfenced></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mi mathvariant='normal'></mi></mrow><mrow><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mn>1</mn></mrow><mrow><mi>$Success</mi></mrow></msubsup></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mi>$SampleSize</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow><mrow><mi>$Pop</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$Success</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mi>$SampleSize</mi></mrow><mrow><mi>$Pop</mi></mrow></msubsup></mrow></mfenced></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Prob</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.</p>@
qu.5.8.editing=useHTML@
qu.5.8.solution=@
qu.5.8.algorithm=$Pop=range(20,22);
$Success=switch(rint(2), 4,5);
$SampleSize=switch(rint(2), 7,8);
$Prob=maple("
X:=Statistics[RandomVariable](Hypergeometric($Pop, $Success, $SampleSize)):
PX:=Statistics[Probability](X < 2):
PXDecimal:=evalf(PX):
PXDecimal
");@
qu.5.8.uid=516da99b-1cfa-4a41-9de5-c98d00cd4386@
qu.5.8.info=  Course=Introductory Statistics;
  Topic=Hypergeometric Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
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qu.5.8.part.1.answer.num=$Prob@
qu.5.8.question=<p>Let <em>X</em> be a discrete random variable that follows a hypergeometric distribution with a population size of <em>N</em> = $Pop,&nbsp;the number of successes in the population&nbsp;is <em>M</em> = $Success, and a random sample of size <em>n</em> = $SampleSize that is taken without replacement.</p><p>&nbsp;</p><p>What 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>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>2</mn></mrow></mfenced></mrow></mstyle></math>?</p><p>&nbsp;</p><p>Round your answer to at least&nbsp;3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

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qu.5.9.name=Job Interview: Calculate probability of at least 1@
qu.5.9.comment=<p>Since the random variable <em>X</em> is the number of successes out of <em>n</em> trials, in a situation in which we are sampling without replacement, then <em>X</em> follows a hypergeometric distribution with a population size of $N, a total of 10 successes within the population, and a randomly selected sample of $n individuals.</p>
<p>To find the probability&nbsp;that at least one applicant has a statistics background, we need to determine <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'>&GreaterEqual;</mo><mn>1</mn></mrow></mfenced></mrow></mstyle></math>.&nbsp; One way to do this would be to use the hypergeometric formula for values of <em>X</em> greater than or equal to 1, up to $n, such that: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n</mi></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.&nbsp; Note that <em>X</em> cannot be larger than the sample size.</p>
<p>An easier way to approach this question would be to use the complement of <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'>&GreaterEqual;</mo><mn>1</mn></mrow></mfenced></mrow></mstyle></math>, 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><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>1</mn></mrow></mfenced><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></mstyle></math>.&nbsp; Therefore, using the hypergeometric formula we 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><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</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'>&lt;</mo><mn>1</mn></mrow></mfenced><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><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><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><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><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><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><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><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><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><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><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><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><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><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><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><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><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mn>0</mn></mrow><mrow><mn>10</mn></mrow></msubsup></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mi>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0</mn></mrow><mrow><mi>$N</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>10</mn></mrow></msubsup></mrow></mfenced></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>C</mi><mrow><mi>$n</mi></mrow><mrow><mi>$N</mi></mrow></msubsup></mrow></mfenced></mrow></mfrac><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><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><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><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><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><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><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$ProbComp</mi><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Prob</mi></mrow></mstyle></math></p>@
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qu.5.9.solution=@
qu.5.9.algorithm=$N=switch(rint(3), 40,45,50);
$n=switch(rint(3), 5,6,7);
$Prob=maple("
X:=Statistics[RandomVariable](Hypergeometric($N, 10, $n)):
PX:=Statistics[Probability](X >= 1):
PXDecimal:=evalf(PX):
PXDecimal
");
$ProbComp=1-$Prob;@
qu.5.9.uid=53750707-96e0-4098-bc18-df028604d4cb@
qu.5.9.info=  Course=Introductory Statistics;
  Topic=Hypergeometric Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Application;
@
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qu.5.9.part.1.err=0.0010@
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qu.5.9.part.1.answer.num=$Prob@
qu.5.9.question=<p>In order to qualify for a job opening, applicants must have a some experience in statistics.&nbsp;&nbsp;$N individuals submit an application for the job, of which only 10 individuals actually have experience with statistics.&nbsp; Suppose that $n applicants are randomly selected to be interviewed on the first day, with the remaining applicants to be interviewed later in the week (we can assume that once an individual has been selected to be interviewed, he or she will not be selected again).&nbsp; What is the probability that of the first $n applicants to be interviewed, at least one of them will have statistical experience?</p><p>&nbsp;</p><p>Round your answer to&nbsp;at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.6.topic=Poisson Distribution@

qu.6.1.mode=Inline@
qu.6.1.name=Dandelion Study: Calculate probability of more than x@
qu.6.1.comment=<p>If the random variable <em>X</em> is the number of occurrences (in this case, dandelions) in a unit of space, then <em>X</em> follows a Poisson distribution with a mean of $lambda.&nbsp; To find the probability of more than $x dandelions in one square metre, we need to find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$x</mi></mrow></mfenced></mrow></mstyle></math>.&nbsp; In this case, it is necessary to work with the complement, 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><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>$x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$x</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p>Using the formula for the Poisson distribution, we can calculate the complement as:</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'>&leq;</mo><mi>$x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$x</mi></mrow></mfenced><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><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><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><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><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><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><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><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msup><mi>$lambda</mi><mrow><mn>0</mn></mrow></msup><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$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><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><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><mi>$lambda</mi></mrow></msup></mrow><mrow><mi>$x</mi><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><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><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><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><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><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><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><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ProbComp</mi></mrow></mstyle></math></p>
<p>Therefore, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>2</mn></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mn>2</mn></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$ProbComp</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Prob</mi></mrow></mstyle></math></p>@
qu.6.1.editing=useHTML@
qu.6.1.solution=@
qu.6.1.algorithm=$lambda=range(4,6);
$x=switch(rint(2), 2,3);
$Prob=maple("
X:=Statistics[RandomVariable](Poisson($lambda)):
PX:=Statistics[Probability](X > $x):
PXDecimal:=evalf(PX):
PXDecimal
");
$ProbComp=1-$Prob;@
qu.6.1.uid=ee5b7609-9b55-458e-a17b-fc3fdf77f9a3@
qu.6.1.info=  Course=Introductory Statistics;
  Topic=Poisson Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Application;
@
qu.6.1.weighting=1@
qu.6.1.numbering=alpha@
qu.6.1.part.1.name=sro_id_1@
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qu.6.1.part.1.numStyle=   @
qu.6.1.part.1.editing=useHTML@
qu.6.1.part.1.showUnits=false@
qu.6.1.part.1.err=0.01@
qu.6.1.part.1.question=(Unset)@
qu.6.1.part.1.mode=Numeric@
qu.6.1.part.1.grading=toler_abs@
qu.6.1.part.1.negStyle=both@
qu.6.1.part.1.answer.num=$Prob@
qu.6.1.question=<p>Suppose the distribution of dandelions in a large meadow is approximately Poisson, with a mean of $lambda dandelions per square metre.&nbsp; In a randomly selected square metre plot&nbsp;of the meadow, what is the&nbsp;probability there are more than $x dandelions?</p><p>&nbsp;</p><p>Round your answer to&nbsp;at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.6.2.mode=Inline@
qu.6.2.name=Calculate P(X < x | X > y)@
qu.6.2.comment=<p>To find <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'>&lt;</mo><mi>$x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$y</mi></mrow></mfenced></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>, we&nbsp;start with the basic conditional probability formula&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><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</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>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi></mrow></mfenced></mrow><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.&nbsp;&nbsp;Here, this becomes&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><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>$x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$y</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>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>$x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$y</mi></mrow></mfenced></mrow><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$y</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>&nbsp;.&nbsp;</p>
<p>In the numerator, value of <em>X</em> that&nbsp;is&nbsp;less than $x and&nbsp;greater than $y&nbsp;is just $xminus1, so we&nbsp;can use the Poisson formula&nbsp;just for that&nbsp;value:&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>$x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$y</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xminus1</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msup><mi>$lambda</mi><mrow><mi>$xminus1</mi></mrow></msup><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$lambda</mi></mrow></msup></mrow><mrow><mi>$xminus1</mi><mo lspace='0.1111111em' rspace='0.1111111em'>&excl;</mo></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ProbNum</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p>In the denominator, we need to use the&nbsp;Poisson formula for the complement of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$y</mi></mrow></mfenced></mrow></mstyle></math>, such that&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'>&gt;</mo><mi>$y</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>$y</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</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='0.2222222em' 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><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$y</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ProbDenom</mi></mrow></mstyle></math></p>
<p>Therefore, <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>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>$x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$y</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>$ProbNum</mi><mrow><mi>$ProbDenom</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$CondProb</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>@
qu.6.2.editing=useHTML@
qu.6.2.solution=@
qu.6.2.algorithm=$lambda=switch(rint(3), 3,4,5);
$k1=rint(2);
$y=switch($k1, 2,3);
$x=switch($k1, 4,5);
$xminus1=$x-1;
$ProbDenom=maple("
Y:=Statistics[RandomVariable](Poisson($lambda)):
PY:=Statistics[Probability](Y>$y):
PYDecimal:=evalf(PY):
PYDecimal
");
$ProbNum=($lambda^$xminus1)*(e^(-1*$lambda))/fact($xminus1);
$CondProb=$ProbNum/$ProbDenom;@
qu.6.2.uid=64acddf9-e780-427b-9a6f-8ed33c8a2f5d@
qu.6.2.info=  Course=Introductory Statistics;
  Topic=Poisson Distribution;
  Author=Lorna Deeth;
  Difficulty=Hard;
  Features=None;
  Type=Calculation;
@
qu.6.2.weighting=1@
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qu.6.2.part.1.editing=useHTML@
qu.6.2.part.1.showUnits=false@
qu.6.2.part.1.err=0.01@
qu.6.2.part.1.question=(Unset)@
qu.6.2.part.1.mode=Numeric@
qu.6.2.part.1.grading=toler_abs@
qu.6.2.part.1.negStyle=both@
qu.6.2.part.1.answer.num=$CondProb@
qu.6.2.question=<p>Let <em>X</em> be a discrete random variable that follows a Poisson distribution with <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><mi>$lambda</mi></mrow></mstyle></math>.</p><p>&nbsp;</p><p>What is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>$x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$y</mi></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>?</p><p>&nbsp;</p><p>Round your answer to&nbsp;at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.6.3.mode=Multiple Selection@
qu.6.3.name=Definitions of Poisson 1@
qu.6.3.comment=@
qu.6.3.editing=useHTML@
qu.6.3.solution=@
qu.6.3.algorithm=@
qu.6.3.uid=86cd91aa-2740-4f77-907f-cb8a9b286bff@
qu.6.3.info=  Course=Introductory Statistics;
  Topic=Poisson Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.6.3.question=<p>Which of the following statements are true?</p>
<p>&nbsp;</p>
<p>Note that there may be more than&nbsp;one correct answer; select all that are true.</p>@
qu.6.3.answer=1, 2@
qu.6.3.choice.1=The smallest value a Poisson random variable can be is 0.@
qu.6.3.choice.2=The mean and variance of a Poisson random variable are always equal to each other.@
qu.6.3.choice.3=The mean of a Poisson random variable can be negative.@
qu.6.3.choice.4=Because there is no fixed upper-limit for the value of a Poisson random variable, it is not a discrete random variable.@
qu.6.3.choice.5=There are no parameters in a Poisson distribution.@
qu.6.3.fixed=@

qu.6.4.mode=Inline@
qu.6.4.name=Definitions of Poisson 1&2: Random Selection of T/F@
qu.6.4.comment=@
qu.6.4.editing=useHTML@
qu.6.4.solution=@
qu.6.4.algorithm=$k1=rint(10);
$k2=rint(10);
$k3=rint(10);
$k4=rint(10);
$k5=rint(10);
$z=maple("S:=$k1,$k2,$k3,$k4,$k5:
floor(nops({S})/nops([S]))
");
condition:$z;
$a=("'The smallest value a Poisson random variable can be is 0.'");
$b=("'The mean and variance of a Poisson random variable are always equal to each other.'");
$c=("'If X is a Poisson random variable, then X represents a count.'");
$d=("'A Poisson distribution can be applied when counting the number of times an event of interest occurs over time, volume, area or distance.'");
$e=("'One condition required of a Poisson random variable is that the events of interest are occurring independently over time, volume, area or distance.'");
$f=("'The mean of a Poisson random variable can be negative.'");
$g=("'Because there is no fixed upper-limit for the value of a Poisson random variable, X, it is not a discrete random variable.'");
$h=("'There are no parameters in a Poisson distribution.'");
$i=("'In a Poisson distribution, the probability of an event occurring in a given unit of time will actually change over time.'");
$j=maple("convert(cat(`In a Poisson distribution,`,MathML[ExportPresentation](lambda),`is the probability of an event occurring within a given unit of time.`),string)");
$Answers=["'True'","'True'","'True'","'True'","'True'","'False'","'False'","'False'","'False'","'False'"];
$Distractors=["'False'","'False'","'False'","'False'","'False'","'True'","'True'","'True'","'True'","'True'"];
$Q1=switch($k1, $a,$b,$c,$d,$e,$f,$g,$h,$i,$j);
$A1=switch($k1, $Answers);
$D1=switch($k1, $Distractors);
$Q2=switch($k2, $a,$b,$c,$d,$e,$f,$g,$h,$i,$j);
$A2=switch($k2, $Answers);
$D2=switch($k2, $Distractors);
$Q3=switch($k3, $a,$b,$c,$d,$e,$f,$g,$h,$i,$j);
$A3=switch($k3, $Answers);
$D3=switch($k3, $Distractors);
$Q4=switch($k4, $a,$b,$c,$d,$e,$f,$g,$h,$i,$j);
$A4=switch($k4, $Answers);
$D4=switch($k4, $Distractors);
$Q5=switch($k5, $a,$b,$c,$d,$e,$f,$g,$h,$i,$j);
$A5=switch($k5, $Answers);
$D5=switch($k5, $Distractors);@
qu.6.4.uid=c887646c-c116-49b3-a073-6800b1090ba4@
qu.6.4.info=  Course=Introductory Statistics;
  Topic=Poisson Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
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qu.6.4.part.3.question=(Unset)@
qu.6.4.part.3.answer.2=$D3@
qu.6.4.part.3.answer.1=$A3@
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qu.6.4.part.4.answer.2=$D4@
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qu.6.4.part.5.grader=exact@
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qu.6.4.part.5.answer.2=$D5@
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qu.6.4.question=<p>Identify each of the following statements as either TRUE or FALSE.</p><p>&nbsp;</p><p>a)&nbsp;<span>&nbsp;</span><1><span>&nbsp;</span> $Q1</p><p>&nbsp;</p><p>b)&nbsp;<span>&nbsp;</span><2><span>&nbsp;</span> $Q2</p><p>&nbsp;</p><p>c)&nbsp;<span>&nbsp;</span><3><span>&nbsp;</span> $Q3</p><p>&nbsp;</p><p>d)&nbsp;<span>&nbsp;</span><4><span>&nbsp;</span> $Q4</p><p>&nbsp;</p><p>e)&nbsp;<span>&nbsp;</span><5><span>&nbsp;</span> $Q5</p>@

qu.6.5.mode=Multiple Selection@
qu.6.5.name=Definitions of Poisson 2@
qu.6.5.comment=@
qu.6.5.editing=useHTML@
qu.6.5.solution=@
qu.6.5.algorithm=@
qu.6.5.uid=aa3c1141-307c-462a-9535-a267266d1e03@
qu.6.5.info=  Course=Introductory Statistics;
  Topic=Poisson Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.6.5.question=<p>Which of the following statements are true?</p>
<p>&nbsp;</p>
<p>Note that there may be more than&nbsp;one correct answer; select all that are true.</p>@
qu.6.5.answer=1, 2, 3@
qu.6.5.choice.1=If X is a Poisson random variable, then X represents a count.@
qu.6.5.choice.2=A Poisson distribution can be applied when counting the number of times an event of interest occurs over time, volume, area or distance.@
qu.6.5.choice.3=One condition required of a Poisson random variable is that the events of interest are occurring independently over time, volume, area or distance.@
qu.6.5.choice.4=In a Poisson distribution, the probability of an event occurring in a given unit of time will actually change over time.@
qu.6.5.choice.5=In a Poisson distribution, <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></mrow></mstyle></math> is the probability of an event occurring over a given unit of time.@
qu.6.5.fixed=@

qu.6.6.mode=Inline@
qu.6.6.name=Calculate P(X < x)@
qu.6.6.comment=<p>To find <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'>&lt;</mo><mi>$x</mi></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>, we&nbsp;can use the Poisson formula&nbsp;for values of&nbsp;<em>X</em> less than $x:&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>$x</mi></mrow></mfenced><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><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><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xminus1</mi></mrow></mfenced></mrow></mstyle></math>.&nbsp;&nbsp;Therefore, we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>$x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msup><mi>$lambda</mi><mrow><mn>0</mn></mrow></msup><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$lambda</mi></mrow></msup></mrow><mrow><mn>0</mn><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</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><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><msup><mi>$lambda</mi><mrow><mi>$xminus1</mi></mrow></msup><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$lambda</mi></mrow></msup></mrow><mrow><mi>$xminus1</mi><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Prob</mi></mrow></mstyle></math>.</p>@
qu.6.6.editing=useHTML@
qu.6.6.solution=@
qu.6.6.algorithm=$lambda=switch(rint(2), 5,6);
$x=switch(rint(2), 2,3);
$xminus1=$x-1;
$Prob=maple("
X:=Statistics[RandomVariable](Poisson($lambda)):
PX:=Statistics[Probability](X<$x):
PXDecimal:=evalf(PX):
PXDecimal
");@
qu.6.6.uid=4fe8218c-ee97-4e4e-80b7-9dfa91463215@
qu.6.6.info=  Course=Introductory Statistics;
  Topic=Poisson Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.6.6.weighting=1@
qu.6.6.numbering=alpha@
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qu.6.6.part.1.numStyle=   @
qu.6.6.part.1.editing=useHTML@
qu.6.6.part.1.showUnits=false@
qu.6.6.part.1.err=0.01@
qu.6.6.part.1.question=(Unset)@
qu.6.6.part.1.mode=Numeric@
qu.6.6.part.1.grading=toler_abs@
qu.6.6.part.1.negStyle=both@
qu.6.6.part.1.answer.num=$Prob@
qu.6.6.question=<p>Let <em>X</em> be a discrete random variable that follows a Poisson distribution with <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><mi>$lambda</mi></mrow></mstyle></math>.</p><p>&nbsp;</p><p>What is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>$x</mi></mrow></mfenced></mrow></mstyle></math>?</p><p>&nbsp;</p><p>Round your answer to&nbsp;at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.6.7.mode=Inline@
qu.6.7.name=Calculate P(X >= x)@
qu.6.7.comment=<p>To find <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'>&GreaterEqual;</mo><mi>$x</mi></mrow></mfenced></mrow></mstyle></math>, we need to use the complement, which is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>$x</mi></mrow></mfenced><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><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><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xminus1</mi></mrow></mfenced></mrow></mstyle></math>.&nbsp; Using the formula for the Poisson distribution, we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>$x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msup><mi>$lambda</mi><mrow><mn>0</mn></mrow></msup><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$lambda</mi></mrow></msup></mrow><mrow><mn>0</mn><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</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><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><msup><mi>$lambda</mi><mrow><mi>$xminus1</mi></mrow></msup><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$lambda</mi></mrow></msup></mrow><mrow><mi>$xminus1</mi><mo lspace='0.1111111em' rspace='0.1111111em'>&excl;</mo></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ProbComp</mi></mrow></mstyle></math>.</p>
<p>Therefore, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mi>$x</mi></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'>&lt;</mo><mi>$x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$ProbComp</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Prob</mi></mrow></mstyle></math></p>@
qu.6.7.editing=useHTML@
qu.6.7.solution=@
qu.6.7.algorithm=$lambda=switch(rint(2), 4,5);
$x=switch(rint(2), 2,3);
$xminus1=$x-1;
$Prob=maple("
X:=Statistics[RandomVariable](Poisson($lambda)):
PX:=Statistics[Probability](X>=$x):
PXDecimal:=evalf(PX):
PXDecimal
");
$ProbComp=1-$Prob;@
qu.6.7.uid=6aa9c89f-6c39-4960-9acf-fd93aca8771d@
qu.6.7.info=  Course=Introductory Statistics;
  Topic=Poisson Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
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qu.6.7.part.1.negStyle=both@
qu.6.7.part.1.answer.num=$Prob@
qu.6.7.question=<p>Let <em>X</em> be a discrete random variable that follows a Poisson distribution with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&lambda;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$lambda</mi></mrow></mstyle></math>.</p><p>&nbsp;</p><p>What 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>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mi>$x</mi></mrow></mfenced></mrow></mstyle></math>?</p><p>&nbsp;</p><p>Round your answer to&nbsp;at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.6.8.mode=Inline@
qu.6.8.name=Doctor's Office: Calculate mean, probability of x@
qu.6.8.comment=<p>a)&nbsp; If the average number of patients in one hour is $HourRate, then in the three hours the clinic is open, the average number of&nbsp;patients would be 3*$HourRate = $lambda.</p>
<p>&nbsp;</p>
<p>b)&nbsp; To&nbsp;calculate the probability that only&nbsp;$x patients are seen in the three hours before lunch, we need to use the formula for the Poisson distribution to calculate <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>$x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msup><mi>$lambda</mi><mrow><mi>$x</mi></mrow></msup><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$lambda</mi></mrow></msup></mrow><mrow><mi>$x</mi><mo lspace='0.1111111em' rspace='0.1111111em'>&excl;</mo></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Prob</mi></mrow></mstyle></math>&nbsp;</p>@
qu.6.8.editing=useHTML@
qu.6.8.solution=@
qu.6.8.algorithm=$HourRate=switch(rint(2), 4,5);
$lambda=3*$HourRate;
$x=range(8,10);
$Prob=($lambda^$x)*(e^(-1*$lambda))/fact($x);@
qu.6.8.uid=a3d18a4f-659a-4ca4-a3d2-1eec36aa5b30@
qu.6.8.info=  Course=Introductory Statistics;
  Topic=Poisson Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Application;
@
qu.6.8.weighting=1,1@
qu.6.8.numbering=alpha@
qu.6.8.part.1.name=sro_id_1@
qu.6.8.part.1.answer.units=@
qu.6.8.part.1.numStyle=   @
qu.6.8.part.1.editing=useHTML@
qu.6.8.part.1.showUnits=false@
qu.6.8.part.1.question=(Unset)@
qu.6.8.part.1.mode=Numeric@
qu.6.8.part.1.grading=exact_value@
qu.6.8.part.1.negStyle=both@
qu.6.8.part.1.answer.num=$lambda@
qu.6.8.part.2.name=sro_id_2@
qu.6.8.part.2.answer.units=@
qu.6.8.part.2.numStyle=   @
qu.6.8.part.2.editing=useHTML@
qu.6.8.part.2.showUnits=false@
qu.6.8.part.2.err=0.0010@
qu.6.8.part.2.question=(Unset)@
qu.6.8.part.2.mode=Numeric@
qu.6.8.part.2.grading=toler_abs@
qu.6.8.part.2.negStyle=both@
qu.6.8.part.2.answer.num=$Prob@
qu.6.8.question=<p>Suppose that the number of patients that a health clinic sees follows a Poisson distribution with an average rate of $HourRate patients per hour.</p><p>&nbsp;</p><p>a)&nbsp; If the clinic opens at 9:00 am, what would be the expected number of patients seen before the clinic closes for lunch at noon.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; What is the probability that the clinic will see exactly $x patients in the hours between when the clinic opens and when it closes for lunch.</span></p><p>&nbsp;</p><p><span>Round your answer to&nbsp;at least 3 decimal places.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.6.9.mode=Inline@
qu.6.9.name=Calculate P(X = x)@
qu.6.9.comment=<p>Using the formula for the Poisson distribution, we get <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>$x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msup><mi>$lambda</mi><mrow><mi>$x</mi></mrow></msup><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$lambda</mi></mrow></msup></mrow><mrow><mi>$x</mi><mo lspace='0.1111111em' rspace='0.0em'>&excl;</mo></mrow></mfrac></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Prob</mi></mrow></mstyle></math></p>@
qu.6.9.editing=useHTML@
qu.6.9.solution=@
qu.6.9.algorithm=$lambda=range(4,7);
$x=range(2,5);
condition:not(eq($lambda,$x));
$Prob=($lambda^$x*e^(-1*$lambda))/fact($x);@
qu.6.9.uid=e9f3bb09-586d-4785-8fe5-6fd084c463cc@
qu.6.9.info=  Course=Introductory Statistics;
  Topic=Poisson Distribution;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
qu.6.9.weighting=1@
qu.6.9.numbering=alpha@
qu.6.9.part.1.name=sro_id_1@
qu.6.9.part.1.answer.units=@
qu.6.9.part.1.numStyle=   @
qu.6.9.part.1.editing=useHTML@
qu.6.9.part.1.showUnits=false@
qu.6.9.part.1.err=0.0010@
qu.6.9.part.1.question=(Unset)@
qu.6.9.part.1.mode=Numeric@
qu.6.9.part.1.grading=toler_abs@
qu.6.9.part.1.negStyle=both@
qu.6.9.part.1.answer.num=$Prob@
qu.6.9.question=<p>Let <em>X</em> be a discrete random variable that follows a Poisson distribution with <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><mi>$lambda</mi></mrow></mstyle></math>.</p><p>&nbsp;</p><p>What is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$x</mi></mrow></mfenced></mrow></mstyle></math>?</p><p>&nbsp;</p><p>Round your answer to&nbsp;at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

