<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">A </span><strong><span style="color:black">Venn diagram </span></strong><span style="color:black">is a picture that represents the outcomes of an experiment. It generally consists of a box that represents the sample space </span><em><span style="color:black">S </span></em><span style="color:black">together with circles or ovals. The circles or ovals represent events.</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Suppose an experiment has the outcomes 1, 2, 3, . . . , 12 where each outcome has an equal chance of occurring.</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Let event </span><em><span style="color:black">A </span></em><span style="color:black">= {1, 2, 3, 4, 5, 6} and event </span><em><span style="color:black">B </span></em><span style="color:black">= {6, 7, 8, 9}. Then </span><em><span style="color:black">A </span></em><span style="color:black">AND </span><em><span style="color:black">B </span></em><span style="color:black">= {6} and </span><em><span style="color:black">A </span></em><span style="color:black">OR </span><em><span style="color:black">B </span></em><span style="color:black">= {1, 2, 3, 4, 5, 6, 7,</span> 8, 9}. The Venn diagram is as follows:</span></p>

<p style="text-align:justify"><span style="font-size:14px"><img alt="" src="https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16239255601623925560.png" style="height:321px; width:800px" /></span></p>

<p style="text-align:justify"><strong><span style="font-size:14px">Fig 3.15</span></strong></p>

<p style="text-align:justify"><strong><span style="font-size:14px">EXAMPLE&nbsp;3.28</span></strong></p>

<p style="text-align:justify"><span style="font-size:14px">Flip two fair coins. Let&nbsp;<em>A</em>&nbsp;= tails on the first coin. Let&nbsp;<em>B</em>&nbsp;= tails on the second coin. Then&nbsp;<em>A</em>&nbsp;= {<em>TT</em>,&nbsp;<em>TH</em>} and&nbsp;<em>B</em>&nbsp;= {<em>TT</em>,&nbsp;<em>HT</em>}. Therefore,&nbsp;<em>A</em>&nbsp;AND&nbsp;<em>B</em>&nbsp;= {<em>TT</em>}.&nbsp;<em>A</em>&nbsp;OR&nbsp;<em>B</em>&nbsp;= {<em>TH</em>,&nbsp;<em>TT</em>,&nbsp;<em>HT</em>}.</span></p>

<p style="text-align:justify"><span style="font-size:14px">The sample space when you flip two fair coins is&nbsp;<em>X</em>&nbsp;= {<em>HH</em>,&nbsp;<em>HT</em>,&nbsp;<em>TH</em>,&nbsp;<em>TT</em>}. The outcome&nbsp;<em>HH</em>&nbsp;is in NEITHER&nbsp;<em>A</em>&nbsp;NOR&nbsp;<em>B</em>. The Venn diagram is as follows:</span></p>

<p style="text-align:justify"><span style="font-size:14px"><img alt="" src="https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16239256241623925624.jpg" style="height:540px; width:800px" /></span></p>

<p style="text-align:justify"><span style="font-size:14px"><strong><span style="color:#424242">Figure&nbsp;3.16</span></strong></span></p>

<p style="text-align:justify"><strong><span style="font-size:14px"><span style="color:#002060">EXAMPLE&nbsp;3.29</span></span></strong></p>

<p style="text-align:justify"><span style="font-size:14px"><strong><span style="color:#424242">Forty percent</span></strong>&nbsp;of the students at a local college belong to a club and&nbsp;<strong><span style="color:#424242">50 percent</span></strong>&nbsp;work part time.&nbsp;<strong><span style="color:#424242">Five percent</span></strong>&nbsp;of the students work part time and belong to a club. Draw a Venn diagram showing the relationships. Let&nbsp;<em><span style="color:#424242">C</span></em>&nbsp;= student belongs to a club and&nbsp;<em><span style="color:#424242">PT</span></em>&nbsp;= student works part time.</span></p>

<p style="text-align:justify"><span style="font-size:14px">Start by drawing a rectangle to represent the sample space. Then draw two circles or ovals inside the rectangle to represent the events of interest: belonging to a club (<em><span style="color:#424242">C</span></em>) and working part time (<em><span style="color:#424242">PT</span></em>). Always draw overlapping shapes to represent outcomes that are in both events.</span></p>

<p style="text-align:justify"><span style="font-size:14px"><img alt="" src="https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16239256761623925676.jpg" style="height:617px; width:800px" /></span></p>

<p style="text-align:justify"><span style="font-size:14px"><strong><span style="color:#424242">Figure&nbsp;3.17</span></strong></span></p>

<p style="text-align:justify"><span style="font-size:14px">Label each piece of the diagram clearly and note the probability or frequency of each part. Start by labeling the overlapping section first. Note that the probabilities in&nbsp;<em><span style="color:#424242">C</span></em>&nbsp;total 0.40 and the sum of the probabilities in&nbsp;<em><span style="color:#424242">PT</span></em>&nbsp;is 0.50. The total of all probabilities displayed must be 1, representing 100 percent of the sample space.</span></p>

<p style="text-align:justify"><span style="font-size:14px">If a student is selected at random, find the following:</span></p>

<ol style="list-style-type:lower-alpha">
	<li>
	<p style="text-align:justify"><span style="font-size:14px">the probability that the student belongs to a club.</span></p>
	</li>
	<li>
	<p style="text-align:justify"><span style="font-size:14px">the probability that the student works part time.</span></p>
	</li>
	<li>
	<p style="text-align:justify"><span style="font-size:14px">the probability that the student belongs to a club AND works part time.</span></p>
	</li>
	<li>
	<p style="text-align:justify"><span style="font-size:14px">the probability that the student belongs to a club&nbsp;<strong><span style="color:#424242">given</span></strong>&nbsp;that the student works part time.</span></p>
	</li>
	<li>
	<p style="text-align:justify"><span style="font-size:14px">the probability that the student belongs to a club&nbsp;<strong><span style="color:#424242">OR</span></strong>&nbsp;works part time.</span></p>
	</li>
</ol>

<p style="text-align:justify"><strong><span style="font-size:14px"><span style="color:#333333">Solution&nbsp;1</span></span></strong></p>

<p style="text-align:justify"><span style="font-size:14px"><em><span style="color:black">P</span></em><span style="color:black">(<em>C</em>) = .40</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><em><span style="color:black">P</span></em><span style="color:black">(<em>PT</em>) = .50</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><em><span style="color:black">P</span></em><span style="color:black">(<em>C</em>&nbsp;AND&nbsp;<em>PT</em>) = .05</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">P(C|PT)</span>&nbsp;<span class="math-tex">\(=\frac{P(C\hspace{0.1cm}AND\hspace{0.1cm}PT)}{P(PT)}=\frac{0.5}{0.50}=1\)</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><em><span style="color:black">P</span></em><span style="color:black">(<em>C</em>&nbsp;OR&nbsp;<em>PT</em>) =&nbsp;<em>P</em>(<em>C</em>) +&nbsp;<em>P</em>(<em>PT</em>) &minus;&nbsp;<em>P</em>(<em>C</em>&nbsp;AND&nbsp;<em>PT</em>) = .40 + .50 &minus; .05 = .85</span></span></p>

<p style="text-align:justify"><strong><span style="font-size:14px"><span style="color:#002060">EXAMPLE&nbsp;3.30</span></span></strong></p>

<p style="text-align:justify"><span style="font-size:14px">A person with type O blood and a negative Rh factor (Rh&ndash;) can donate blood to any person with any blood type. Four percent of African Americans have type O blood and a negative Rh factor, 5&minus;10 percent of African Americans have the Rh&ndash; factor, and 51 percent have type O blood.</span></p>

<p style="text-align:justify"><span style="font-size:14px"><img alt="" src="https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16239259511623925951.jpg" style="height:601px; width:800px" /></span></p>

<p style="text-align:justify"><span style="font-size:14px"><strong><span style="color:#424242">Figure&nbsp;3.18</span></strong></span></p>

<p style="text-align:justify"><span style="font-size:14px">The &ldquo;O&rdquo; circle represents the African Americans with type O blood. The &ldquo;Rh&ndash;&ndash;&quot; oval represents the African Americans with the Rh&ndash; &ndash;factor.</span></p>

<p style="text-align:justify"><span style="font-size:14px">We will use the average of 5 percent and 10 percent, 7.5 percent, as the percentage of African Americans who have the Rh&ndash;&ndash; factor. Let&nbsp;<em><span style="color:#424242">O</span></em>&nbsp;= African American with Type O blood and&nbsp;<em><span style="color:#424242">R</span></em>&nbsp;= African American with Rh&ndash;&ndash;factor.</span></p>

<ol style="list-style-type:lower-alpha">
	<li>
	<p style="text-align:justify"><span style="font-size:14px"><em><span style="color:black">P</span></em><span style="color:black">(<em>O</em>) = ___________</span></span></p>
	</li>
	<li>
	<p style="text-align:justify"><span style="font-size:14px"><em><span style="color:black">P</span></em><span style="color:black">(<em>R</em>) = ___________</span></span></p>
	</li>
	<li>
	<p style="text-align:justify"><span style="font-size:14px"><em><span style="color:black">P</span></em><span style="color:black">(<em>O</em>&nbsp;AND&nbsp;<em>R</em>) = ___________</span></span></p>
	</li>
	<li>
	<p style="text-align:justify"><span style="font-size:14px"><em><span style="color:black">P</span></em><span style="color:black">(<em>O</em>&nbsp;OR&nbsp;<em>R</em>) = ____________</span></span></p>
	</li>
	<li>
	<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">In the Venn Diagram, describe the overlapping area using a complete sentence.</span></span></p>
	</li>
	<li>
	<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">In the Venn Diagram, describe the area in the rectangle but outside both the circle and the oval using a complete sentence.</span></span></p>
	</li>
</ol>

<p style="text-align:justify"><span style="font-size:14px"><strong>Solution&nbsp;3.30</strong></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">a.&nbsp;<em>P</em>(<em>O</em>) = .51</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">b.&nbsp;<em>P</em>(<em>R</em>) = .075 because an average of 7.5 percent of African Americans have the Rh&ndash; &ndash;factor.</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">c.&nbsp;<em>P</em>(<em>O</em>&nbsp;AND&nbsp;<em>R</em>) = 0.04 because 4 percent of African Americans have both Type O blood and the Rh&ndash; &ndash;factor.</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">d.&nbsp;<em>P</em>(<em>O</em>&nbsp;OR&nbsp;<em>R</em>) =&nbsp;<em>P</em>(<em>O</em>) +&nbsp;<em>P</em>(<em>R</em>) -&nbsp;<em>P</em>(<em>O</em>&nbsp;AND&nbsp;<em>R</em>) = .51 + .075 &minus; .04 = .545</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">e. The area represents the African Americans that have type O blood and the Rh&ndash;&ndash; factor.</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">f. The area represents the African Americans that have neither type O blood nor the Rh&ndash;&ndash; factor.</span></span></p>