<p style="text-align:justify"><span style="font-size:14px"><span style="color:#000000"><strong>EXAMPLE&nbsp;3.14&nbsp;&nbsp; </strong></span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Krishna is trying to choose where to go on vacation. His two choices are:&nbsp;<em>A</em>&nbsp;= New Zealand and&nbsp;<em>B</em>&nbsp;= Alaska.</span></span></p>

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

<ul>
	<li style="text-align:justify">
	<p><span style="font-size:14px"><span style="color:black">Krishna can only afford one vacation. The probability that he chooses&nbsp;<em>A</em>&nbsp;is&nbsp;<em>P</em>(<em>A</em>) = .6 and the probability that he chooses&nbsp;<em>B</em>&nbsp;is&nbsp;<em>P</em>(<em>B</em>) = .35.</span></span></p>
	</li>
	<li style="text-align:justify">
	<p><span style="font-size:14px"><span style="color:black"><em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) = 0 because Klaus can only afford to take one vacation.</span></span></p>
	</li>
	<li style="text-align:justify">
	<p><span style="font-size:14px"><span style="color:black">Therefore, the probability that he chooses either New Zealand or Alaska is&nbsp;<em>P</em>(<em>A</em>&nbsp;OR&nbsp;<em>B</em>) =&nbsp;<em>P</em>(<em>A</em>) +&nbsp;<em>P</em>(<em>B</em>) = .6 + .35 = .95. Note that the probability that he does not choose to go anywhere on vacation must be .05.</span></span></p>
	</li>
</ul>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:#000000"><strong>EXAMPLE&nbsp;3.15</strong></span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Carlos plays college soccer. He makes a goal 65 percent of the time he shoots. Carlos is going to attempt two goals in a row in the next game.&nbsp;<em>A</em>&nbsp;= the event Carlos is successful on his first attempt.&nbsp;<em>P</em>(<em>A</em>) = .65.&nbsp;<em>B</em>&nbsp;= the event Carlos is successful on his second attempt.&nbsp;<em>P</em>(<em>B</em>) = .65. Carlos tends to shoot in streaks. The probability that he makes the second goal&nbsp;<strong>given</strong>&nbsp;that he made the first goal is .90.</span><br />
<span style="color:#424242">a. </span><span style="color:black">What is the probability that he makes both goals?</span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:#000000"><strong>Solution&nbsp;3.15</strong></span></span></p>

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

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

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">a. The problem is asking you to find&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) =&nbsp;<em>P</em>(<em>B</em>&nbsp;AND&nbsp;<em>A</em>). Since&nbsp;<em>P</em>(<em>B</em>|<em>A</em>) = :&nbsp;<em>P</em>(<em>B</em>&nbsp;AND&nbsp;<em>A</em>) =&nbsp;<em>P</em>(<em>B</em>|<em>A</em>)&nbsp;<em>P</em>(<em>A</em>)&nbsp;=&nbsp;(.90)(.65) = .585.</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Carlos makes the first and second goals with probability .585.<br />
b. What is the probability that Carlos makes either the first goal or the second goal?</span></span></p>

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

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

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">b.The problem is asking you to find&nbsp;<em>P</em>(<em>A</em>&nbsp;OR&nbsp;<em>B</em>).<em> </em></span></span></p>

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

<p style="margin-left:24px; text-align:justify"><span style="font-size:14px"><em><span style="color:black">P</span></em><span style="color:black">(<em>A</em>&nbsp;OR&nbsp;<em>B</em>) =&nbsp;<em>P</em>(<em>A</em>) +&nbsp;<em>P</em>(<em>B</em>) &minus;&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) = .65 + .65 &minus; .585 = .715</span></span></p>

<p style="margin-left:24px; text-align:justify">&nbsp;</p>

<div>
<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Carlos makes either the first goal or the second goal with probability .715.</span></span></p>
</div>

<p style="text-align:justify"><br />
<br />
<span style="font-size:14px"><span style="color:black">c. Are&nbsp;<em>A</em>&nbsp;and&nbsp;<em>B</em>&nbsp;independent? Solution&nbsp;3</span></span></p>

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

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

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">c.No, they are not, because&nbsp;<em>P</em>(<em>B</em>&nbsp;AND&nbsp;<em>A</em>) = .585.</span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><em><span style="color:black">P</span></em><span style="color:black">(<em>B</em>)<em>P</em>(<em>A</em>) = (.65)(.65) = .423</span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">.423 &ne; .585 =&nbsp;<em>P</em>(<em>B</em>&nbsp;AND&nbsp;<em>A</em>)</span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">So,&nbsp;<em>P</em>(<em>B</em>&nbsp;AND&nbsp;<em>A</em>) is&nbsp;<strong>not</strong>&nbsp;equal to&nbsp;<em>P</em>(<em>B</em>)<em>P</em>(<em>A</em>).</span></span><br />
<br />
&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:#424242">d. </span>Are&nbsp;<em>A</em>&nbsp;and&nbsp;<em>B</em>&nbsp;mutually exclusive?</span></p>

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

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

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

<p style="text-align:justify"><span style="font-size:14px">d. No, they are not because&nbsp;<em>P</em>(<em>A</em>&nbsp;and&nbsp;<em>B</em>) = .585.</span></p>

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

<p style="text-align:justify"><span style="font-size:14px">To be mutually exclusive,&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) must equal zero.</span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:#000000"><strong>EXAMPLE&nbsp;3.16</strong></span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px">A community swim team has&nbsp;150&nbsp;members.&nbsp;Seventy-five&nbsp;of the members are advanced swimmers.&nbsp;Forty-seven&nbsp;of the members are intermediate swimmers. The remainder are novice swimmers.&nbsp;Forty&nbsp;of the advanced swimmers practice four times a week.&nbsp;Thirty&nbsp;of the intermediate swimmers practice four times a week.&nbsp;Ten&nbsp;of the novice swimmers practice four times a week. Suppose<span style="color:#424242"> one member of the swim team is chosen randomly.</span></span></p>

<p style="text-align:justify"><br />
<span style="font-size:14px"><span style="color:#424242">a. </span><span style="color:black">What is the probability that the member is a novice swimmer?</span></span></p>

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

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

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">a. There are&nbsp;<strong>150</strong>&nbsp;members; 75 of these are advanced, and 47 of these are intermediate swimmers. So there are 150 &minus; 75 &minus; 47 =&nbsp;<strong>28</strong>&nbsp;novice swimmers. The probability that a randomly selected swimmer is a novice is&nbsp;28150.28150.</span></span></p>

<p style="text-align:justify"><br />
<br />
<span style="font-size:14px"><span style="color:#424242">b. What is the probability that the member practices four times a week?</span></span></p>

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

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

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:#424242">b.&nbsp;</span><span style="color:black">40 + 30 + 10150=80150</span></span><br />
<br />
&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">c. What is the probability that the member is an advanced swimmer and practices four times a week?</span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:#000000"><strong>Solution&nbsp;c</strong></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. There are 40 advanced swimmers who practice four times per week, so the probability is&nbsp;</span><span class="math-tex">\(\frac{40}{150}\)</span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">d. What is the probability that a member is an advanced swimmer and an intermediate swimmer? Are being an advanced swimmer and being an intermediate swimmer mutually exclusive? Why or why not?</span></span></p>

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

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

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

<p style="text-align:justify"><span style="font-size:14px"><em><span style="color:black">P</span></em><span style="color:black">(advanced AND intermediate)&nbsp;=&nbsp;0, so these are mutually exclusive events. A swimmer cannot be an advanced swimmer and an intermediate swimmer at the same time.</span></span></p>

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

<p style="text-align:justify"><br />
<span style="font-size:14px"><span style="color:black">e. Are being a novice swimmer and practicing four times a week independent events? Why or why not?</span></span></p>

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

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

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">e. No, these are not independent events.<br />
<em>P</em>(novice AND practices four times per week) = .0667<br />
<em>P</em>(novice)<em>P</em>(practices four times per week) = .0996<br />
.0667 &ne; .0996</span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:#000000"><strong>EXAMPLE&nbsp;3.17</strong></span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Felicity attends a school in Modesto, CA. The probability that Felicity enrolls in a math class is .2 and the probability that she enrolls in a speech class is .65. The probability that she enrolls in a math class GIVEN that she enrolls in speech class is .25.</span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Let&nbsp;<em>M</em>&nbsp;= math class,&nbsp;<em>S</em>&nbsp;= speech class, and&nbsp;<em>M</em>|<em>S</em>&nbsp;= math given speech.</span></span></p>

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

<ol style="list-style-type:lower-alpha">
	<li>
	<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">What is the probability that Felicity enrolls in math and speech?<br />
	Find&nbsp;<em>P</em>(<em>M</em>&nbsp;AND&nbsp;<em>S</em>) =&nbsp;<em>P</em>(<em>M</em>|<em>S</em>)<em>P</em>(<em>S</em>).</span></span></p>
	</li>
	<li>
	<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">What is the probability that Felicity enrolls in math or speech classes?<br />
	Find&nbsp;<em>P</em>(<em>M</em>&nbsp;OR&nbsp;<em>S</em>) =&nbsp;<em>P</em>(<em>M</em>) +&nbsp;<em>P</em>(<em>S</em>) &minus;&nbsp;<em>P</em>(<em>M</em>&nbsp;AND&nbsp;<em>S</em>).</span></span></p>
	</li>
	<li>
	<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Are&nbsp;<em>M</em>&nbsp;and&nbsp;<em>S</em>&nbsp;independent? Is&nbsp;<em>P</em>(<em>M</em>|<em>S</em>) =&nbsp;<em>P</em>(<em>M</em>)?</span></span></p>
	</li>
	<li>
	<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Are&nbsp;<em>M</em>&nbsp;and&nbsp;<em>S</em>&nbsp;mutually exclusive? Is&nbsp;<em>P</em>(<em>M</em>&nbsp;AND&nbsp;<em>S</em>) = 0?</span></span></p>
	</li>
</ol>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:#000000"><strong>Solution&nbsp;3.17</strong></span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">a.&nbsp;<em>P</em>(<em>M</em>&nbsp;AND&nbsp;<em>S</em>) =&nbsp;<em>P</em>(<em>M</em>|<em>S</em>)<em>P</em>(<em>S</em>) = .25(.65) = .1625</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">b.&nbsp;<em>P</em>(<em>M</em>&nbsp;OR&nbsp;<em>S</em>) =&nbsp;<em>P</em>(<em>M</em>) +&nbsp;<em>P</em>(<em>S</em>) &minus;&nbsp;<em>P</em>(<em>M</em>&nbsp;AND&nbsp;<em>S</em>) = .2 + .65 &minus; .1625 = .6875</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">c. No,&nbsp;<em>P</em>(<em>M</em>|<em>S</em>) = .25 and&nbsp;<em>P</em>(<em>M</em>) = .2.</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">d. No,&nbsp;<em>P</em>(<em>M</em>&nbsp;AND&nbsp;<em>S</em>) = .1625.</span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:#000000"><strong>EXAMPLE&nbsp;3.18</strong></span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Researchers are studying one particular type of disease that affects women more often than men. Studies show that about one woman in seven (approximately 14.3 percent) who live to be 90 will develop the disease. Suppose that of those women who develop this disease, a test is negative 2 percent of the time. Also suppose that in the general population of women, the test for the disease is negative about 85 percent of the time. Let&nbsp;<em>B</em>&nbsp;= woman develops the disease and let&nbsp;<em>N</em>&nbsp;= tests negative. Suppose one woman is selected at random.</span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">a. What is the probability that the woman develops the disease? What is the probability that woman tests negative?</span></span></p>

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

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

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">a.&nbsp;<em>P</em>(<em>B</em>) = .143;&nbsp;<em>P</em>(<em>N</em>) = .85</span><br />
<span style="color:black">b. Given that the woman develops the disease, what is the probability that she tests negative?</span></span></p>

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

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

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">b. Among women who develop the disease, the test is negative 2 percent of the time, so </span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">N</span></em><span style="color:black">|</span><em><span style="color:black">B</span></em><span style="color:black">) = .02</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. What is the probability that the woman has the disease AND tests negative?</span></span></p>

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

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

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

<p style="text-align:justify"><span style="font-size:14px"><em><span style="color:black">c.P</span></em><span style="color:black">(</span><em><span style="color:black">B </span></em><span style="color:black">AND </span><em><span style="color:black">N</span></em><span style="color:black">) = </span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">B</span></em><span style="color:black">)</span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">N</span></em><span style="color:black">|</span><em><span style="color:black">B</span></em><span style="color:black">) = (.143)(.02) = .0029</span></span></p>

<p style="margin-left:24px; text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">d. What is the probability that the woman has the disease OR tests negative?</span></span></p>

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

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

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">d. </span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">B </span></em><span style="color:black">OR </span><em><span style="color:black">N</span></em><span style="color:black">) = </span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">B</span></em><span style="color:black">) + </span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">N</span></em><span style="color:black">) &minus; </span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">B </span></em><span style="color:black">AND </span><em><span style="color:black">N</span></em><span style="color:black">) = .143 + .85 &minus; .0029 = .9901</span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">e. Are having the disease and testing negative independent events?</span></span></p>

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

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

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">e. No. </span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">N</span></em><span style="color:black">) = .85; </span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">N</span></em><span style="color:black">|</span><em><span style="color:black">B</span></em><span style="color:black">) = .02. So, </span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">N</span></em><span style="color:black">|</span><em><span style="color:black">B</span></em><span style="color:black">) does not equal </span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">N</span></em><span style="color:black">).</span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">f. Are having the disease and testing negative mutually exclusive?</span></span></p>

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

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

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">f. No. </span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">B </span></em><span style="color:black">AND </span><em><span style="color:black">N</span></em><span style="color:black">) = .0029. For </span><em><span style="color:black">B </span></em><span style="color:black">and </span><em><span style="color:black">N </span></em><span style="color:black">to be mutually exclusive, </span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">B </span></em><span style="color:black">AND </span><em><span style="color:black">N</span></em><span style="color:black">) must be zero.</span></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><strong>EXAMPLE 3.19Exame</strong></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:white">3.19</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Refer to the information in </span><strong><span style="color:#26397f">Example 3.18</span></strong><span style="color:black">. </span><em><span style="color:black">P </span></em><span style="color:black">= tests positive.</span></span></p>

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

<ol style="list-style-type:lower-alpha">
	<li>
	<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Given that a woman develops the disease, what is the probability that she tests positive? Find </span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">P</span></em><span style="color:black">|</span><em><span style="color:black">B</span></em><span style="color:black">) = 1 &minus;</span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">N</span></em><span style="color:black">|</span><em><span style="color:black">B</span></em><span style="color:black">).</span></span></p>
	</li>
	<li>
	<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">What is the probability that a woman develops the disease and tests positive? </span></span></p>
	</li>
</ol>

<p style="margin-left:24px; text-align:justify"><span style="font-size:14px"><span style="color:black">Find </span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">B </span></em><span style="color:black">AND </span><em><span style="color:black">P</span></em><span style="color:black">) =</span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">P</span></em><span style="color:black">|</span><em><span style="color:black">B</span></em><span style="color:black">)</span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">B</span></em><span style="color:black">).</span></span></p>

<ol start="3" style="list-style-type:lower-alpha">
	<li>
	<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">What is the probability that a woman does not develop the disease? Find </span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">B&prime;</span></em><span style="color:black">) = 1 &minus; </span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">B</span></em><span style="color:black">).</span></span></p>
	</li>
	<li>
	<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">What is the probability that a woman tests positive for the disease? Find </span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">P</span></em><span style="color:black">) = 1 &minus; </span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">N</span></em><span style="color:black">).</span></span></p>
	</li>
</ol>

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

<p style="text-align:justify"><span style="font-size:14px"><strong>Solution 3.19</strong></span></p>

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

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">a. </span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">P</span></em><span style="color:black">|</span><em><span style="color:black">B</span></em><span style="color:black">) = 1 &minus; </span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">N</span></em><span style="color:black">|</span><em><span style="color:black">B</span></em><span style="color:black">) = 1 &minus; .02 = .98</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">b. </span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">B </span></em><span style="color:black">AND </span><em><span style="color:black">P</span></em><span style="color:black">) = </span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">P</span></em><span style="color:black">|</span><em><span style="color:black">B</span></em><span style="color:black">)</span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">B</span></em><span style="color:black">) = .98(.143) = .1401</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">c. </span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">B&#39;</span></em><span style="color:black">) = 1 &minus; </span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">B</span></em><span style="color:black">) = 1 &minus; .143 = .857</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">d. </span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">P</span></em><span style="color:black">) = 1 &minus; </span><em><span style="color:black">P</span></em><span style="color:black">(</span><em><span style="color:black">N</span></em><span style="color:black">) = 1 &minus; .85 = .15</span></span></p>

<p>&nbsp;</p>