prob-indep.xml

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<!-- This file is part of the book    							-->
<!--                                               				-->
<!--   Inquiries into Probability and Statistics - Math S–304	-->
<!--                                                			-->
<!-- Copyright (C) 2016  Thomas W. Judson      					-->

<chapter xml:id="prob-indep" xmlns:xi="http://www.w3.org/2001/XInclude">

	<title>Conditional Probability and Independent Events</title>

        <introduction>
        <p>The learning objectives for this chapter are:
            <ul>
                <li>To understand the basic ideas of independence.</li>

                <li>To understand and be able to use conditional probability.</li>

                <li>To understand and be able to apply Bayes' Theorem.</li>

            </ul></p>
        </introduction>

	<section xml:id="section-prob-indep-indep">
		<title>Independent Events</title>

        <exercise>
            <statement>
                <p>Suppose that I have two coins, a penny and a nickel. If I flip the two coins, what is the probability that the nickel comes up heads?  If I show you that the penny came up heads, does this change the probability of the nickel being heads?</p>
             </statement>
         </exercise>

         <p>This exercise is an example of <term>independent events</term>.  Given two events <m>A</m> and <m>B</m>, the events are independent if have knowledge of the outcome of one gives no additional knowledge of the outcome of the other.  If <m>A</m> and <m>B</m> are independent events, then
            <me>P(A \cap B) = P(A) P(B).</me>
        For three events to be independent, they must be <term>pairwise independent</term>.  That is, for <m>A_1, A_2, A_3</m> to be independent, it must be the case that
            <me>P(A_i \cap A_j) = P(A_i)P(A_j)</me>
        for <m>i \neq j</m>, where <m>i, j = 1, 2, 3</m>.</p>

        <exercise>
            <statement>
                <p>A fair coin is tossed twice.  
                	<md>
                		<mrow>A \amp =  \text{heads on the first toss}</mrow>
                		<mrow>B \amp = \text{heads on the second toss}</mrow>
                		<mrow>C \amp = \text{heads on both tosses}</mrow>
                	</md>
				Tell whether each pair of events below is independent or dependent.
                    <ol>
                        <li><m>A</m> and <m>B</m></li>

                        <li><m>A</m> and <m>C</m></li>

                        <li><m>B</m> and <m>C</m></li>

                    </ol></p>
            </statement>
        </exercise>

        <exercise>
        	<statement>
        		<p>An executive committee consisted of 10 members: 4 women and 6 men.  Three members were selected at random to be sent to a meeting in Hawaii.  A blindfolded woman drew 3 names of the 10 names from a hat.  Three women were chosen.  What is the probability of such luck?</p>
        	</statement>
        </exercise>

        <exercise>
            <statement>
                <p>Assume that the probability is <m>1/2</m> that a child will be born a boy. If a family has 4 children, what is the probability that they are all boys?</p>
             </statement>
         </exercise>

        <exercise>
            <statement>
                <p>In recent decades, home pregnancy tests have become quite popular. The tests have become very accurate, but occassionally these tests give wrong results. The test could come back negative while a woman is actually pregnant or postive if a woman is not pregant. Suppose a particular brand of home pregnancy test kits has a rate of 3 per cent of false negatives. That is, there is  probability of 0.03 that a woman could have a negative test and actually be pregnant. Suppose 150 women are tested and all the results come up negative. What is the probability that at least one of the women is actually pregnant?
                	<ol>

                		<li><p>What is the probability that one woman will accurately test as negative?</p></li>

                		<li><p>What is the probability that two women will accurately test as negative?</p></li>

                		<li><p>What is the probability that all 150 women will accurately test as negative?</p></li>

                		<li><p>What is the probability that there will be at least one of 150 women will have a false negative?</p></li>

                		<li><p>What are we assuming for this problem?</p></li>

                	</ol>
                </p>
             </statement>
             <hint>
             	<p><m>P(\text{at least one pregnancy}) = 1 - P(\text{no pregnancies}) = 1 - (0.97)^{150} \approx 0.98963</m></p>
             </hint>
         </exercise>

         <exercise>
            <statement>
                <p>What is the probability that two people in our class have the same birthday?</p>
             </statement>
             <hint>
             	<p>The probability that three people will have different birthdays is <m>365/365 \cdot 364/365 \cdot 363/365</m>.</p>
             </hint>
         </exercise>

         <exercise>
            <statement>
                <p>Suppose that a box contains 6 red balls, 4 white balls, and 5 blue balls.  If a ball is drawn at random, what is the probability that
                    <ol>
                        <li>The ball is white?</li>

                        <li>The ball is red?</li>

                        <li>The ball is blue?</li>

                        <li>The ball is not blue?</li>

                        <li>The ball is red or white?</li>

                    </ol></p>

                <p>If three balls are drawn at random, what is the probability that that they drawn in the order of red, white, and blue, if replacment is allowed?  What if the balls are not replace?</p>
            </statement>
        </exercise>

	</section>

	<section xml:id="section-prob-indep-conditional">
        <title>Conditional Probability</title>

        <exercise>
            <statement>
                <p>Suppose that I have two coins, a penny and a nickel. If I flip the two coins, what is the probability that the nickel comes up heads?  If I show you that the penny came up heads, does this change the probability of the nickel being heads?</p>
             </statement>
         </exercise>

         <p>This exercise is an example of <term>independent events</term>.  Given two events <m>A</m> and <m>B</m>, the events are independent if have knowledge of the outcome of one gives no additional knowledge of the outcome of the other.  If <m>A</m> and <m>B</m> are independent events, then
            <me>P(A \text{ and } B) = P(A) P(B).</me>
        For three events to be independent, they must be <term>pairwise independent</term>.  That is, for <m>A_1, A_2, A_3</m> to be independent, it must be the case that
            <me>P(A_i \cap A_j) = P(A_i)P(A_j)</me>
        for <m>i \neq j</m>, where <m>i, j = 1, 2, 3</m>.</p>

        <exercise>
            <statement>
                <p>You are playing matchmaker and set up your two friends Alice and Bob on a date.
                    <ol>
                        <li><p>What is the probability that the date will be successful?</p></li>

                        <li><p>Alice calls you right after the date and tells you that she is absolutely enamoured with Bob.  What is the probability that the date was successful?</p></li>

                    </ol></p>
             </statement>
        </exercise>

         <p>This exercise is an example of <term>conditional probability</term>.  Given two events <m>A</m> and <m>B</m>, 
            <me>P(A \text{ given } B) = P(A|B) = \frac{P(A \cap B)}{P(B)}</me>
        or equivalently
            <me>P(A \cap B)= P(B) P(A|B).</me>
        In other words, if event <m>B</m> is known to be true, then we may say something further about the probability of <m>A</m>.</p>

        <exercise>
            <statement>
                <p>For any three events <m>A</m>, <m>B</m>, and <m>C</m>, prove that 
                    <me>P(A \cap B \cap C) = P(A) P(B|A) P(C|A \cap B).</me></p>
             </statement>
         </exercise>

         <exercise>
            <statement>
                <p>Suppose that a box contains 6 red balls, 4 white balls, and 5 red balls.  If a ball is drawn at random, what is the probability that
                    <ol>
                        <li>The ball is white?</li>

                        <li>The ball is red?</li>

                        <li>The ball is blue?</li>

                        <li>The ball is not blue?</li>

                        <li>The ball is red or white?</li>

                    </ol></p>

                <p>If three balls are drawn at random, what is the probability that that they drawn in the order of red, white, and blue, if replacment is allowed?  What if the balls are not replace?</p>
            </statement>
        </exercise>

        <exercise>
            <statement>
                <p>A gumball machine contains 3 red gumballs and 2 green ones.  Kaylee really wants to have a red gumball as would her little brother Connor.  One gumball is purchased at random and <em>not replaced</em>.  Then a second gumball is purchased.  
                    <ol>
                        <li>What is the probability that both gumballs will be red?</li>

                        <li>What is the probability that both gumballs will be the same color?</li>

                    </ol></p>
            </statement>
            <hint>
                <p>A tree diagram might prove very helpful.</p>
            </hint>
        </exercise>


    </section>

	<section xml:id="section-prob-indep-bayes">
        <title>Bayes' Theorem</title>

        <example xml:id="example-breast-cancer">
            <p>Approximately 1<percent /> of women aged 40<ndash />50 have breast cancer. A woman with breast cancer has a 90<percent /> chance of a positive test from a mammogram, while a woman without has a 10<percent /> chance of a false positive result. What is the probability a woman has breast cancer given that she just had a positive test?
            <ol>
                <li>20 <percent /></li>

                <li>40 <percent /></li>

                <li>60 <percent /></li>

                <li>80 <percent /></li>

            </ol></p>
        </example>


        <theorem xml:id="theorem-prob-cond">
            <statement>
                <p>If an event <m>A</m> must result in one of the mutually exclusive events <m>A_1, A_2, \ldots, A_n</m>, then
            		<me>P(A) = P(A_1)P(A | A_1) + P(A_2)P(A | A_2) + \cdots + P(A_n)P(A | A_n).</me></p>
             </statement>
        </theorem>

        <exercise>
        	<statement>
        		<p>Prove Theorem<nbsp /><xref ref="theorem-prob-cond" /> for <m>n = 2</m>.</p>
            </statement>
        </exercise>


        <theorem xml:id="theorem-prob-bayes">
        	<title>Bayes' Theorem</title>
            <statement>
                <p>Suppose that we have mutually exclusive events <m>A_1, A_2, \ldots, A_n</m> and that <m>S = A_1 \cup A_2 \cup \cdots A_n</m>, where <m>S</m> is the sample space.  If <m>A</m> is any event, then
            		<me>P(A_k | A) = \frac{P(A_k)P(A_k| A)}{\sum_{i = 1}^n P(A_i)P(A_i| A)}.</me></p>
             </statement>
         </theorem>

         <p>The statement of this theorem is a bit confusing, so let's see how it works for <m>n = 2</m>.  Suppose that <m>S = A \cup A'</m>.  Then Bayes' Theorem says
            <me>P(B | A) = \frac{P(B \cap A)}{P(A)} = \frac{P(B \cap A)}{P(B \cap A) + P(B' \cap A)} = \frac{P(B \cap A)}{P(B)P(A|B) + P(B') P(A|B')}.</me></p>

        <exercise>
            <statement>
                <p>Prove Theorem<nbsp /><xref ref="theorem-prob-bayes" /> for <m>n = 2</m>.</p>
            </statement>
        </exercise>

        <exercise>
            <statement>
                <p>Use Bayes' Theorem to determine the probability a woman has breast cancer given that she just had a positive test in Example<nbsp /><xref ref="example-breast-cancer" />?</p>
            </statement>
            <hint>
                <p>Let <m>B</m> be the event that the woman has breast cancer and <m>A</m> the event of  a positive test and calculate <m>P(B|A)</m>.</p>
            </hint>
        </exercise>

        <exercise>
            <statement>
                <p>Nicole Brown was murdered at her home in Los Angeles on the night of June 12, 1994. The Prime suspect was her husband 0.J.Simpson, at the time a well-known celebrity famous both as a TV actor and as a retired professional football star. This murder led to one of the most heavily publicized murder trial in U.S. during the last century. The fact that the murder suspect had previously physically abused his wife played an important role in the trial. The famous defense lawyer Alan Dershowitz, a member of the team of lawyers defending the accused, tried to belittle the relevence of the fact by stating that only 0.1<percent /> of the men who physically abuse their wives actually end up murdering them.  Was the fact that O.J.Simpson had previously physically abused his wife irrelevant to the case?</p>
            </statement>
            <hint>
                <p>Let <m>E</m> be all the evidence, that Nicole Brown was murdered and was previously physically abused by her husband, and <m>G</m> be the event that O.J.Simpson is guilty. Calculate <m>P(G|E)</m>.</p>
            </hint>
        </exercise>



    </section>





		<!--
			\question
If $A$ is the set of students taking Math 128 at SFA and $B$ is the set of students taking FRE 435 (French Film) at SFA, describe in words what is meant by each of the following.
\begin{multicols}{3}
\begin{parts}

\part 
$P(A \cup B)$

\part 
$P(A \cap B)$

\part 
$1 - P(B)$

\end{parts}
\end{multicols}



\question 
\begin{parts}

\part 
Your sock drawer contains three red socks and two black socks.  A sock is chosen at random from the drawer and not replaced.  Then a second sock is chosen.  Draw a tree diagram for this experiment and find the probability that you do not choose a matching pair of socks.

\vfill

\part
Suppose that you replace the first sock, and then chose a second sock.  Draw a tree diagram for this experiment, and find the probability that you do not choose a matching pair of socks.

\vfill

\part
What is the probability in (a) and (b) that you chose a matching pair of socks?

\vfill

\end{parts}


\question
Three boxes contain a number of black and/or white balls.  
\begin{itemize}

\item
Box 1 contains 4 white balls and 1 black ball.

\item
Box 2 contains 2 white balls and 2 black balls.

\item
Box 3 contains 2 white balls and 0 black balls.

\end{itemize}
Draw a ball from Box 1 and place it in Box 2.  Then draw a ball from Box 2 and place it in Box 3.  Finally, draw a ball from Box 3.  What is the probability that the last ball drawn from Box 3 is white?

\vfill

\question
A drawer has three black socks and two red socks.  A sock is drawn at random and not replaced.  Then a second sock is drawn.  
\begin{parts}

\part
If the first sock was black, what is the probability that the second sock is red?

\vfill

\part
If the first sock was red, what is the probability that the second sock is red?

\vfill

\end{parts}

-->


</chapter>