qu.1.topic=Basic Probability@

qu.1.1.mode=Inline@
qu.1.1.name=P(B|A^c)@
qu.1.1.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><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>B</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><msubsup><mi>A</mi><mi></mi><mrow><mi>c</mi></mrow></msubsup><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mstyle></math>, we need to make use of the general formula for 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>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, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>B</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><msup><mi>A</mi><mrow><mi>c</mi></mrow></msup><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>B</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><msup><mi>A</mi><mrow><mi>c</mi></mrow></msup></mrow></mfenced></mrow><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>A</mi><mrow><mi>c</mi></mrow></msup></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Since <em>P(A) = $A, </em>then <em>P(A<sup>c</sup>) = 1 - P(A) = 1 - $A = $AComp</em>.</p>
<p>To find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>A</mi><mrow><mi>c</mi></mrow></msup></mrow></mfenced></mrow></mstyle></math>, it is easiest to display the given probabilities in a Venn diagram:</p>
<p>&nbsp;</p>
<p><div align="center">
	<applet width="400" height="300" code="applets.labelImage.LabelImage" archive="modules/applets.jar">
		<param name="image" value="__BASE_URI__Pictures/Venn1.jpg" />
		<param name="size" value="3" />
		<param name="label.1.x" value="150" />
		<param name="label.1.y" value="160" />
		<param name="label.1.text" value="$AIntersectBComp" />
		<param name="label.2.x" value="220" />
		<param name="label.2.y" value="160" />
		<param name="label.2.text" value="$Intersection" />    
		<param name="label.3.x" value="280" />
		<param name="label.3.y" value="160" />
		<param name="label.3.text" value="$ACompIntersectB" />
	</applet>
</div></p>
<p>From the Venn diagram, it can be seen 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>B</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><msup><mi>A</mi><mrow><mi>c</mi></mrow></msup></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ACompIntersectB</mi></mrow></mstyle></math>.&nbsp; Therefore, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>B</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>A</mi><mrow><mi>c</mi></mrow></msup><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$ACompIntersectB</mi></mrow><mrow><mi>$AComp</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$BGivenAComp</mi></mrow></mstyle></math></p>@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$A=rand(0.2, 0.3, 2);
$B=rand(0.6, 0.8, 2);
$Intersection=rand(0.11, 0.19, 2);
$AComp=1-$A;
$ACompIntersectB=$B-$Intersection;
$BGivenAComp=$ACompIntersectB/$AComp;
$AIntersectBComp=$A-$Intersection;
condition:not(eq($AComp,$ACompIntersectB));
condition:lt($BGivenAComp,1.0);@
qu.1.1.uid=268e99c0-d254-4c87-b4ad-5434ccb9537a@
qu.1.1.info=  Course=Introductory Statistics;
  Topic=Basic Probability;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.1.1.weighting=1@
qu.1.1.numbering=alpha@
qu.1.1.part.1.name=sro_id_1@
qu.1.1.part.1.answer.units=@
qu.1.1.part.1.numStyle=   @
qu.1.1.part.1.editing=useHTML@
qu.1.1.part.1.showUnits=false@
qu.1.1.part.1.err=0.0010@
qu.1.1.part.1.question=(Unset)@
qu.1.1.part.1.mode=Numeric@
qu.1.1.part.1.grading=toler_abs@
qu.1.1.part.1.negStyle=both@
qu.1.1.part.1.answer.num=$BGivenAComp@
qu.1.1.question=<p>Given that <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$A</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</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><mi>$B</mi></mrow></mstyle></math>and the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mi>B</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Intersection</mi></mrow></mstyle></math>,&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>B</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><msup><mi>A</mi><mrow><mi>c</mi></mrow></msup></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mi mathvariant='normal'></mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>?</p><p>&nbsp;</p><p>&nbsp;</p><p>Round&nbsp;your answer to&nbsp;at least 3 decimal places.</p><p>&nbsp;</p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi><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><msup><mi>A</mi><mrow><mi>c</mi></mrow></msup></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>&nbsp;</span><1><span>&nbsp;</span>&nbsp;</p>@

qu.1.2.mode=Inline@
qu.1.2.name=P((A Intersect B)^c)@
qu.1.2.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><msup><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><mi>c</mi></mrow></msup></mrow></mfenced></mrow></mstyle></math>, we make use of the fact 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>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><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><msup><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><mi>c</mi></mrow></msup></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></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><msup><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><mi>c</mi></mrow></msup></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>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><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$Intersection</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$InterComp</mi></mrow></mstyle></math>.</p>@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=$A=rand(0.2, 0.3, 2);
$B=rand(0.6, 0.8, 2);
$Intersection=rand(0.11, 0.19, 2);
$InterComp=1 - $Intersection;@
qu.1.2.uid=6c21c1c9-854f-41cf-b323-336bf0d17e5f@
qu.1.2.info=  Course=Introductory Statistics;
  Topic=Basic Probability;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
qu.1.2.weighting=1@
qu.1.2.numbering=alpha@
qu.1.2.part.1.name=sro_id_1@
qu.1.2.part.1.answer.units=@
qu.1.2.part.1.numStyle=   @
qu.1.2.part.1.editing=useHTML@
qu.1.2.part.1.showUnits=false@
qu.1.2.part.1.question=(Unset)@
qu.1.2.part.1.mode=Numeric@
qu.1.2.part.1.grading=exact_value@
qu.1.2.part.1.negStyle=both@
qu.1.2.part.1.answer.num=$InterComp@
qu.1.2.question=<p>Given 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>A</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><mi>$A</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$B</mi></mrow></mstyle></math>and 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>A</mi><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mi>B</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Intersection</mi></mrow></mstyle></math>,&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=','><msup><mrow><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&cap;</mo></mrow><mi>B</mi></mrow></mfenced></mrow><mrow><mi>c</mi></mrow></msup></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>?</p><p>&nbsp;</p><p>&nbsp;</p><p>Enter&nbsp;your answer to&nbsp;2 decimal places.</p><p>&nbsp;</p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><msup><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&cap;</mo></mrow><mi>B</mi></mrow></mfenced><mrow><mi>c</mi></mrow></msup></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>&nbsp;</span><1><span>&nbsp;</span>&nbsp;</p>@

qu.1.3.mode=Inline@
qu.1.3.name=Dice Roll: Calculate P(-), P(A U B), P(A^c Int B^c) with Venn Diagram@
qu.1.3.comment=<p>a)&nbsp; To calculate $Question, we simply add up the number of simple events in $Event (in this case, there are $Numerator simple events), and divide by 6, which is the total number of simple events in the sample space.&nbsp; Therefore, $Question = $Answer.</p>
<p>&nbsp;</p>
<p>b)&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>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi></mrow></mfenced></mrow></mstyle></math>, we can 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>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'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</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>B</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mi>B</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.&nbsp; Because there are no simple events common to both <em>A</em> and <em>B</em>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>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><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></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>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</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><mfrac><mn>3</mn><mrow><mn>6</mn></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><mn>6</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Union</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>c)&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><msup><mi>A</mi><mrow><mi>c</mi></mrow></msup><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><msup><mi>B</mi><mrow><mi>c</mi></mrow></msup></mrow></mfenced></mrow></mstyle></math>, we need to count the number of simple events that are <strong>not</strong> in <em>A</em> and <strong>not</strong> in <em>B</em>.&nbsp; Here, there are 2 simple events in <em>A<sup>c</sup> </em>and in <em>B<sup>c</sup>, </em>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><msup><mi>A</mi><mrow><mi>c</mi></mrow></msup><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><msup><mi>B</mi><mrow><mi>c</mi></mrow></msup></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>2</mn><mrow><mn>6</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ACompIntBComp</mi></mrow></mstyle></math>.</p>@
qu.1.3.editing=useHTML@
qu.1.3.solution=@
qu.1.3.algorithm=$k=rint(2);
$QA=("'P(A)'");
$QB=("'P(B)'");
$Event=switch($k, "A", "B");
$Question=switch($k, $QA, $QB);
$Answer=switch($k, 3/6, 1/6);
$Numerator=switch($k, 3, 1);
$Union=4/6;
$ACompIntBComp=2/6;@
qu.1.3.uid=80c79992-1863-4d7b-80ed-7bb691083e57@
qu.1.3.info=  Course=Introductory Statistics;
  Topic=Basic Probability;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=Venn Diagram;
  Type=Calculation;
@
qu.1.3.weighting=1,1,1@
qu.1.3.numbering=alpha@
qu.1.3.part.1.name=sro_id_1@
qu.1.3.part.1.answer.units=@
qu.1.3.part.1.numStyle=   @
qu.1.3.part.1.editing=useHTML@
qu.1.3.part.1.showUnits=false@
qu.1.3.part.1.err=0.0010@
qu.1.3.part.1.question=(Unset)@
qu.1.3.part.1.mode=Numeric@
qu.1.3.part.1.grading=toler_abs@
qu.1.3.part.1.negStyle=both@
qu.1.3.part.1.answer.num=$Answer@
qu.1.3.part.2.name=sro_id_2@
qu.1.3.part.2.answer.units=@
qu.1.3.part.2.numStyle=   @
qu.1.3.part.2.editing=useHTML@
qu.1.3.part.2.showUnits=false@
qu.1.3.part.2.err=0.0010@
qu.1.3.part.2.question=(Unset)@
qu.1.3.part.2.mode=Numeric@
qu.1.3.part.2.grading=toler_abs@
qu.1.3.part.2.negStyle=both@
qu.1.3.part.2.answer.num=$Union@
qu.1.3.part.3.name=sro_id_3@
qu.1.3.part.3.answer.units=@
qu.1.3.part.3.numStyle=   @
qu.1.3.part.3.editing=useHTML@
qu.1.3.part.3.showUnits=false@
qu.1.3.part.3.err=0.0010@
qu.1.3.part.3.question=(Unset)@
qu.1.3.part.3.mode=Numeric@
qu.1.3.part.3.grading=toler_abs@
qu.1.3.part.3.negStyle=both@
qu.1.3.part.3.answer.num=$ACompIntBComp@
qu.1.3.question=<p>The following Venn Diagram illustrates the sample space when a simple die is rolled once.&nbsp; Event <em>A</em> occurs when the number shown on the die is even, and event <em>B</em> occurs when the number shown is 5.</p><p align="center"><img alt="" width="400" height="300" src="__BASE_URI__Pictures/Venn2.jpg" /></p><p>a)&nbsp; What is $Question ?</p><p>&nbsp;</p><p>Do <strong>not</strong> express your answer as a fraction.&nbsp;</p><p>Round your response 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 <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi></mrow></mfenced></mrow></mstyle></math>?</span></p><p>&nbsp;</p><p><span>Do <strong>not</strong> express your answer as a fraction.</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><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><msup><mi>A</mi><mrow><mi>c</mi></mrow></msup><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><msup><mi>B</mi><mrow><mi>c</mi></mrow></msup></mrow></mfenced></mrow></mstyle></math>?</span></span></p><p>&nbsp;</p><p><span><span>Do not express your answer as a fraction.</span></span></p><p><span><span>Round your response to&nbsp;at least&nbsp;3 decimal places.</span></span></p><p><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></p>@

qu.1.4.mode=Inline@
qu.1.4.name=Calculate P(B|A)@
qu.1.4.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><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>B</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>A</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mstyle></math>, we can use the conditional probability equation <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>B</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>A</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>B</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>A</mi></mrow></mfenced></mrow><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.&nbsp; All of these values were given to us in the question (note 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>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><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>B</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>A</mi></mrow></mfenced></mrow></mstyle></math>), therefore we can just substitute them into the equation: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>B</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>A</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>$Intersection</mi><mrow><mi>$A</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$BGivenA</mi></mrow></mstyle></math>.</p>@
qu.1.4.editing=useHTML@
qu.1.4.solution=@
qu.1.4.algorithm=$A=rand(0.2, 0.3, 2);
$B=rand(0.6, 0.8, 2);
$Intersection=rand(0.11, 0.19, 2);
$BGivenA=$Intersection/$A;
condition:not(eq($Intersection,$A));@
qu.1.4.uid=b9827097-44bd-4c0d-afa2-a12102b4ec3e@
qu.1.4.info=  Course=Introductory Statistics;
  Topic=Basic Probability;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
qu.1.4.weighting=1@
qu.1.4.numbering=alpha@
qu.1.4.part.1.name=sro_id_1@
qu.1.4.part.1.answer.units=@
qu.1.4.part.1.numStyle=   @
qu.1.4.part.1.editing=useHTML@
qu.1.4.part.1.showUnits=false@
qu.1.4.part.1.err=0.0010@
qu.1.4.part.1.question=(Unset)@
qu.1.4.part.1.mode=Numeric@
qu.1.4.part.1.grading=toler_abs@
qu.1.4.part.1.negStyle=both@
qu.1.4.part.1.answer.num=$BGivenA@
qu.1.4.question=<p>Given 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>A</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><mi>$A</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$B</mi></mrow></mstyle></math>and 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>A</mi><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mi>B</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Intersection</mi></mrow></mstyle></math>,&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>B</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>A</mi></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>?</p><p>&nbsp;</p><p>&nbsp;</p><p>Round&nbsp;your answer to&nbsp;at least&nbsp;3 decimal places.</p><p>&nbsp;</p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi><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>A</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>&nbsp;</span><1><span>&nbsp;</span>&nbsp;</p>@

qu.1.5.mode=Inline@
qu.1.5.name=P(A^c Intersect B^c)@
qu.1.5.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><msup><mi>A</mi><mrow><mi>c</mi></mrow></msup><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><msup><mi>B</mi><mrow><mi>c</mi></mrow></msup></mrow></mfenced></mrow></mstyle></math>, we need to find the probability of events that are not&nbsp;in <em>A</em> <strong>and</strong> not in <em>B</em>.&nbsp; This is equivalent to finding 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>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi></mrow></mfenced></mrow></mstyle></math>; that 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><msup><mi>A</mi><mrow><mi>c</mi></mrow></msup><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><msubsup><mi>B</mi><mi></mi><mrow><mi>c</mi></mrow></msubsup></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mi>B</mi></mrow></mfenced><mrow><mi>c</mi></mrow></msup></mrow></mfenced></mrow></mstyle></math>.&nbsp; 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>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'>&cup;</mo><mi>B</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><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</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>B</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><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><mi>B</mi></mrow></mfenced></mrow></mstyle></math>, we find 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>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mi>B</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$B</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Intersection</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Union</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><msup><mi>A</mi><mrow><mi>c</mi></mrow></msup><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><msup><mi>B</mi><mrow><mi>c</mi></mrow></msup></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$Union</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ACompIntersectBComp</mi></mrow></mstyle></math></p>@
qu.1.5.editing=useHTML@
qu.1.5.solution=@
qu.1.5.algorithm=$A=rand(0.2, 0.3, 2);
$B=rand(0.6, 0.8, 2);
$Intersection=rand(0.11, 0.19, 2);
$Union=$A + $B - $Intersection;
$ACompIntersectBComp=1-$Union;
condition:lt($Union,1);@
qu.1.5.uid=7d446816-49be-4612-889d-a93efbf0fa1f@
qu.1.5.info=  Course=Introductory Statistics;
  Topic=Basic Probability;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.1.5.weighting=1@
qu.1.5.numbering=alpha@
qu.1.5.part.1.name=sro_id_1@
qu.1.5.part.1.answer.units=@
qu.1.5.part.1.numStyle=   @
qu.1.5.part.1.editing=useHTML@
qu.1.5.part.1.showUnits=false@
qu.1.5.part.1.question=(Unset)@
qu.1.5.part.1.mode=Numeric@
qu.1.5.part.1.grading=exact_value@
qu.1.5.part.1.negStyle=both@
qu.1.5.part.1.answer.num=$ACompIntersectBComp@
qu.1.5.question=<p>Given 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>A</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><mi>$A</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$B</mi></mrow></mstyle></math>and 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>A</mi><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mi>B</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Intersection</mi></mrow></mstyle></math>,&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><msup><mi>A</mi><mrow><mi>c</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><msup><mi>B</mi><mrow><mi>c</mi></mrow></msup></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>?</p><p>&nbsp;</p><p>&nbsp;</p><p>Enter&nbsp;your answer to&nbsp;2 decimal places.</p><p>&nbsp;</p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><msup><mi>A</mi><mrow><mi>c</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><msup><mi>B</mi><mrow><mi>c</mi></mrow></msup></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>&nbsp;</span><1><span>&nbsp;</span>&nbsp;</p>@

qu.1.6.mode=Inline@
qu.1.6.name=P(A^c Intersect B)@
qu.1.6.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><msup><mi>A</mi><mrow><mi>c</mi></mrow></msup><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></mstyle></math>, it is easiest to diplay the given probabilities in a Venn diagram:</p>
<p>&nbsp;</p>
<p><div align="center">
	<applet width="400" height="300" code="applets.labelImage.LabelImage" archive="modules/applets.jar">
		<param name="image" value="__BASE_URI__Pictures/Venn1.jpg" />
		<param name="size" value="3" />
		<param name="label.1.x" value="150" />
		<param name="label.1.y" value="160" />
		<param name="label.1.text" value="$AIntersectBComp" />
		<param name="label.2.x" value="220" />
		<param name="label.2.y" value="160" />
		<param name="label.2.text" value="$Intersection" />    
		<param name="label.3.x" value="280" />
		<param name="label.3.y" value="160" />
		<param name="label.3.text" value="$ACompIntersectB" />
	</applet>
</div></p>
<p>From the Venn diagram, it can be seen 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><msup><mi>A</mi><mrow><mi>c</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mi>B</mi></mrow></mfenced></mrow></mstyle></math>, the area that is <strong>not</strong> in <em>A</em> but is in <em>B</em>, is $ACompIntersectB.</p>@
qu.1.6.editing=useHTML@
qu.1.6.solution=@
qu.1.6.algorithm=$A=rand(0.2, 0.3, 2);
$B=rand(0.6, 0.8, 2);
$Intersection=rand(0.11, 0.19, 2);
$ACompIntersectB=$B-$Intersection;
$AIntersectBComp=$A-$Intersection;@
qu.1.6.uid=96c33b2a-24fc-4419-bf0c-32dbd958aa5a@
qu.1.6.info=  Course=Introductory Statistics;
  Topic=Basic Probability;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.1.6.weighting=1@
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qu.1.6.part.1.name=sro_id_1@
qu.1.6.part.1.answer.units=@
qu.1.6.part.1.numStyle=   @
qu.1.6.part.1.editing=useHTML@
qu.1.6.part.1.showUnits=false@
qu.1.6.part.1.question=(Unset)@
qu.1.6.part.1.mode=Numeric@
qu.1.6.part.1.grading=exact_value@
qu.1.6.part.1.negStyle=both@
qu.1.6.part.1.answer.num=$ACompIntersectB@
qu.1.6.question=<p>Given 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>A</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><mi>$A</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$B</mi></mrow></mstyle></math>and 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>A</mi><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mi>B</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Intersection</mi></mrow></mstyle></math>,&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><msup><mi>A</mi><mrow><mi>c</mi></mrow></msup><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><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>?</p><p>&nbsp;</p><p>&nbsp;</p><p>Enter&nbsp;your answer to&nbsp;2 decimal places.</p><p>&nbsp;&nbsp;</p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><msup><mi>A</mi><mrow><mi>c</mi></mrow></msup><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><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>&nbsp;</span><1><span>&nbsp;</span>&nbsp;</p>@

qu.1.7.mode=Inline@
qu.1.7.name=Calculate P(A|B)@
qu.1.7.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><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></mrow></mstyle></math>, we can use the conditional probability equation <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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; Because these values are all given to us in the question, we can simply substitute them into the equation.&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>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><mi>$Intersection</mi><mrow><mi>$B</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$AGivenB</mi></mrow></mstyle></math>.</p>@
qu.1.7.editing=useHTML@
qu.1.7.solution=@
qu.1.7.algorithm=$A=rand(0.2, 0.3, 2);
$B=rand(0.6, 0.8, 2);
$Intersection=rand(0.11, 0.19, 2);
$AGivenB=$Intersection/$B;
condition:not(eq($Intersection,$B));@
qu.1.7.uid=9eda61cc-8282-47c4-bb8e-d7475bf329cf@
qu.1.7.info=  Course=Introductory Statistics;
  Topic=Basic Probability;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
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qu.1.7.part.1.editing=useHTML@
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qu.1.7.part.1.answer.num=$AGivenB@
qu.1.7.question=<p>Given 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>A</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><mi>$A</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$B</mi></mrow></mstyle></math>and 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>A</mi><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mi>B</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Intersection</mi></mrow></mstyle></math>,&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>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></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mi mathvariant='normal'></mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>?</p><p>&nbsp;</p><p>&nbsp;</p><p>Round&nbsp;your answer to at least&nbsp;3 decimal places.</p><p>&nbsp;&nbsp;</p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>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></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>&nbsp;</span><1><span>&nbsp;</span>&nbsp;</p>@

qu.1.8.mode=Inline@
qu.1.8.name=Coffee Cup odds and probability@
qu.1.8.comment=<p>a)&nbsp; To find the probability of winning, we need to convert the given odds to a probability.&nbsp; We can read the odds 1:$odds as "on average, there will be 1 win for every&nbsp;$odds losses", or equivalently "for every&nbsp;$Trials attempts, on average there will be 1 win and&nbsp;$odds losses".&nbsp; Therefore, to find the probability of winning we can 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>p</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mn>1</mn><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$odds</mi></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; Therefore, the probability of winning is $win.</p>
<p>&nbsp;</p>
<p>b)&nbsp; To find P(<em>At least one win</em>), it is easiest to work with the complement.&nbsp; In this case, the complement of P(<em>At least one win</em>) is P(<em>All three are losses</em>).&nbsp; Therefore, P(<em>At least one win</em>) = 1 - P(<em>All three are losses</em>).&nbsp; In part (a), we found that the probability of winning on a single trial is&nbsp;<em>p</em> = $win, therefore the probability of losing on a single trial is <em>q</em> = 1 -&nbsp;<em>p</em> = 1 -&nbsp;$win = $lose.&nbsp; Because the trials are independent, we can find P(<em>All three are losses</em>)&nbsp;= <em>q*q*q</em>&nbsp; = $LLL.&nbsp; Finally, P(<em>At least one win</em>) = 1 -&nbsp;$LLL = $OneWin.</p>@
qu.1.8.editing=useHTML@
qu.1.8.solution=@
qu.1.8.algorithm=$odds=range(5,8);
$Trials=1+$odds;
$win=1/(1+$odds);
$lose=1-$win;
$LLL=$lose^3;
$OneWin=1-$LLL;@
qu.1.8.uid=17ed9c6b-8704-428b-b451-01f7687bfb34@
qu.1.8.info=  Course=Introductory Statistics;
  Topic=Basic Probabiliy;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Application;
@
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qu.1.8.part.1.answer.num=$win@
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qu.1.8.part.2.answer.num=$OneWin@
qu.1.8.question=<p>In a coffee cup contest, participants can roll up the&nbsp;brim of their coffee cup to see if they have won a prize.&nbsp; On the coffee cup, it states that the odds of winning&nbsp;a prize are 1:$odds.</p><p>&nbsp;</p><p>a)&nbsp; For any&nbsp;random coffee purchase, what is the probability of winning a prize?</p><p>&nbsp;</p><p>Round&nbsp;your answer to at least&nbsp;3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p><span>b)&nbsp; If you were to buy 3 coffees in one day, what is the probability that at least one of the three would be a winning cup? (Assume that the cups can be considered independent.)</span></p><p>&nbsp;</p><p><span>Round&nbsp;your answer to&nbsp;at least&nbsp;3 decimal places.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.1.9.mode=Inline@
qu.1.9.name=Calculate P(B), P(A^c Int B), Independence with Venn Diagram@
qu.1.9.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>B</mi></mrow></mfenced></mrow></mstyle></math>, we use the fact that the sum of all the probabilities in the sample space must be 1.&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>$AintBComp</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$UnionComp</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><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</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><mn>1</mn></mrow></mstyle></math>, which when rearranged gives us <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced><mo lspace='0.0em' 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='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$AintBComp</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$UnionComp</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>$B</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; Because event <em>B</em> is a subset of event <em>A </em>(i.e. <em>B</em> is completely contained within <em>A</em>), there are no points where <em>B</em> intersects with the complement of <em>A</em>.&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><msup><mi>A</mi><mrow><mi>c</mi></mrow></msup><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><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>c)&nbsp; To determine if events <em>A</em> and <em>B</em> are independent, we can use the check for independence<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi></mrow></mfenced></mrow></mstyle></math>.&nbsp; From part (a), we can determine 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>A</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><mi>$A</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><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></mrow></mstyle></math>, we can use the 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='0.0em' 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; However, since <em>B</em> is a subset of <em>A</em>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>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><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>B</mi></mrow></mfenced></mrow></mstyle></math>, 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>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><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced></mrow><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></mstyle></math>.&nbsp; Thus, events <em>A</em> and <em>B</em> are not independent.</p>@
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qu.1.9.algorithm=$A=rand(0.5, 0.8, 2);
$B=rand(0.1, 0.3, 2);
$AintBComp=$A-$B;
$UnionComp=1-$A;@
qu.1.9.uid=41ee48d9-e305-430d-a376-444e249a903e@
qu.1.9.info=  Course=Introductory Statistics;
  Topic=Basic Probability;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=Venn Diagram;
  Type=Calculation;
@
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qu.1.9.question=<p>Consider the following Venn Diagram:</p><p>&nbsp;</p><div align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="400" height="300"><param name="image" value="__BASE_URI__Pictures/Venn3.jpg" /><param name="size" value="2" /><param name="label.1.x" value="170" /><param name="label.1.y" value="120" /><param name="label.1.text" value="$AintBComp" /><param name="label.2.x" value="290" /><param name="label.2.y" value="250" /><param name="label.2.text" value="$UnionComp" /></applet></div><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>B</mi></mrow></mfenced></mrow></mstyle></math>?</p><p>&nbsp;</p><p>Enter&nbsp;your response to&nbsp;2 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><msup><mi>A</mi><mrow><mi>c</mi></mrow></msup><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></mstyle></math>?</span></p><p>&nbsp;</p><p><span>Enter&nbsp;your response to&nbsp;2 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; Are events&nbsp;<em>A</em> and <em>B</em> independent?</span></span></p><p><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></p>@

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qu.1.10.name=Calculate P(A Int B), Independence with Venn Diagram@
qu.1.10.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>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></mstyle></math>, we can use the fact that the sum of all the probabilities within a sample space must be one.&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>$AintBComp</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$BintAComp</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><mi>$UnionComp</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>1</mn></mrow></mstyle></math>.&nbsp; Rearranging this equation, we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>1</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$AintBComp</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$BintAComp</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$UnionComp</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$AintB</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>
<p>b)&nbsp; To determine if events <em>A</em> and <em>B</em> are independent, we can check using the test for independence <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced></mrow></mstyle></math>.&nbsp; In part (a), we determined the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$AintB</mi></mrow></mstyle></math>, 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>A</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><mi>$A</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$B</mi></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>A</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</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><mi>$AintB</mi></mrow></mstyle></math>.&nbsp; We can then conclude that events <em>A</em> and <em>B</em> are independent.</p>@
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qu.1.10.algorithm=$A=rand(0.1, 0.3, 1);
$B=rand(0.4, 0.6, 1);
$AintB=$A*$B;
$Union=$A+$B-$AintB;
$UnionComp=1-$Union;
$AintBComp=$A-$AintB;
$BintAComp=$B-$AintB;
condition:lt($Union,1.0);@
qu.1.10.uid=80fee585-51ce-4721-a188-e2b60f96a522@
qu.1.10.info=  Course=Introductory Statistics;
  Topic=Basic Probability;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=Venn Diagram;
  Type=Calculation;
@
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qu.1.10.part.1.mode=Numeric@
qu.1.10.part.1.grading=exact_value@
qu.1.10.part.1.negStyle=both@
qu.1.10.part.1.answer.num=$AintB@
qu.1.10.part.2.grader=exact@
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qu.1.10.part.2.answer.2=No@
qu.1.10.part.2.answer.1=Yes@
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qu.1.10.question=<p>Consider the following Venn Diagram:</p><p>&nbsp;</p><div align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="400" height="300"><param name="image" value="__BASE_URI__Pictures/Venn1.jpg" /><param name="size" value="3" /><param name="label.1.x" value="150" /><param name="label.1.y" value="170" /><param name="label.1.text" value="$AintBComp" /><param name="label.2.x" value="270" /><param name="label.2.y" value="170" /><param name="label.2.text" value="$BintAComp" /><param name="label.3.x" value="290" /><param name="label.3.y" value="250" /><param name="label.3.text" value="$UnionComp" /></applet></div><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>A</mi><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mi>B</mi></mrow></mfenced></mrow></mstyle></math>?</p><p>&nbsp;</p><p>Enter&nbsp;your&nbsp;response to&nbsp;2 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; Are events&nbsp;<em>A</em> and <em>B</em> independent?</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.1.11.mode=Inline@
qu.1.11.name=Calculate P((A U B)^c)@
qu.1.11.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><msup><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi></mrow></mfenced><mrow><mi>c</mi></mrow></msup></mrow></mfenced></mrow></mstyle></math>, we can start by finding <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi></mrow></mfenced></mrow></mstyle></math>&nbsp;with 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>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'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</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><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</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>B</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><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></mstyle></math>&nbsp;= $A + $B - $Intersection = $Union.</p>
<p>We can then use the fact 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>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</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><msup><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi></mrow></mfenced><mrow><mi>c</mi></mrow></msup></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><msup><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi></mrow></mfenced><mrow><mi>c</mi></mrow></msup></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='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><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'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi></mrow></mfenced></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><msup><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi></mrow></mfenced><mrow><mi>c</mi></mrow></msup></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='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Union</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>$UnionCompliment</mi></mrow></mstyle></math></p>@
qu.1.11.editing=useHTML@
qu.1.11.solution=@
qu.1.11.algorithm=$A=rand(0.2, 0.3, 2);
$B=rand(0.6, 0.8, 2);
$Intersection=rand(0.11, 0.19, 2);
$Union=$A + $B - $Intersection;
$UnionCompliment=1-$Union;
condition:lt($Union,1);@
qu.1.11.uid=f2563830-a7ee-4bd6-b020-9078a4e16071@
qu.1.11.info=  Course=Introductory Statistics;
  Topic=Basic Probability;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
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qu.1.11.part.1.negStyle=both@
qu.1.11.part.1.answer.num=$UnionCompliment@
qu.1.11.question=<p>Given 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>A</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><mi>$A</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$B</mi></mrow></mstyle></math>and 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>A</mi><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mi>B</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Intersection</mi></mrow></mstyle></math>,&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><msup><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mi>B</mi></mrow></mfenced><mrow><mi>c</mi></mrow></msup></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mi></mi></mrow></mstyle></math>?</p><p>&nbsp;</p><p>Enter&nbsp;your answer to 2 decimal places.</p><p>&nbsp;</p><p>&nbsp;</p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><msup><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mi>B</mi></mrow></mfenced><mrow><mi>c</mi></mrow></msup></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>&nbsp;</span><1><span>&nbsp;</span>&nbsp;</p>@

qu.1.12.mode=Inline@
qu.1.12.name=P(A|B^c)@
qu.1.12.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><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><msup><mi>B</mi><mrow><mi>c</mi></mrow></msup><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mstyle></math>, we need to make use of the general formula for 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>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><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfrac></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><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><msup><mi>B</mi><mrow><mi>c</mi></mrow></msup><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><msup><mi>B</mi><mrow><mi>c</mi></mrow></msup></mrow></mfenced></mrow><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>B</mi><mrow><mi>c</mi></mrow></msup></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Since <em>P(B)</em> = $B, we get <em>P(B<sup>c</sup>) = 1 - P(B) = 1 - $B = $BComp</em>.&nbsp; To find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>B</mi><mrow><mi>c</mi></mrow></msup></mrow></mfenced></mrow></mstyle></math>, it is easiest to display the given probabilities in a Venn diagram:</p>
<p>&nbsp;</p>
<p><div align="center">
	<applet width="400" height="300" code="applets.labelImage.LabelImage" archive="modules/applets.jar">
		<param name="image" value="__BASE_URI__Pictures/Venn1.jpg" />
		<param name="size" value="3" />
		<param name="label.1.x" value="150" />
		<param name="label.1.y" value="160" />
		<param name="label.1.text" value="$AIntersectBComp" />
		<param name="label.2.x" value="220" />
		<param name="label.2.y" value="160" />
		<param name="label.2.text" value="$Intersection" />    
		<param name="label.3.x" value="280" />
		<param name="label.3.y" value="160" />
		<param name="label.3.text" value="$BIntersectAComp" />
	</applet>
</div></p>
<p>It can be seen from the Venn diagram 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>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><msup><mi>B</mi><mrow><mi>c</mi></mrow></msup></mrow></mfenced></mrow></mstyle></math>&nbsp;is $AIntersectBComp.&nbsp; Therefore, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>B</mi><mrow><mi>c</mi></mrow></msup><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mi>$AIntersectBComp</mi><mrow><mi>$BComp</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$AGivenBComp</mi></mrow></mstyle></math></p>@
qu.1.12.editing=useHTML@
qu.1.12.solution=@
qu.1.12.algorithm=$A=rand(0.2, 0.3, 2);
$B=rand(0.6, 0.8, 2);
$Intersection=rand(0.11, 0.19, 2);
$BComp=1-$B;
$AIntersectBComp=$A-$Intersection;
$AGivenBComp=$AIntersectBComp/($BComp);
$BIntersectAComp=$B-$Intersection;
condition:not(eq($AIntersectBComp,$BComp));@
qu.1.12.uid=a7efafba-86c2-457b-9547-b7ed4bff8ab2@
qu.1.12.info=  Course=Introductory Statistics;
  Topic=Basic Probability;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.1.12.weighting=1@
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qu.1.12.part.1.showUnits=false@
qu.1.12.part.1.err=0.0010@
qu.1.12.part.1.question=(Unset)@
qu.1.12.part.1.mode=Numeric@
qu.1.12.part.1.grading=toler_abs@
qu.1.12.part.1.negStyle=both@
qu.1.12.part.1.answer.num=$AGivenBComp@
qu.1.12.question=<p>Given 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>A</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><mi>$A</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$B</mi></mrow></mstyle></math>and 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>A</mi><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mi>B</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Intersection</mi></mrow></mstyle></math>,&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>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><msup><mi>B</mi><mrow><mi>c</mi></mrow></msup></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>?</p><p>&nbsp;</p><p>&nbsp;</p><p>Round&nbsp;your answer to at least&nbsp;3 decimal places.</p><p>&nbsp;</p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>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><msup><mi>B</mi><mrow><mi>c</mi></mrow></msup></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>&nbsp;</span><1><span>&nbsp;</span>&nbsp;</p>@

qu.1.13.mode=Inline@
qu.1.13.name=P(A Intersect B^c)@
qu.1.13.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>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><msup><mi>B</mi><mrow><mi>c</mi></mrow></msup></mrow></mfenced></mrow></mstyle></math>, it is easiest to display the given probabilities in a Venn diagram:</p>
<p>&nbsp;</p>
<p><div align="center">
	<applet width="400" height="300" code="applets.labelImage.LabelImage" archive="modules/applets.jar">
		<param name="image" value="__BASE_URI__Pictures/Venn1.jpg" />
		<param name="size" value="3" />
		<param name="label.1.x" value="150" />
		<param name="label.1.y" value="160" />
		<param name="label.1.text" value="$AIntersectBComp" />
		<param name="label.2.x" value="210" />
		<param name="label.2.y" value="160" />
		<param name="label.2.text" value="$Intersection" />    
		<param name="label.3.x" value="280" />
		<param name="label.3.y" value="160" />
		<param name="label.3.text" value="$BIntersectAComp" />
	</applet>
</div></p>
<p>From the Venn diagram, it can be seen 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>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><msup><mi>B</mi><mrow><mi>c</mi></mrow></msup></mrow></mfenced></mrow></mstyle></math>, the area that is in <em>A</em> but <strong>not</strong> in <em>B</em>, is $AIntersectBComp.</p>@
qu.1.13.editing=useHTML@
qu.1.13.solution=@
qu.1.13.algorithm=$A=rand(0.2, 0.3, 2);
$B=rand(0.6, 0.8, 2);
$Intersection=rand(0.11, 0.19, 2);
$AIntersectBComp=$A-$Intersection;
$BIntersectAComp=$B-$Intersection;@
qu.1.13.uid=767af2b4-7d3e-48c0-a3ca-f69dddf69c7a@
qu.1.13.info=  Course=Introductory Statistics;
  Topic=Basic Probability;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.1.13.weighting=1@
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qu.1.13.part.1.showUnits=false@
qu.1.13.part.1.question=(Unset)@
qu.1.13.part.1.mode=Numeric@
qu.1.13.part.1.grading=exact_value@
qu.1.13.part.1.negStyle=both@
qu.1.13.part.1.answer.num=$AIntersectBComp@
qu.1.13.question=<p>Given 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>A</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><mi>$A</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$B</mi></mrow></mstyle></math>and 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>A</mi><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mi>B</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Intersection</mi></mrow></mstyle></math>,&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>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><msup><mi>B</mi><mrow><mi>c</mi></mrow></msup></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mi mathvariant='normal'></mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>?</p><p>&nbsp;</p><p>&nbsp;</p><p>Enter&nbsp;your answer to&nbsp;2 decimal places.</p><p>&nbsp;</p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>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><msup><mi>B</mi><mrow><mi>c</mi></mrow></msup></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>&nbsp;</span><1><span>&nbsp;</span>&nbsp;</p>@

qu.1.14.mode=Inline@
qu.1.14.name=Calculate P(A|B), Independence with Venn Diagram@
qu.1.14.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><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></mrow></mstyle></math>, we can use the 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; The numerator is given to us in the question: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>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><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$AintB</mi></mrow></mstyle></math>, but we need to determine the value of the denominator, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced></mrow></mstyle></math>.&nbsp; To do this, we can use the additional information, that <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Union</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.&nbsp; 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>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'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</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><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</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>B</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><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></mstyle></math>, we can rearrange it to 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><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><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'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</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>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><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Union</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$AintB</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$B</mi></mrow></mstyle></math>.&nbsp; Finally, 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><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><mi>$AintB</mi><mrow><mi>$B</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$AGivenB</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; To determine if the events <em>A</em> and <em>B</em> are independent, we can use the check for independence <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi></mrow></mfenced></mrow></mstyle></math>.&nbsp; Using the answer from part (a), we can conclude that the events <em>A</em> and <em>B</em> are not independent.</p>@
qu.1.14.editing=useHTML@
qu.1.14.solution=@
qu.1.14.algorithm=$AintB=rand(0.1, 0.2, 2);
$AintBComp=rand(0.4, 0.5, 2);
$BintAComp=rand(0.2, 0.3, 2);
$A=$AintB + $AintBComp;
$B=$AintB + $BintAComp;
condition:not(eq($A*$B,$AintB));
$Union=$A + $B - $AintB;
condition:lt($Union,0.90);
$AGivenB=$AintB/$B;@
qu.1.14.uid=0f9fc3d4-d180-43a6-9d5a-ea0830e4b195@
qu.1.14.info=  Course=Introductory Statistics;
  Topic=Basic Probability;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=Venn Diagram;
  Type=Calculation;
@
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qu.1.14.part.1.numStyle=   @
qu.1.14.part.1.editing=useHTML@
qu.1.14.part.1.showUnits=false@
qu.1.14.part.1.err=0.01@
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qu.1.14.part.1.grading=toler_abs@
qu.1.14.part.1.negStyle=both@
qu.1.14.part.1.answer.num=$AGivenB@
qu.1.14.part.2.grader=exact@
qu.1.14.part.2.name=sro_id_2@
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qu.1.14.part.2.display.permute=false@
qu.1.14.part.2.question=(Unset)@
qu.1.14.part.2.answer.2=No@
qu.1.14.part.2.answer.1=Yes@
qu.1.14.part.2.mode=List@
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qu.1.14.question=<p>Consider the following Venn Diagram:</p><p>&nbsp;</p><div align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="400" height="300"><param name="image" value="__BASE_URI__Pictures/Venn1.jpg" /><param name="size" value="2" /><param name="label.1.x" value="150" /><param name="label.1.y" value="170" /><param name="label.1.text" value="$AintBComp" /><param name="label.2.x" value="210" /><param name="label.2.y" value="170" /><param name="label.2.text" value="$AintB" /></applet></div><p>&nbsp;</p><p>a)&nbsp; Given the additional information 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>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi></mrow></mfenced></mrow></mstyle></math>=$Union, 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>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></mrow></mfenced></mrow></mstyle></math>?</p><p>&nbsp;</p><p>Round your&nbsp;response to at least&nbsp;3 decimal places.</p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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></mrow></mstyle></math>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; Are events&nbsp;<em>A</em> and <em>B</em> independent?</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.1.15.mode=Inline@
qu.1.15.name=Calculate P(A U B)@
qu.1.15.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>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi></mrow></mfenced></mrow></mstyle></math>, we can use the&nbsp;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>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</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><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</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>B</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><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></mstyle></math>.</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><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi></mrow></mfenced></mrow></mstyle></math>&nbsp;= $A + $B -$Intersection = $Union</p>@
qu.1.15.editing=useHTML@
qu.1.15.solution=@
qu.1.15.algorithm=$A=rand(0.2, 0.3, 2);
$B=rand(0.6, 0.8, 2);
$Intersection=rand(0.11, 0.19, 2);
$Union=$A + $B - $Intersection;
condition:lt($Union,1);@
qu.1.15.uid=b96452a6-f63c-4037-a299-ce48bfca2ebb@
qu.1.15.info=  Course=Introductory Statistics;
  Topic=Basic Probability;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
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qu.1.15.part.1.negStyle=both@
qu.1.15.part.1.answer.num=$Union@
qu.1.15.question=<p>Given that <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$A</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</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><mi>$B</mi></mrow></mstyle></math>and the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mi>B</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Intersection</mi></mrow></mstyle></math>,&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>A</mi><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mi>B</mi></mrow></mfenced></mrow></mstyle></math>?</p><p>&nbsp;</p><p>Enter&nbsp;your answer to 2 decimal places.</p><p>&nbsp;</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mi>B</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><1><span>&nbsp;</span>&nbsp;</p>@

qu.2.topic=Introduction to Probability@

qu.2.1.mode=Multiple Selection@
qu.2.1.name=Definitions4: Mutually exclusive, independence.@
qu.2.1.comment=@
qu.2.1.editing=useHTML@
qu.2.1.solution=@
qu.2.1.algorithm=@
qu.2.1.uid=dfb36c4c-da32-430a-86c9-7fca8dc73fa3@
qu.2.1.info=  Course=Introductory Statistics;
  Topic=Introduction to Probability;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.2.1.question=<p>Which of the following statements are TRUE?</p>
<p>(Assume for the purposes of this question that the events have a non-zero probability of occurring).</p>
<p>&nbsp;</p>
<p>There may be more than one correct answer; please select all that are true.</p>@
qu.2.1.answer=2, 3@
qu.2.1.choice.1=If two events (A and B) are independent, then the conditional probability of B given A is 1.@
qu.2.1.choice.2=Mutually exclusive events are never independent, but dependent events are not always mutually exclusive.@
qu.2.1.choice.3=If the conditional probability of event A, given event B, is the same as the probability event A, then events A and B are independent.@
qu.2.1.choice.4=The union of events A and B is the event that either A occurs or B occurs, but not that both occur.@
qu.2.1.choice.5=If event A is a subset of event B, then the probability of their intersection is 1.@
qu.2.1.fixed=@

qu.2.2.mode=Multiple Selection@
qu.2.2.name=Definitions2: events, sample spaces, simple events@
qu.2.2.comment=@
qu.2.2.editing=useHTML@
qu.2.2.solution=@
qu.2.2.algorithm=@
qu.2.2.uid=3436b8c8-403e-45ca-afc5-6adcdc4e958e@
qu.2.2.info=  Course=Introductory Statistics;
  Topic=Introduction to Probability;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.2.2.question=<p>Which of the following statements are TRUE?</p>
<p>&nbsp;</p>
<p>There may be more than one correct answer; select all that are true.</p>@
qu.2.2.answer=1, 2@
qu.2.2.choice.1=More than one event can be defined on the same sample space.@
qu.2.2.choice.2=If all simple events are equally likely, the probability of an event is the number of simple events within it, divided by the total number of simple events in the sample space.@
qu.2.2.choice.3=The probability of a simple event can be any value between -1 and 1.@
qu.2.2.choice.4=The sum of all the probabilities for all simple events within a sample space is less than 1.@
qu.2.2.choice.5=If two events are independent then they have no simple events in common.@
qu.2.2.fixed=@

qu.2.3.mode=Multiple Selection@
qu.2.3.name=Definitions1: events, sample spaces, simple events@
qu.2.3.comment=@
qu.2.3.editing=useHTML@
qu.2.3.solution=@
qu.2.3.algorithm=@
qu.2.3.uid=18dc8eee-8351-4baa-80de-102229f7ced4@
qu.2.3.info=  Course=Introductory Statistics;
  Topic=Introduction to Probability;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.2.3.question=<p>Which of the following statements are TRUE?</p>
<p>&nbsp;</p>
<p>There may be more than one correct answer; select all that are true.</p>@
qu.2.3.answer=1, 3, 5@
qu.2.3.choice.1=An event is a collection of simple events.@
qu.2.3.choice.2=An event is the most basic outcome of an experiment.@
qu.2.3.choice.3=An event is a subset of the sample space.@
qu.2.3.choice.4=Two events, defined on the same sample space, cannot be mutually exclusive.@
qu.2.3.choice.5=The intersection of two events occurs when they have one or more simple events in common.@
qu.2.3.fixed=@

qu.2.4.mode=Multiple Selection@
qu.2.4.name=Definitions10: Mutually exclusive, independence@
qu.2.4.comment=@
qu.2.4.editing=useHTML@
qu.2.4.solution=@
qu.2.4.algorithm=@
qu.2.4.uid=7221d5eb-c5a9-48f9-acfa-0e1b97519d00@
qu.2.4.info=  Course=Introductory Statistics;
  Topic=Introduction to Probability;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.2.4.question=<p>Which of the following statements are TRUE?</p>
<p>(Assume for the purposes of this question that the events have a non-zero proability of occuring, unless otherwise stated).</p>
<p>&nbsp;</p>
<p>There may be more than one correct answer; select all that are true.</p>
<p>&nbsp;</p>@
qu.2.4.answer=1, 4, 5@
qu.2.4.choice.1=If the intersection of A and B is non-zero, then A and B could be independent or dependent.@
qu.2.4.choice.2=The sum of all probabilities in a sample space is less than 1.@
qu.2.4.choice.3=A probability can take on a value less than 0.@
qu.2.4.choice.4=The intersection of two events is the set of all the sample points they have in common.@
qu.2.4.choice.5=If events A and B are disjoint, then they are also dependent.@
qu.2.4.fixed=@

qu.2.5.mode=Multiple Selection@
qu.2.5.name=Definitions3: Mutually exclusive, independence.@
qu.2.5.comment=@
qu.2.5.editing=useHTML@
qu.2.5.solution=@
qu.2.5.algorithm=@
qu.2.5.uid=79f734eb-af3c-40fa-90b5-32f94d21190b@
qu.2.5.info=  Course=Introductory Statistics;
  Topic=Introduction to Probability;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.2.5.question=<p>Which of the following statements are TRUE?</p>
<p>(Assume for the purposes of this question that the events have a non-zero probability of occurring).</p>
<p>&nbsp;</p>
<p>There may be more than one correct answer; please select all that are true.</p>@
qu.2.5.answer=3, 4@
qu.2.5.choice.1=If two events are independent, then they are also mutually exclusive.@
qu.2.5.choice.2=Independent events cannot occur at the same time.@
qu.2.5.choice.3=Mutually exclusive events have no simple events in common.@
qu.2.5.choice.4=If two events (A and B) are mutually exclusive, then the conditional probability of A given B is 0.@
qu.2.5.choice.5=Independent events are also known as "disjoint" events.@
qu.2.5.fixed=@

qu.2.6.mode=Multiple Choice@
qu.2.6.name=Simple events: Deterimine if mutually exclusive, independent@
qu.2.6.comment=<p>Simple events are the most basic outcome of an experiment.&nbsp; Since there cannot be more than one outcome each time an experiment is conducted, the simple events are mutually exclusive.&nbsp; By definition, mutually exclusive events cannot be independent of each other; they are always dependent.</p>@
qu.2.6.editing=useHTML@
qu.2.6.solution=@
qu.2.6.algorithm=@
qu.2.6.uid=5bdbce90-a4be-4dc2-9635-314cca02f54a@
qu.2.6.info=  Course=Introductory Statistics;
  Topic=Introduction to Probability;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.2.6.question=<p>The simple events within a sample space are:</p>@
qu.2.6.answer=3@
qu.2.6.choice.1=Mutually exclusive and independent.@
qu.2.6.choice.2=Not mutually exclusive but are independent.@
qu.2.6.choice.3=Mutually exclusive but not independent.@
qu.2.6.choice.4=Not mutually exclusive and not independent.@
qu.2.6.choice.5=Not mutually exclusive, but can be independent.@
qu.2.6.fixed=@

qu.2.7.mode=Inline@
qu.2.7.name=Definitions9&10 Combined: Random Selection of T/F Statements@
qu.2.7.comment=@
qu.2.7.editing=useHTML@
qu.2.7.solution=@
qu.2.7.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=("'If A and B are mutually exclusive, then P(A|B) = P(B|A).'");
$b=("'If the intersection of A and B is non-zero, then A and B could be independent or dependent.'");
$c=("'The sum of all probabilities in a sample space is < 1.'");
$d=("'A probability can take on a value less than 0.'");
$e=("'The intersection of two events is the set of all the sample points they have in common.'");
$f=("'If events A and B are disjoint, then they are also dependent.'");
$g=("'If P(A) = 0.7 and P(B) = 0.5, then the probability of the intersection of A and B is less than 0.'");
$hij=maple("
H1:=convert(cat(`The `,MathML[ExportPresentation](P(A intersect B^c) = P(A) - P(A intersect B)),`.`),string):
A1:=convert(cat(`The P(A U B) + P(`,MathML[ExportPresentation]((A union B)^c),`) = 1.`),string):
J1:=convert(cat(`The P`,MathML[ExportPresentation](((A union B)^c) = P(A^c union B^c)),`.`),string):
H1, A1, J1
");
$h=switch(0, $hij);
$i=switch(1, $hij);
$j=switch(2, $hij);
$Answers=["'True'","'True'","'False'","'False'","'True'","'True'","'False'","'True'","'True'","'False'"];
$Distractors=["'False'","'False'","'True'","'True'","'False'","'False'","'True'","'False'","'False'","'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, $Distractors);@
qu.2.7.uid=8f97e3fc-4bb1-4108-97f7-4805c0079e2e@
qu.2.7.info=  Course=Introductory Statistics;
  Topic=Introduction to Probability;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=Random selection of statements;
  Type=Concept;
@
qu.2.7.weighting=1,1,1,1,1@
qu.2.7.numbering=alpha@
qu.2.7.part.1.grader=exact@
qu.2.7.part.1.name=sro_id_1@
qu.2.7.part.1.editing=useHTML@
qu.2.7.part.1.display.permute=true@
qu.2.7.part.1.question=(Unset)@
qu.2.7.part.1.answer.2=$D1@
qu.2.7.part.1.answer.1=$A1@
qu.2.7.part.1.mode=List@
qu.2.7.part.1.display=menu@
qu.2.7.part.1.credit.2=0.0@
qu.2.7.part.1.credit.1=1.0@
qu.2.7.part.2.grader=exact@
qu.2.7.part.2.name=sro_id_2@
qu.2.7.part.2.editing=useHTML@
qu.2.7.part.2.display.permute=true@
qu.2.7.part.2.question=(Unset)@
qu.2.7.part.2.answer.2=$D2@
qu.2.7.part.2.answer.1=$A2@
qu.2.7.part.2.mode=List@
qu.2.7.part.2.display=menu@
qu.2.7.part.2.credit.2=0.0@
qu.2.7.part.2.credit.1=1.0@
qu.2.7.part.3.grader=exact@
qu.2.7.part.3.name=sro_id_3@
qu.2.7.part.3.editing=useHTML@
qu.2.7.part.3.display.permute=true@
qu.2.7.part.3.question=(Unset)@
qu.2.7.part.3.answer.2=$D3@
qu.2.7.part.3.answer.1=$A3@
qu.2.7.part.3.mode=List@
qu.2.7.part.3.display=menu@
qu.2.7.part.3.credit.2=0.0@
qu.2.7.part.3.credit.1=1.0@
qu.2.7.part.4.grader=exact@
qu.2.7.part.4.name=sro_id_4@
qu.2.7.part.4.editing=useHTML@
qu.2.7.part.4.display.permute=true@
qu.2.7.part.4.question=(Unset)@
qu.2.7.part.4.answer.2=$D4@
qu.2.7.part.4.answer.1=$A4@
qu.2.7.part.4.mode=List@
qu.2.7.part.4.display=menu@
qu.2.7.part.4.credit.2=0.0@
qu.2.7.part.4.credit.1=1.0@
qu.2.7.part.5.grader=exact@
qu.2.7.part.5.name=sro_id_5@
qu.2.7.part.5.editing=useHTML@
qu.2.7.part.5.display.permute=true@
qu.2.7.part.5.question=(Unset)@
qu.2.7.part.5.answer.2=$D5@
qu.2.7.part.5.answer.1=$A5@
qu.2.7.part.5.mode=List@
qu.2.7.part.5.display=menu@
qu.2.7.part.5.credit.2=0.0@
qu.2.7.part.5.credit.1=1.0@
qu.2.7.question=<p>Identify each of the following statements as either&nbsp;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; $Q5</span></p>@

qu.2.8.mode=Multiple Selection@
qu.2.8.name=Definitions7: Mutually exclusive, independence@
qu.2.8.comment=@
qu.2.8.editing=useHTML@
qu.2.8.solution=@
qu.2.8.algorithm=@
qu.2.8.uid=d10605e9-00fd-4ef0-b4b5-0752af3b2a41@
qu.2.8.info=  Course=Introductory Statistics;
  Topic=Introduction to Probability;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.2.8.question=<p>Which of the following statements are TRUE?</p>
<p>(Assume for the purposes of this question that the events have a non-zero probability of occurring, unless otherwise stated).</p>
<p>&nbsp;</p>
<p>There may be more than one correct answer; select all that are true.</p>
<p>&nbsp;</p>@
qu.2.8.answer=2, 5@
qu.2.8.choice.1=If A and B do not intersect, then A and B are independent.@
qu.2.8.choice.2=If A is a subset of B, 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>A</mi><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mi>B</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced></mrow></mstyle></math>.@
qu.2.8.choice.3=If A and B are mutually exclusive and P(A) = 0.9, then P(B) = 0.1.@
qu.2.8.choice.4=If the intersection of events A and B is non-zero, then A and B are mutually exclusive.@
qu.2.8.choice.5=If P(A) + P(B) = 1, 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>B</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>A</mi><mrow><mi>c</mi></mrow></msup></mrow></mfenced></mrow></mstyle></math>.@
qu.2.8.fixed=@

qu.2.9.mode=Multiple Selection@
qu.2.9.name=Definitions6: Mutually exclusive, independence.@
qu.2.9.comment=@
qu.2.9.editing=useHTML@
qu.2.9.solution=@
qu.2.9.algorithm=@
qu.2.9.uid=be725a8c-17e3-4e7b-a23e-587bbfc14207@
qu.2.9.info=  Course=Introductory Statistics;
  Topic=Introduction to Probability;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.2.9.question=<p>Which of the following statements are TRUE?</p>
<p>(Assume for the purposes of this question that the events have a non-zero probability of occurring).</p>
<p>&nbsp;</p>
<p>There may be more than one correct answer; please select all that are true.</p>@
qu.2.9.answer=1, 3, 4@
qu.2.9.choice.1=If the probability of event A is influenced by the occurrence of event B, then the two events are dependent.@
qu.2.9.choice.2=If event A equals event B, then the probability of their intersection is 1.@
qu.2.9.choice.3=If A and B are independent, 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><msup><mi>A</mi><mrow><mi>c</mi></mrow></msup></mrow></mstyle></math> and B are also independent.@
qu.2.9.choice.4=Event A and its complement, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>A</mi><mrow><mi>c</mi></mrow></msup></mrow></mstyle></math>, are mutually exclusive events.@
qu.2.9.choice.5=A 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><msup><mi>A</mi><mrow><mi>c</mi></mrow></msup></mrow></mstyle></math> are independent events.@
qu.2.9.fixed=@

qu.2.10.mode=Multiple Selection@
qu.2.10.name=Definitions5: Mutually exclusive, independence.@
qu.2.10.comment=@
qu.2.10.editing=useHTML@
qu.2.10.solution=@
qu.2.10.algorithm=@
qu.2.10.uid=e099bad6-4ee2-4b69-a831-e24676e3dfb5@
qu.2.10.info=  Course=Introductory Statistics;
  Topic=Introduction to Probability;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.2.10.question=<p>Which of the following statements are TRUE?</p>
<p>(Assume for the purposes of this question that the events have a non-zero probability of occurring).</p>
<p>&nbsp;</p>
<p>There may be more than one correct answer; please select all that are true.</p>@
qu.2.10.answer=2, 3, 4@
qu.2.10.choice.1=If event A is a subset of event B, then the conditional probability of event A, given event B, is 1.@
qu.2.10.choice.2=If event A equals event B, then the conditional probability of event B, given event A, is 1.@
qu.2.10.choice.3=If two events are independent, then their complements are also independent.@
qu.2.10.choice.4=If two events are mutually exclusive, then the union of those two events is the sum of the individual probabilities of the two events.@
qu.2.10.choice.5=If A and B are mutually exclusive, then their complements are also mutually exclusive.@
qu.2.10.fixed=@

qu.2.11.mode=Multiple Choice@
qu.2.11.name=Determine independence/dependence of mutually exclusive events@
qu.2.11.comment=<p>If events are mutually exclusive, it means that they have no points in their intersection, and therefore cannot occur at the same time.&nbsp; Thus, if one event occurs, another event cannot occur, which implies dependence between mutually exclusive events.</p>@
qu.2.11.editing=useHTML@
qu.2.11.solution=@
qu.2.11.algorithm=@
qu.2.11.uid=86ea3192-d9cb-4588-8181-2303531466c7@
qu.2.11.info=  Course=Introductory Statistics;
  Topic=Introduction to Probability;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.2.11.question=<p>Which one of the following statements is TRUE?</p>
<p>(Assume for the purposes of this question that the events have a non-zero probability of occurring).</p>@
qu.2.11.answer=3@
qu.2.11.choice.1=Mutually exclusive events are always independent.@
qu.2.11.choice.2=Mutually exclusive events can be independent, but are not always independent.@
qu.2.11.choice.3=Mutually exclusive events are never independent.@
qu.2.11.choice.4=Dependent events are always mutually exclusive.@
qu.2.11.choice.5=Dependent events are never mutually exclusive.@
qu.2.11.fixed=@

qu.2.12.mode=Inline@
qu.2.12.name=Definitions7&8 Combined: Random Selection of T/F Statements@
qu.2.12.comment=@
qu.2.12.editing=useHTML@
qu.2.12.solution=@
qu.2.12.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=("'If A and B are mutually exclusive and P(A) = 0.9, then P(B) = 0.1.'");
$b=("'If P(A|B) = 1, then A is a subset of B.'");
$c=("'If A and B are independent, then P(B)*P(A|B) = P(A)*P(B).'");
$d=("'If A and B do not intersect, then A and B are independent.'");
$e=("'If the intersection of the events A and B is non-zero, then A and B are mutually exclusive.'");
$f=("'If A is a subset of B, then P(A U B) = P(B).'");
$ghij=maple("
G1:=convert(cat(`If P(A) + P(B) = 1, then P(B) = P(`,MathML[ExportPresentation](A^c),`).`),string):
H1:=convert(cat(`If P(A) = 0.6, P(B) = 0.8, then P(`,MathML[ExportPresentation](A intersect B),`) is between 0.4 and 0.6, inclusive.`),string):
A1:=convert(cat(`If P(A) = 0.5, P(B) = 0.5 and A and B do not intersect, then P(`,MathML[ExportPresentation](A^c intersect B^c),`) = 0.`),string):
J1:=convert(cat(`If P(`,MathML[ExportPresentation](A^c intersect B^c),`) = 0, then P(A U B) = 1.`),string):
G1, H1, A1, J1
");
$g=switch(0, $ghij);
$h=switch(1, $ghij);
$i=switch(2, $ghij);
$j=switch(3, $ghij);
$Answers=["'False'","'False'","'True'","'False'","'False'","'True'","'True'","'True'","'True'","'True'"];
$Distractors=["'True'","'True'","'False'","'True'","'True'","'False'","'False'","'False'","'False'","'False'"];
$Q1=switch($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, $Distractors);@
qu.2.12.uid=8f7b188d-e18f-4ea1-a51f-ce6982b429ff@
qu.2.12.info=  Course=Introductory Statistics;
  Topic=Introduction to Probability;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=Random selection of statements;
  Type=Concept;
@
qu.2.12.weighting=1,1,1,1,1@
qu.2.12.numbering=alpha@
qu.2.12.part.1.grader=exact@
qu.2.12.part.1.name=sro_id_1@
qu.2.12.part.1.editing=useHTML@
qu.2.12.part.1.display.permute=true@
qu.2.12.part.1.question=(Unset)@
qu.2.12.part.1.answer.2=$D1@
qu.2.12.part.1.answer.1=$A1@
qu.2.12.part.1.mode=List@
qu.2.12.part.1.display=menu@
qu.2.12.part.1.credit.2=0.0@
qu.2.12.part.1.credit.1=1.0@
qu.2.12.part.2.grader=exact@
qu.2.12.part.2.name=sro_id_2@
qu.2.12.part.2.editing=useHTML@
qu.2.12.part.2.display.permute=true@
qu.2.12.part.2.question=(Unset)@
qu.2.12.part.2.answer.2=$D2@
qu.2.12.part.2.answer.1=$A2@
qu.2.12.part.2.mode=List@
qu.2.12.part.2.display=menu@
qu.2.12.part.2.credit.2=0.0@
qu.2.12.part.2.credit.1=1.0@
qu.2.12.part.3.grader=exact@
qu.2.12.part.3.name=sro_id_3@
qu.2.12.part.3.editing=useHTML@
qu.2.12.part.3.display.permute=true@
qu.2.12.part.3.question=(Unset)@
qu.2.12.part.3.answer.2=$D3@
qu.2.12.part.3.answer.1=$A3@
qu.2.12.part.3.mode=List@
qu.2.12.part.3.display=menu@
qu.2.12.part.3.credit.2=0.0@
qu.2.12.part.3.credit.1=1.0@
qu.2.12.part.4.grader=exact@
qu.2.12.part.4.name=sro_id_4@
qu.2.12.part.4.editing=useHTML@
qu.2.12.part.4.display.permute=true@
qu.2.12.part.4.question=(Unset)@
qu.2.12.part.4.answer.2=$D4@
qu.2.12.part.4.answer.1=$A4@
qu.2.12.part.4.mode=List@
qu.2.12.part.4.display=menu@
qu.2.12.part.4.credit.2=0.0@
qu.2.12.part.4.credit.1=1.0@
qu.2.12.part.5.grader=exact@
qu.2.12.part.5.name=sro_id_5@
qu.2.12.part.5.editing=useHTML@
qu.2.12.part.5.display.permute=true@
qu.2.12.part.5.question=(Unset)@
qu.2.12.part.5.answer.2=$D5@
qu.2.12.part.5.answer.1=$A5@
qu.2.12.part.5.mode=List@
qu.2.12.part.5.display=menu@
qu.2.12.part.5.credit.2=0.0@
qu.2.12.part.5.credit.1=1.0@
qu.2.12.question=<p>Identify each of the following statements as either&nbsp;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; $Q5</span></p>@

qu.2.13.mode=Multiple Choice@
qu.2.13.name=Definition of the sample space@
qu.2.13.comment=<p>Since a simple event is the most basic outcome of the experiment, the <em>sample space</em> is the collection of all the simple events that result from an experiment.</p>@
qu.2.13.editing=useHTML@
qu.2.13.solution=@
qu.2.13.algorithm=@
qu.2.13.uid=e3c669da-59c0-4a48-ace7-8997230d7bce@
qu.2.13.info=  Course=Introductory Statistics;
  Topic=Introduction to Probability;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Concept;
@
qu.2.13.question=<p>The sample space is:</p>@
qu.2.13.answer=1@
qu.2.13.choice.1=The collection of all simple events resulting from an experiment.@
qu.2.13.choice.2=A subset of all simple events resulting from an experiment.@
qu.2.13.choice.3=A collection of all events resulting from an experiment.@
qu.2.13.choice.4=A sample of some of the events resulting from an experiment.@
qu.2.13.choice.5=The probability of the simple events resulting from an experiment.@
qu.2.13.fixed=@

qu.2.14.mode=Inline@
qu.2.14.name=Definitions3&4 Combined: Random Selection of T/F Statements@
qu.2.14.comment=@
qu.2.14.editing=useHTML@
qu.2.14.solution=@
qu.2.14.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=("'If two events are independent, then they are also mutually exclusive.'");
$b=("'Independent events cannot occur at the same time.'");
$c=("'Mutually exclusive events have no simple events in common.'");
$d=("'If two events (A and B) are mutually exclusive, then the conditional probability of A given B is 0.'");
$e=("'If two events (A and B) are independent, then the conditional probability of B given A is 1.'");
$f=("'Mutually exclusive events are never independent, but dependent events are not always mutually exclusive.'");
$g=("'If the conditional probability of an event A, given event B, is the same as the probability of event A, then events A and B are independent.'");
$h=("'Independent events are also known as disjoint events.'");
$i=("'If event A is a subset of event B, then the probability of their intersection is 1.'");
$j=("'The union of events A and B is the event that either A occurs or B occurs, but not that both occur.'");
$Answers=["'False'","'False'","'True'","'True'","'False'","'True'","'True'","'False'","'False'","'False'"];
$Distractors=["'True'","'True'","'False'","'False'","'True'","'False'","'False'","'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, $Distractors);@
qu.2.14.uid=f3bf3378-f525-4c2a-96b5-15e7a32033a0@
qu.2.14.info=  Course=Introductory Statistics;
  Topic=Introduction to Probability;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=Random selection of statements;
  Type=Concept;
@
qu.2.14.weighting=1,1,1,1,1@
qu.2.14.numbering=alpha@
qu.2.14.part.1.grader=exact@
qu.2.14.part.1.name=sro_id_1@
qu.2.14.part.1.editing=useHTML@
qu.2.14.part.1.display.permute=true@
qu.2.14.part.1.question=(Unset)@
qu.2.14.part.1.answer.2=$D1@
qu.2.14.part.1.answer.1=$A1@
qu.2.14.part.1.mode=List@
qu.2.14.part.1.display=menu@
qu.2.14.part.1.credit.2=0.0@
qu.2.14.part.1.credit.1=1.0@
qu.2.14.part.2.grader=exact@
qu.2.14.part.2.name=sro_id_2@
qu.2.14.part.2.editing=useHTML@
qu.2.14.part.2.display.permute=true@
qu.2.14.part.2.question=(Unset)@
qu.2.14.part.2.answer.2=$D2@
qu.2.14.part.2.answer.1=$A2@
qu.2.14.part.2.mode=List@
qu.2.14.part.2.display=menu@
qu.2.14.part.2.credit.2=0.0@
qu.2.14.part.2.credit.1=1.0@
qu.2.14.part.3.grader=exact@
qu.2.14.part.3.name=sro_id_3@
qu.2.14.part.3.editing=useHTML@
qu.2.14.part.3.display.permute=true@
qu.2.14.part.3.question=(Unset)@
qu.2.14.part.3.answer.2=$D3@
qu.2.14.part.3.answer.1=$A3@
qu.2.14.part.3.mode=List@
qu.2.14.part.3.display=menu@
qu.2.14.part.3.credit.2=0.0@
qu.2.14.part.3.credit.1=1.0@
qu.2.14.part.4.grader=exact@
qu.2.14.part.4.name=sro_id_4@
qu.2.14.part.4.editing=useHTML@
qu.2.14.part.4.display.permute=true@
qu.2.14.part.4.question=(Unset)@
qu.2.14.part.4.answer.2=$D4@
qu.2.14.part.4.answer.1=$A4@
qu.2.14.part.4.mode=List@
qu.2.14.part.4.display=menu@
qu.2.14.part.4.credit.2=0.0@
qu.2.14.part.4.credit.1=1.0@
qu.2.14.part.5.grader=exact@
qu.2.14.part.5.name=sro_id_5@
qu.2.14.part.5.editing=useHTML@
qu.2.14.part.5.display.permute=true@
qu.2.14.part.5.question=(Unset)@
qu.2.14.part.5.answer.2=$D5@
qu.2.14.part.5.answer.1=$A5@
qu.2.14.part.5.mode=List@
qu.2.14.part.5.display=menu@
qu.2.14.part.5.credit.2=0.0@
qu.2.14.part.5.credit.1=1.0@
qu.2.14.question=<p>Identify each of the following statements as either&nbsp;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; $Q5</span></p>@

qu.2.15.mode=Multiple Selection@
qu.2.15.name=Definitions8: Mutually exclusive, independence.@
qu.2.15.comment=@
qu.2.15.editing=useHTML@
qu.2.15.solution=@
qu.2.15.algorithm=@
qu.2.15.uid=503f16b9-8c15-4094-bad0-530bb931dda1@
qu.2.15.info=  Course=Introductory Statistics;
  Topic=Introduction to Probability;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.2.15.question=<p>Which of the following statements are TRUE?</p>
<p>(Assume for the purposes of this question that the events have a non-zero probability of occurring, unless otherwise stated).</p>
<p>&nbsp;</p>
<p>There may be more than one correct answer; select all that are true.</p>@
qu.2.15.answer=1, 2, 3, 5@
qu.2.15.choice.1=If P(A) = 0.6, P(B) = 0.8, 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>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></mstyle></math> is between 0.4 and 0.6, inclusive.@
qu.2.15.choice.2=If P(A) = 0.5, P(B) = 0.5, and A and B do not intersect, 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><msup><mi>A</mi><mrow><mi>c</mi></mrow></msup><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><msup><mi>B</mi><mrow><mi>c</mi></mrow></msup></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math>.@
qu.2.15.choice.3=If <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><msup><mi>A</mi><mrow><mi>c</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><msup><mi>B</mi><mrow><mi>c</mi></mrow></msup></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='bold' fontweight='bold' lspace='0.0em' rspace='0.0em'>then</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mi>B</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></mstyle></math>.@
qu.2.15.choice.4=If <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>A</mi><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mi>B</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></mstyle></math>, then A is a subset of B.@
qu.2.15.choice.5=If A and B are independent, 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>B</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>A</mi><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mi>B</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced></mrow></mstyle></math>.@
qu.2.15.fixed=@

qu.2.16.mode=Inline@
qu.2.16.name=Definitions5&6 Combined: Random Selection of T/F Statements@
qu.2.16.comment=@
qu.2.16.editing=useHTML@
qu.2.16.solution=@
qu.2.16.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=("'If the probability of event A is influenced by the occurrence of event B, then the two events are dependent.'");
$b=("'If event A is a subset of event B, then the conditional probability of event A, given event B, is 1.'");
$c=("'If event A equals event B, then the conditional probability of event B, given event A, is 1.'");
$d=("'If event A equals event B, then the probability of their intersection is 1.'");
$e=("'If two events are independent, then their complements are also independent.'");
$f=("'If A and B are mutually exclusive, then their complements are also mutually exclusive.'");
$g=("'If two events are mutually exclusive, then the union of those two events is the sum of the individual probabilities of these two events.'");
$hij=maple("
H1:=convert(cat(`If A and B are independent, then`,MathML[ExportPresentation](A^c),`and B are also independent.`),string):
A1:=convert(cat(`Event A and its complement,`,MathML[ExportPresentation](A^c),`,are mutually exclusive.`),string):
J1:=convert(cat(`A and `,MathML[ExportPresentation](A^c),`are independent events.`),string):
H1, A1, J1
");
$h=switch(0, $hij);
$i=switch(1, $hij);
$j=switch(2, $hij);
$Answers=["'True'","'False'","'True'","'False'","'True'","'False'","'True'","'True'","'True'","'False'"];
$Distractors=["'False'","'True'","'False'","'True'","'False'","'True'","'False'","'False'","'False'","'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, $Distractors);@
qu.2.16.uid=3caa6fc8-a7bf-4fb4-bd57-e78baf8b1184@
qu.2.16.info=  Course=Introductory Statistics;
  Topic=Introduction to Probability;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=Random selection of statements;
  Type=Concept;
@
qu.2.16.weighting=1,1,1,1,1@
qu.2.16.numbering=alpha@
qu.2.16.part.1.grader=exact@
qu.2.16.part.1.name=sro_id_1@
qu.2.16.part.1.editing=useHTML@
qu.2.16.part.1.display.permute=true@
qu.2.16.part.1.question=(Unset)@
qu.2.16.part.1.answer.2=$D1@
qu.2.16.part.1.answer.1=$A1@
qu.2.16.part.1.mode=List@
qu.2.16.part.1.display=menu@
qu.2.16.part.1.credit.2=0.0@
qu.2.16.part.1.credit.1=1.0@
qu.2.16.part.2.grader=exact@
qu.2.16.part.2.name=sro_id_2@
qu.2.16.part.2.editing=useHTML@
qu.2.16.part.2.display.permute=true@
qu.2.16.part.2.question=(Unset)@
qu.2.16.part.2.answer.2=$D2@
qu.2.16.part.2.answer.1=$A2@
qu.2.16.part.2.mode=List@
qu.2.16.part.2.display=menu@
qu.2.16.part.2.credit.2=0.0@
qu.2.16.part.2.credit.1=1.0@
qu.2.16.part.3.grader=exact@
qu.2.16.part.3.name=sro_id_3@
qu.2.16.part.3.editing=useHTML@
qu.2.16.part.3.display.permute=true@
qu.2.16.part.3.question=(Unset)@
qu.2.16.part.3.answer.2=$D3@
qu.2.16.part.3.answer.1=$A3@
qu.2.16.part.3.mode=List@
qu.2.16.part.3.display=menu@
qu.2.16.part.3.credit.2=0.0@
qu.2.16.part.3.credit.1=1.0@
qu.2.16.part.4.grader=exact@
qu.2.16.part.4.name=sro_id_4@
qu.2.16.part.4.editing=useHTML@
qu.2.16.part.4.display.permute=true@
qu.2.16.part.4.question=(Unset)@
qu.2.16.part.4.answer.2=$D4@
qu.2.16.part.4.answer.1=$A4@
qu.2.16.part.4.mode=List@
qu.2.16.part.4.display=menu@
qu.2.16.part.4.credit.2=0.0@
qu.2.16.part.4.credit.1=1.0@
qu.2.16.part.5.grader=exact@
qu.2.16.part.5.name=sro_id_5@
qu.2.16.part.5.editing=useHTML@
qu.2.16.part.5.display.permute=true@
qu.2.16.part.5.question=(Unset)@
qu.2.16.part.5.answer.2=$D5@
qu.2.16.part.5.answer.1=$A5@
qu.2.16.part.5.mode=List@
qu.2.16.part.5.display=menu@
qu.2.16.part.5.credit.2=0.0@
qu.2.16.part.5.credit.1=1.0@
qu.2.16.question=<p>Identify each of the following statements as either&nbsp;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; $Q5</span></p>@

qu.2.17.mode=Inline@
qu.2.17.name=Definitions1&2 Combined: Random Selection of T/F Statements@
qu.2.17.comment=@
qu.2.17.editing=useHTML@
qu.2.17.solution=@
qu.2.17.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=("'An event is a collection of simple events.'");
$b=("'An event is a subset of the sample space.'");
$c=("'More than one event can be defined on the same sample space.'");
$d=("'The intersection of two events occurs when they have one or more simple events in common.'");
$e=("'If all simple events are equally likely, the probability of an event is the number of simple events within it, divided by the total number of simple events in the sample space.'");
$f=("'The probability of a simple event can be any value between -1 and 1.'");
$g=("'The sum of all the probabilities for all simple events within a sample space is less than 1.'");
$h=("'If two events are independent, then they have no simple events in common.'");
$i=("'Two events, defined on the same sample space, cannot be mutually exclusive.'");
$j=("'An event is the most basic outcome of an experiment.'");
$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, $Distractors);@
qu.2.17.uid=a6a895a8-36e7-4c3c-99db-ccef3e7c0ac4@
qu.2.17.info=  Course=Introductory Statistics;
  Topic=Introduction to Probability;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=Random selection of statements;
  Type=Concept;
@
qu.2.17.weighting=1,1,1,1,1@
qu.2.17.numbering=alpha@
qu.2.17.part.1.grader=exact@
qu.2.17.part.1.name=sro_id_1@
qu.2.17.part.1.editing=useHTML@
qu.2.17.part.1.display.permute=true@
qu.2.17.part.1.question=(Unset)@
qu.2.17.part.1.answer.2=$D1@
qu.2.17.part.1.answer.1=$A1@
qu.2.17.part.1.mode=List@
qu.2.17.part.1.display=menu@
qu.2.17.part.1.credit.2=0.0@
qu.2.17.part.1.credit.1=1.0@
qu.2.17.part.2.grader=exact@
qu.2.17.part.2.name=sro_id_2@
qu.2.17.part.2.editing=useHTML@
qu.2.17.part.2.display.permute=true@
qu.2.17.part.2.question=(Unset)@
qu.2.17.part.2.answer.2=$D2@
qu.2.17.part.2.answer.1=$A2@
qu.2.17.part.2.mode=List@
qu.2.17.part.2.display=menu@
qu.2.17.part.2.credit.2=0.0@
qu.2.17.part.2.credit.1=1.0@
qu.2.17.part.3.grader=exact@
qu.2.17.part.3.name=sro_id_3@
qu.2.17.part.3.editing=useHTML@
qu.2.17.part.3.display.permute=true@
qu.2.17.part.3.question=(Unset)@
qu.2.17.part.3.answer.2=$D3@
qu.2.17.part.3.answer.1=$A3@
qu.2.17.part.3.mode=List@
qu.2.17.part.3.display=menu@
qu.2.17.part.3.credit.2=0.0@
qu.2.17.part.3.credit.1=1.0@
qu.2.17.part.4.grader=exact@
qu.2.17.part.4.name=sro_id_4@
qu.2.17.part.4.editing=useHTML@
qu.2.17.part.4.display.permute=true@
qu.2.17.part.4.question=(Unset)@
qu.2.17.part.4.answer.2=$D4@
qu.2.17.part.4.answer.1=$A4@
qu.2.17.part.4.mode=List@
qu.2.17.part.4.display=menu@
qu.2.17.part.4.credit.2=0.0@
qu.2.17.part.4.credit.1=1.0@
qu.2.17.part.5.grader=exact@
qu.2.17.part.5.name=sro_id_5@
qu.2.17.part.5.editing=useHTML@
qu.2.17.part.5.display.permute=true@
qu.2.17.part.5.question=(Unset)@
qu.2.17.part.5.answer.2=$D5@
qu.2.17.part.5.answer.1=$A5@
qu.2.17.part.5.mode=List@
qu.2.17.part.5.display=menu@
qu.2.17.part.5.credit.2=0.0@
qu.2.17.part.5.credit.1=1.0@
qu.2.17.question=<p>Identify each of the following statements as either&nbsp;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; $Q5</span></p>@

qu.2.18.mode=Multiple Selection@
qu.2.18.name=Definitions9: Mutually exclusive, independence@
qu.2.18.comment=@
qu.2.18.editing=useHTML@
qu.2.18.solution=@
qu.2.18.algorithm=@
qu.2.18.uid=f299a755-193c-44d2-925e-a2e9c1da2460@
qu.2.18.info=  Course=Introductory Statistics;
  Topic=Introduction to Probability;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.2.18.question=<p>Which of the following statements are TRUE?</p>
<p>(Assume for the purposes of this question that the events have a non-zero probability of occurring, unless otherwise stated).</p>
<p>&nbsp;</p>
<p>There may be more than one correct answer; select all that are true.</p>
<p>&nbsp;</p>@
qu.2.18.answer=1, 3, 4@
qu.2.18.choice.1=If A and B are mutually exclusive events, 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><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>A</mi><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mi>B</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>B</mi><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mi>A</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mstyle></math>.@
qu.2.18.choice.2=If P(A) = 0.7, P(B) = 0.5, then the probability of their intersection is less than 0.@
qu.2.18.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><msup><mi>B</mi><mrow><mi>c</mi></mrow></msup></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mi>B</mi></mrow></mfenced></mrow></mstyle></math>.@
qu.2.18.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mi>B</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mi>B</mi></mrow></mfenced><mrow><mi>c</mi></mrow></msup></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></mstyle></math>.@
qu.2.18.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mi>B</mi></mrow></mfenced><mrow><mi>c</mi></mrow></msup></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>A</mi><mrow><mi>c</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><msup><mi>B</mi><mrow><mi>c</mi></mrow></msup></mrow></mfenced></mrow></mstyle></math>.@
qu.2.18.fixed=@

