<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:black">A&nbsp;<strong>two-way table</strong>&nbsp;provides a way of portraying data that can facilitate calculating probabilities. When used to calculate probabilities, a two-way table is often called a&nbsp;<strong>contingency table</strong>. The table helps in determining conditional probabilities quite easily. The table displays sample values in relation to two different variables that may be dependent or contingent on one another. We used two-way tables in Chapters 1 and 2 to calculate marginal and conditional distributions. These tables organize data in a way that supports the calculation of relative frequency and, therefore, experimental (empirical) probability. Later on, we will use contingency tables again, but in another manner.</span></span></span></p>

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

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

<p><span style="font-size:14px"><strong>EXAMPLE&nbsp;3.20</strong></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:black">Suppose a study of speeding violations and drivers who use cell phones produced the following fictional data</span><span style="color:#002060">:</span></span></p>

<p>&nbsp;</p>

<table cellspacing="0" class="Table" style="border-collapse:collapse; width:900px">
	<caption>
	<p><span style="font-size:18px"><strong>Table3.3</strong></span></p>
	</caption>
	<thead>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; height:57px; vertical-align:bottom">
			<div style="text-align:center">&nbsp;</div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; height:57px; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Speeding Violation in the Last Year</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; height:57px; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>No Speeding Violation in the Last Year</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; height:57px; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Total</strong></span></div>
			</td>
		</tr>
	</thead>
	<tbody>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:41px">
			<div style="text-align:center"><span style="font-size:28px">Uses a cell phone while driving</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:41px">
			<div style="text-align:center"><span style="font-size:28px">25</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:41px">
			<div style="text-align:center"><span style="font-size:28px">280</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:41px">
			<div style="text-align:center"><span style="font-size:28px">305</span></div>
			</td>
		</tr>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:58px">
			<div style="text-align:center"><span style="font-size:28px">Does not use a cell phone while driving</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:58px">
			<div style="text-align:center"><span style="font-size:28px">45</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:58px">
			<div style="text-align:center"><span style="font-size:28px">405</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:58px">
			<div style="text-align:center"><span style="font-size:28px">450</span></div>
			</td>
		</tr>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:25px">
			<div style="text-align:center"><span style="font-size:28px">Total</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:25px">
			<div style="text-align:center"><span style="font-size:28px">70</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:25px">
			<div style="text-align:center"><span style="font-size:28px">685</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:25px">
			<div style="text-align:center"><span style="font-size:28px">755</span></div>
			</td>
		</tr>
	</tbody>
</table>

<p>&nbsp;</p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:#424242">The total number of people in the sample is 755. The row totals are 305 and 450. The column totals are 70 and 685. Notice that 305 + 450 = 755 and 70 + 685 = 755.</span></span></p>

<p>&nbsp;</p>

<ol style="list-style-type:lower-alpha">
	<li>
	<p><span style="font-size:14px"><span style="color:#424242">Using the table, calculate the following probabilities:<br />
	Find&nbsp;<em>P</em>(Person uses a cell phone while driving).</span></span></p>
	</li>
	<li>
	<p><span style="font-size:14px"><span style="color:#424242">Find&nbsp;<em>P</em>(Person had no violation in the last year).</span></span></p>
	</li>
	<li>
	<p><span style="font-size:14px"><span style="color:#424242">Find&nbsp;<em>P</em>(Person had no violation in the last year&nbsp;<em>and</em>&nbsp;uses a cell phone while driving).</span></span></p>
	</li>
	<li>
	<p><span style="font-size:14px"><span style="color:#424242">Find&nbsp;<em>P</em>(Person uses a cell phone while driving&nbsp;<em>or</em>&nbsp;person had no violation in the last year).</span></span></p>
	</li>
	<li>
	<p><span style="font-size:14px"><span style="color:#424242">Find&nbsp;<em>P</em>(Person uses a cell phone while driving&nbsp;<em>given</em>&nbsp;person had a violation in the last year).</span></span></p>
	</li>
	<li>
	<p><span style="font-size:14px"><span style="color:#424242">Find&nbsp;<em>P</em>(Person had no violation last year&nbsp;<em>given</em>&nbsp;person does not use a cell phone while driving).</span></span></p>
	</li>
</ol>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:#000000"><strong>Solution&nbsp;3.20</strong></span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:#424242">a. </span><span style="color:black">This is the same as the marginal distribution (</span><a href="https://openstax.org/books/statistics/pages/1-2-data-sampling-and-variation-in-data-and-sampling" style="color:blue; text-decoration:underline"><span style="color:black">Section 1.2</span></a><span style="color:black">).</span></span></p>

<p><span style="font-size:14px"><span style="color:black">P(Person uses a cell phone while driving)=<span class="math-tex">\(\frac{number\hspace{0.1cm}who\hspace{0.1cm}had\hspace{0.1cm}no\hspace{0.1cm}violation}{number\hspace{0.1cm}in\hspace{0.1cm}study}=\frac{685}{755}\)</span>&nbsp;&asymp;.9073.</span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">c. Find the number of participants who satisfy&nbsp;<em>both</em>&nbsp;conditions.</span></span></p>

<p><span style="font-size:14px"><span style="color:black">P(Person&nbsp;had&nbsp;no&nbsp;violation&nbsp;in&nbsp;the&nbsp;last&nbsp;year&nbsp;AND&nbsp;uses&nbsp;a&nbsp;cell&nbsp;phone&nbsp;while&nbsp;driving)= number&nbsp;in&nbsp;study=<span class="math-tex">\(\frac{number\hspace{0.1cm}who\hspace{0.1cm}had\hspace{0.1cm}no\hspace{0.1cm}violation\hspace{0.1cm}AND\hspace{0.1cm}uses\hspace{0.1cm}cell\hspace{0.1cm}phone\hspace{0.1cm}while\hspace{0.1cm}driving}{number\hspace{0.1cm}in\hspace{0.1cm}study}\)</span></span></span></p>

<p><span style="font-size:14px"><span style="color:black">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<span class="math-tex">\(=\frac{280}{755}\)</span></span></span></p>

<p><span style="font-size:14px">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span style="color:black">&asymp;.3709</span></span></p>

<p style="text-align:justify"><span style="font-size:14px">d. To find this probability, you need to identify how many participants use a cell phone while driving OR have no violation in the past year OR both.</span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:#424242">P(Person uses a cell phone while driving OR had no violation in the last year)</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:#424242"><span class="math-tex">\(\frac{25+405+280}{755}=\frac{710}{755}\)</span></span>&nbsp;<span style="color:black">&asymp;.9404</span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px">e. This is a conditional probability. You are&nbsp;<em><span style="color:#424242">given</span></em>&nbsp;that the person had no violation in the last year, so you need only consider the values in that column of data.</span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:#424242">(Person&nbsp;uses&nbsp;a&nbsp;cell&nbsp;phone&nbsp;while&nbsp;driving&nbsp;GIVEN&nbsp;the&nbsp;person&nbsp;had&nbsp;a&nbsp;violation&nbsp;in&nbsp;the&nbsp;last&nbsp;year)</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black"><span class="math-tex">\(\frac{number\hspace{0.1cm}who\hspace{0.1cm}used\hspace{0.1cm}cell\hspace{0.1cm}phone\hspace{0.1cm}AND\hspace{0.1cm}had\hspace{0.1cm}a\hspace{0.1cm}violation}{number\hspace{0.1cm}in\hspace{0.1cm}study\hspace{0.1cm}who\hspace{0.1cm}had\hspace{0.1cm}a\hspace{0.1cm}violation\hspace{0.1cm}in\hspace{0.1cm}the\hspace{0.1cm}last\hspace{0.1cm}year}=\frac{25}{70}\)</span>&nbsp;</span><span style="color:#424242">&asymp;.3571</span> </span></p>

<p style="text-align:justify"><span style="font-size:14px">f. For this conditional probability, consider only values in the row labeled &ldquo;Does not use a cell phone while driving.&rdquo;</span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">P(Person had no violation last year GIVEN person does not use cell phone while driving)<span class="math-tex">\(=\frac{405}{450}=.9\)</span></span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:#000000"><strong>EXAMPLE&nbsp;3.21</strong></span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><a href="https://openstax.org/books/statistics/pages/3-4-contingency-tables#M05_ch03-tbl003" style="color:blue; text-decoration:underline"><span style="color:#027eb5">Table 3.5</span></a><span style="color:#424242">&nbsp;shows a random sample of 100 hikers and the areas of hiking they prefer.</span></span></p>

<table cellspacing="0" class="Table" style="border-collapse:collapse; border:none; width:900px">
	<caption>
	<p><span style="font-size:18px"><strong>Table3.5 Hiking Area Preference</strong></span></p>
	</caption>
	<thead>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><span style="color:black">Sex</span></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><span style="color:black">The Coastline</span></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><span style="color:black">Near Lakes and Streams</span></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><span style="color:black">On Mountain Peaks</span></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><span style="color:black">Total</span></span></div>
			</td>
		</tr>
	</thead>
	<tbody>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px"><span style="color:black">Female</span></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px"><span style="color:black">18</span></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px"><span style="color:black">16</span></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px"><span style="color:black">___</span></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px"><span style="color:black">45</span></span></div>
			</td>
		</tr>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px"><span style="color:black">Male</span></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px"><span style="color:black">___</span></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px"><span style="color:black">___</span></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px"><span style="color:black">14</span></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px"><span style="color:black">55</span></span></div>
			</td>
		</tr>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px"><span style="color:black">Total</span></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px"><span style="color:black">___</span></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px"><span style="color:black">41</span></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px"><span style="color:black">___</span></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px"><span style="color:black">___</span></span></div>
			</td>
		</tr>
	</tbody>
</table>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:black">a. Complete the table.</span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:#000000">Solution&nbsp;3.21</span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:black">a. There are 45 females in the sample; 18 prefer the coastline and 16 prefer hiking near lakes and streams. So, we know there are 45 &minus; 18 &minus; 16 = 11 female students who prefer hiking on mountain peaks.</span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:black">Continue reasoning in this way to complete the table.</span></span></p>

<p>&nbsp;</p>

<table cellspacing="0" class="Table" style="border-collapse:collapse; border:none; width:900px">
	<caption>
	<p><span style="font-size:18px"><strong>Table3.6 Hiking Area Preference</strong></span></p>
	</caption>
	<thead>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Sex</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>The Coastline</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Near Lakes and Streams</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>On Mountain Peaks</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Total</strong></span></div>
			</td>
		</tr>
	</thead>
	<tbody>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">Female</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">18</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">16</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px"><strong>11</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">45</span></div>
			</td>
		</tr>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">Male</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px"><strong>16</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px"><strong>25</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">14</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">55</span></div>
			</td>
		</tr>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">Total</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px"><strong>34</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">41</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px"><strong>25</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px"><strong>100</strong></span></div>
			</td>
		</tr>
	</tbody>
</table>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:black">b. Are the events&nbsp;<em>being female</em>&nbsp;and&nbsp;<em>preferring the coastline</em>&nbsp;independent events?</span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:black">Let&nbsp;<em>F</em>&nbsp;= being female and let&nbsp;<em>C</em>&nbsp;= preferring the coastline.</span></span></p>

<p>&nbsp;</p>

<ol>
	<li>
	<p><span style="font-size:14px"><span style="color:black">Find&nbsp;<em>P</em>(<em>F</em>&nbsp;AND&nbsp;<em>C</em>).</span></span></p>
	</li>
	<li>
	<p><span style="font-size:14px"><span style="color:black">Find&nbsp;<em>P</em>(<em>F</em>)<em>P</em>(<em>C</em>).</span></span></p>
	</li>
</ol>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:black">Are these two numbers the same? If they are, then&nbsp;<em>F</em>&nbsp;and&nbsp;<em>C</em>&nbsp;are independent. If they are not, then&nbsp;<em>F</em>&nbsp;and&nbsp;<em>C</em>&nbsp;are not independent.</span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:#000000"><strong>Solution3.&nbsp;21</strong></span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:#424242">b.</span></span></p>

<ol>
	<li>
	<p><span style="font-size:14px"><em><span style="color:black">P</span></em><span style="color:black">(<em>F</em>&nbsp;AND&nbsp;<em>C</em>)&nbsp;<span class="math-tex">\(=\frac{18}{100}=.18\)</span> </span></span></p>
	</li>
	<li>
	<p><span style="font-size:14px"><em><span style="color:black">P</span></em><span style="color:black">(<em>F</em>)<em>P</em>(<em>C</em>)&nbsp;<span class="math-tex">\(=(\frac{45}{100})(\frac{34}{100})=(.45)(.34)=.153\)</span></span></span></p>
	</li>
</ol>

<p>&nbsp;</p>

<p><span style="font-size:14px"><em><span style="color:black">P</span></em><span style="color:black">(<em>F</em>&nbsp;AND&nbsp;<em>C</em>) &ne;&nbsp;<em>P</em>(<em>F</em>)<em>P</em>(<em>C</em>), so the events&nbsp;<em>F</em>&nbsp;and&nbsp;<em>C</em>&nbsp;are not independent.</span></span><br />
<br />
&nbsp;</p>

<p><span style="font-size:14px"><span style="color:#424242">c</span><span style="color:#002060">. </span><span style="color:black">Find the probability that a person is male given that the person prefers hiking near lakes and streams. Let&nbsp;<em>M</em>&nbsp;= being male, and let&nbsp;<em>L</em>&nbsp;= prefers hiking near lakes and streams.</span></span></p>

<p>&nbsp;</p>

<ol>
	<li>
	<p><span style="font-size:14px"><span style="color:black">What word tells you this is a conditional?</span></span></p>
	</li>
	<li>
	<p><span style="font-size:14px"><span style="color:black">Is the sample space for this problem all 100 hikers? If not, what is it?</span></span></p>
	</li>
	<li>
	<p><span style="font-size:14px"><span style="color:black">Fill in the blanks and calculate the probability:&nbsp;<em>P</em>(_____|_____) = _____.</span></span></p>
	</li>
</ol>

<p>&nbsp;</p>

<p><span style="font-size:14px"><strong><span style="color:#333333">Solution&nbsp;3</span></strong></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:#424242">c.</span></span></p>

<ol>
	<li>
	<p><span style="font-size:14px"><span style="color:black">The word&nbsp;<em>given</em>&nbsp;tells you that this is a conditional.</span></span></p>
	</li>
	<li>
	<p><span style="font-size:14px"><span style="color:black">No, the sample space for this problem is the 41 hikers who prefer lakes and streams.</span></span></p>
	</li>
	<li>
	<p><span style="font-size:14px"><span style="color:black">Find the conditional probability&nbsp;<em>P</em>(<em>M</em>|<em>L</em>). Because it is given that the person prefers hiking near lakes and streams, you need only consider the values in the column labeled &quot;Near Lakes and Streams.&quot;&nbsp;<em>P</em>(<em>M</em>|<em>L</em>)<span class="math-tex">\(=\frac{25}{41}\)</span> </span></span></p>
	</li>
</ol>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:#424242">d. </span><span style="color:black">Find the probability that a person is female or prefers hiking on mountain peaks. Let&nbsp;<em>F</em>&nbsp;= being female, and let&nbsp;<em>P</em>&nbsp;= prefers mountain peaks.</span></span></p>

<ol>
	<li>
	<p><span style="font-size:14px"><span style="color:black">Find&nbsp;<em>P</em>(<em>F</em>).</span></span></p>
	</li>
	<li>
	<p><span style="font-size:14px"><span style="color:black">Find&nbsp;<em>P</em>(<em>P</em>).</span></span></p>
	</li>
	<li>
	<p><span style="font-size:14px"><span style="color:black">Find&nbsp;<em>P</em>(<em>F</em>&nbsp;AND&nbsp;<em>P</em>).</span></span></p>
	</li>
	<li>
	<p><span style="font-size:14px"><span style="color:black">Find&nbsp;<em>P</em>(<em>F</em>&nbsp;OR&nbsp;<em>P</em>).</span></span></p>
	</li>
</ol>

<p>&nbsp;</p>

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

<p>&nbsp;</p>

<ol>
	<li>
	<p><span style="font-size:14px"><strong><span style="color:#333333"><span class="math-tex">\(P(F)=\frac{45}{100}\)</span></span></strong></span></p>
	</li>
	<li>
	<p><span style="font-size:14px"><strong><span style="color:#333333"><span class="math-tex">\(P(P)=\frac{25}{100}\)</span></span></strong></span></p>
	</li>
	<li>
	<p><span style="font-size:14px"><strong><span style="color:#333333"><span class="math-tex">\(P(F\hspace{0.1cm}AND\hspace{0.1cm}P)=\frac{number\hspace{0.1cm}of\hspace{0.1cm}hikers\hspace{0.1cm}that\hspace{0.1cm}are\hspace{0.1cm}both\hspace{0.1cm}female\hspace{0.1cm}AND\hspace{0.1cm}prefers\hspace{0.1cm}mountain\hspace{0.1cm}peaks}{number\hspace{0.1cm}of\hspace{0.1cm}hikers\hspace{0.1cm}in\hspace{0.1cm}study}=\frac{11}{100}\)</span>&nbsp;</span></strong></span></p>
	</li>
	<li>
	<p><span style="font-size:14px"><em><span style="color:black">P</span></em><span style="color:black">(<em>F</em>&nbsp;OR&nbsp;<em>P</em>) =&nbsp;<em>P</em>(<em>F</em>) +&nbsp;<em>P</em>(<em>P</em>) &minus;&nbsp;<em>P</em>(<em>F</em>&nbsp;AND&nbsp;<em>P</em>) </span>=<span class="math-tex">\(\frac{45}{100}+\frac{25}[100}-\frac{11}{100}=\frac{59}{100}\)</span></span></p>
	</li>
</ol>

<p>&nbsp;</p>

<p><strong><span style="font-size:14px"><span style="color:#002060">EXAMPLE&nbsp;3.22</span></span></strong></p>

<p>&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">Muddy Mouse lives in a cage with three doors. If Muddy goes out the first door, the probability that he gets caught by Alissa the cat is&nbsp;1515&nbsp;and the probability he is not caught is&nbsp;4545. If he goes out the second door, the probability he gets caught by Alissa is&nbsp;1414&nbsp;and the probability he is not caught is&nbsp;3434. The probability that Alissa catches Muddy coming out of the third door is&nbsp;1212&nbsp;and the probability she does not catch Muddy is&nbsp;1212. It is equally likely that Muddy will choose any of the three doors, so the probability of choosing each door is&nbsp;1313.</span></p>

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

<table cellspacing="0" class="Table" style="border-collapse:collapse; border:none; width:900px">
	<caption>
	<p><span style="font-size:18px"><strong>Table3.8 Door Choice</strong></span></p>
	</caption>
	<thead>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Caught or Not</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Door One</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Door Two</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Door Three</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Total</strong></span></div>
			</td>
		</tr>
	</thead>
	<tbody>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">Caught</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center">
			<div><span style="font-size:28px"><span class="math-tex">\(\frac{1}{15}\)</span></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center">
			<div><span style="font-size:28px"><span class="math-tex">\(\frac{1}{12}\)</span></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center">
			<div><span style="font-size:28px"><span class="math-tex">\(\frac{1}{6}\)</span></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center">
			<div>&nbsp;</div>
			</td>
		</tr>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">Not Caught</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center; vertical-align:top">
			<div><span style="font-size:28px"><span class="math-tex">\(\frac{4}{15}\)</span></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center; vertical-align:top">
			<div><span style="font-size:28px"><span class="math-tex">\(\frac{3}{12}\)</span></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center; vertical-align:top">
			<div><span style="font-size:28px"><span class="math-tex">\(\frac{1}{6}\)</span></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center">
			<div>&nbsp;</div>
			</td>
		</tr>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">Total</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">____</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">____</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">____</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">1</span></div>
			</td>
		</tr>
	</tbody>
</table>

<p>&nbsp;</p>

<ul>
	<li>
	<p><span style="font-size:14px"><span style="color:#424242">The first entry&nbsp;<span class="math-tex">\(\frac{1}{15}=(\frac{1}{5})(\frac{1}{3})\)</span></span>&nbsp;<em><span style="color:#424242">P</span></em><span style="color:#424242">(Door One AND Caught).</span></span></p>
	</li>
	<li>
	<p><span style="font-size:14px"><span style="color:#424242">The entry&nbsp;</span><span class="math-tex">\(\frac{1}{15}=(\frac{!}{5})(\frac{1}{3})\)</span>&nbsp;<span style="color:#424242"><em>P</em>(Door One AND Not Caught).</span></span></p>
	</li>
</ul>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:#000000">Verify the remaining entries.</span></span></p>

<p><br />
<br />
<span style="font-size:14px"><span style="color:#002060">a. </span><span style="color:black">Complete the probability contingency table. Calculate the entries for the totals. Verify that the lower-right corner entry is 1.</span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:#000000"><strong>Solution&nbsp;3.22</strong></span></span></p>

<table cellspacing="0" class="Table" style="border-collapse:collapse; border:none; width:900px">
	<caption>
	<p><span style="font-size:18px"><strong>Table3.9 Door Choice</strong></span></p>
	</caption>
	<thead>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Caught or Not</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Door One</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Door Two</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Door Three</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Total</strong></span></div>
			</td>
		</tr>
	</thead>
	<tbody>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">Caught</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">115115</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">112112</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">1616</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">19601960</span></div>
			</td>
		</tr>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">Not Caught</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">415415</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">312312</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">1616</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">41604160</span></div>
			</td>
		</tr>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">Total</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">515515</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">412412</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">2626</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">1</span></div>
			</td>
		</tr>
	</tbody>
</table>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:#424242">b</span><span style="color:black">. What is the probability that Alissa does not catch Muddy?</span></span></p>

<h4>&nbsp;</h4>

<p><strong><span style="font-size:14px"><span style="color:#2e74b5"><span style="color:black">Solution&nbsp;b</span></span></span></strong></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:black">b.&nbsp;41604160</span></span></p>

<p><br />
<br />
<span style="font-size:14px"><span style="color:black">c. What is the probability that Muddy chooses Door One OR Door Two given that Muddy is caught by Alissa?</span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><strong><span style="color:#2e74b5"><span style="color:#333333">Solution&nbsp;c</span></span></strong></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:#424242">c. This is a conditional probability, so consider only probabilities in the row labeled &quot;Caught.&quot; Choosing Door One and choosing Door Two are mutually exclusive, so</span>&nbsp;P(Choosing Door One OR Choosing Door Two AND Caught)<span class="math-tex">\(= \frac{1}{15}+\frac{1}{12}=\frac{9}{60}\)</span>.</span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:#424242">Use the formula for conditional probability&nbsp;P(A??B)&nbsp;<span class="math-tex">\(=\frac{P(A\hspace{0.1cm}AND\hspace{0.1cm}B)}{P(B)}\)</span></span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px">P(Door One OR Door Two|Caught)&nbsp;<span class="math-tex">\(=\frac{P(DOOR\hspace{0.1cm}one\hspace{0.1cm}OR\hspace{0.1cm}DOOR\hspace{0.1cm}TWO\hspace{0.1cm}AND\hspace{0.1cm}Caught)}{P(Caught)}=\frac{\frac{9}{60}}{\frac{19}{60}}=\frac{9}{19}\)</span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:#000000"><strong>EXAMPLE&nbsp;3.23</strong></span></span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><a href="https://openstax.org/books/statistics/pages/3-4-contingency-tables#Ch03_M04_tbl007" style="color:blue; text-decoration:underline"><span style="color:#027eb5">Table 3.10</span></a><span style="color:#424242">&nbsp;contains the number of crimes per 100,000 inhabitants from 2008 to 2011 in the United States.\</span></span></p>

<p>&nbsp;</p>

<table cellspacing="0" class="Table" style="border-collapse:collapse; border:none; width:900px">
	<thead>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Year</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Crime A</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Crime B</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Crime C</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Crime D</strong></span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom">
			<div style="text-align:center"><span style="font-size:28px"><strong>Total</strong></span></div>
			</td>
		</tr>
	</thead>
	<tbody>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">2008</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">145.7</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">732.1</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">29.7</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">314.7</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center">
			<div>&nbsp;</div>
			</td>
		</tr>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">2009</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">133.1</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">717.7</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">29.1</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">259.2</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center">
			<div>&nbsp;</div>
			</td>
		</tr>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">2010</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">119.3</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">701</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">27.7</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">239.1</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center">
			<div>&nbsp;</div>
			</td>
		</tr>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">2011</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">113.7</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">702.2</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">26.8</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">229.6</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center">
			<div>&nbsp;</div>
			</td>
		</tr>
		<tr>
			<td style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none">
			<div style="text-align:center"><span style="font-size:28px">Total</span></div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center">
			<div>&nbsp;</div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center">
			<div>&nbsp;</div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center">
			<div>&nbsp;</div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center">
			<div>&nbsp;</div>
			</td>
			<td style="border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center">
			<div>&nbsp;</div>
			</td>
		</tr>
	</tbody>
</table>

<p>&nbsp;</p>

<p><span style="font-size:14px"><strong>Table3.10</strong>&nbsp;U.S. Crime Index Rates Per 100,000 Inhabitants 2008&ndash;2011</span></p>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:#424242">TOTAL each column and each row. Total data = 4,520.7.</span></span></p>

<p>&nbsp;</p>

<ol style="list-style-type:lower-alpha">
	<li>
	<p><span style="font-size:14px"><span style="color:#424242">Find&nbsp;<em>P</em>(2009 AND Crime A).</span></span></p>
	</li>
	<li>
	<p><span style="font-size:14px"><span style="color:#424242">Find&nbsp;<em>P</em>(2010 AND Crime B).</span></span></p>
	</li>
	<li>
	<p><span style="font-size:14px"><span style="color:#424242">Find&nbsp;<em>P</em>(2010 OR Crime B).</span></span></p>
	</li>
	<li>
	<p><span style="font-size:14px"><span style="color:#424242">Find&nbsp;<em>P</em>(2011|Crime A).</span></span></p>
	</li>
	<li>
	<p><span style="font-size:14px"><span style="color:#424242">Find&nbsp;<em>P</em>(Crime D|2008).</span></span></p>
	</li>
</ol>

<p>&nbsp;</p>

<p><span style="font-size:14px"><span style="color:#000000"><strong>Solution&nbsp;3.23</strong></span></span></p>

<p>&nbsp;</p>

<ol start="1" style="list-style-type:lower-alpha">
	<li>
	<p><span style="font-size:14px"><strong><span style="color:#002060"><span class="math-tex">\(\frac{133.14}{520.7}=.0294\)</span></span></strong></span></p>
	</li>
	<li>
	<p><span style="font-size:14px"><span class="math-tex">\(\frac{7014}{520.7}=.1551\)</span></span></p>
	</li>
	<li>
	<p><span style="font-size:14px"><em><span style="color:black">P</span></em><span style="color:black">(2010 OR Crime B) =&nbsp;<em>P</em>(2010) +&nbsp;<em>P</em>(Crime B) &ndash;&nbsp;<em>P</em>(2010 AND Crime B) <span class="math-tex">\(\frac{1,087.14}{520.7}+\frac{2,852.94}{520.7}-\frac{7014}{520.7}=.7165\)</span></span>\</span></p>
	</li>
	<li>
	<p><span style="font-size:14px"><span class="math-tex">\(\frac{113.7}{511.8}=.2222\)</span></span></p>
	</li>
	<li>
	<p><span style="font-size:14px"><span class="math-tex">\(\frac{314.7}{1,222.2}=.2575\)</span></span></p>
	</li>
</ol>