prob-intro.xml

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<!--   Inquiries into Probability and Statistics - Math S–304	-->
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<chapter xml:id="prob-intro" xmlns:xi="http://www.w3.org/2001/XInclude">

	<title>Introduction to Probability</title>

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

                <li>To understand and be able to use sample spaces and events.</li>

                <li>To understand and be able to apply the probability of equally likely events.</li>

                <li>To understand and be able to use the axioms of probability.</li>

            </ul></p>



    </introduction>

    <section xml:id="section-prob-intro">
        <title>Probability and Random Processes</title>

        <p>Although probability has its origins in gambling, the grading of evidence and testimony in medieval law, and estimating risk for maritime insurance premiums, applications of probability have become far more reaching today.  Probability can be used to describe traffic on the Internet, the best course of a medical treatment, the energy states of subatomic particles, and much more.  A knowledge of probability is essential when we study statistics.  In addition, probability is an active area of mathematical research.</p>

        <p>If we consider a simple random phenomenon, such as rolling a pair of dice or flipping a coin, we do not know what will happen in advance.  However, we can make predictions.  If we flip a fair coin, it will come up heads about half the time.  If we roll a pair of dice, we can expect to roll a 7 a certain precentage of the time.  The description of either one of these to phenomena consists of the following:
            <ul>
                <li><p>A list of all possible outcomes.</p></li>

                <li><p>A probability for each outcome.</p></li>
            </ul>
        These two items provide a basis for all probability models.</p>

        <p>Tossing a coin or rolling a pair of dice are examples of <term>random processes</term>. In other words, we have no knowledge of how the coin or the pair of dice will land prior to each toss or roll.  The possible <term>outcomes</term> for tossing a coin are heads or tails, and the outcomes for rolling a pair of dice are getting a two through twleve.  We define the <term>probability</term> of an outcome of a random process to be the proportion of times that the outcome would occur if we could do an infinite number of observations.  Thus, the probability of obtaining a heads on any particular toss of a coin is <m>1/2</m>, provided the coin is fair.  Usually, we assign numbers between 0 and 1 for probabilites, although we will sometimes using percentages.</p>

        <exercise>
            <statement>
                <p>Assume that we have a pair of dice and that it is equally likely that a one through six will occur on each die for any particular roll.
                    <ol>
                        <li><p>What is the probability of rolling a 3 if you roll only one die?</p></li>

                        <li><p>What is the probability of rolling a 3 if you roll a pair of dice?</p></li>

                        <li><p>What is the probability of rolling a 7 if you roll a pair of dice?</p></li>

                        <li><p>What is the probability of rolling a 7 if you roll only one die?</p></li>

                        <li><p>What is the probability of not rolling a 3 if you roll only one die?</p></li>

                        <li><p>What is the probability of not rolling a 3 if you roll a pair of dice?</p></li>

                        <li><p>What is the probability of not rolling a 7 if you roll a pair of dice?</p></li>

                        <li><p>What is the probability of not rolling a 7 if you roll only one die?</p></li>

                        <li><p>What is the probability of rolling a 3 if you roll a pair of dice and one of the dice is a 1?</p></li>

                        <li><p>What is the probability of rolling a 3 if you roll a pair of dice and one of the dice is not a 1?</p></li>

                     </ol></p>
             </statement>
             <hint>
                <p>Decide what rolling a 3 means when you roll a pair of dice.</p>
            </hint>
         </exercise>

         

         <p>To understand why probability works, we must understand the Law of Large Numbers.  Suppose the we toss a fair coin <m>n</m> times.  We will denote the proportion of heads that occur by <m>\hat{p}_n</m>.  If we toss a coin 100 times and it comes up heads 52 times, then <m>\hat{p}_{100} = 0.52</m>.  The Law of Large Numbers says that <m>\hat{p}_n \to p</m> as <m>n \to \infty</m>.  Although <m>\hat{p}_n</m> may deviate from <m>p</m> even as <m>n</m> grows large, these deviations become smaller and smaller.  The actual Law of Large Numbers needs to be stated a bit more carefully, and can be proven using advanced calculus, but our discussion here catches the essence.<fn>See <url href="https://en.wikipedia.org/wiki/Law_of_large_numbers" />.</fn></p>

         <exercise>
            <statement>
                <p>Find several examples of random processes and processes that are not random.  Are there examples of processes that are not random but can be modeled by random processes?</p>
             </statement>
         </exercise>

        <p>We can sometimes simulate a probability distribution experimentally.</p>

        <exercise>
            <statement>
                <p>Suppose the chance of rain for each of the next three days is 50<percent />.  You are planning an outdoor activity and want to estimate the chance that it will not rain on at least one of these three days.
                    <ol>

                        <li><p>What is the probability that it will not rain at least one of these three days.</p></li>

                        <li><p>Use a coin to simulate the weather each day.  Make 20 sets of three days and record your results.</p></li>

                        <li><p>Make a histogram of the frequency of 0, 1, 2, 3, days with rain.</p></li>

                        <li><p>What is the experimental probability that it will not rain at least one of the three days?</p></li>

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

    </section>

    <section xml:id="section-prob-finite">
    	<title>Sample Spaces and Events</title>

        <p>A <term>sample space</term> is the set of all possible outcomes for a random process.  We will usually denote the sample space by <m>S</m>.  An <term>event</term> is any possible outcome.  For example, we can have the following events for rolling a pair of dice,
            <md>
                <mrow>A \amp = \{ \text{rolling a 7}\}</mrow>
                <mrow>B \amp = \{ \text{rolling an even number}\}</mrow>
                <mrow>C \amp = \{ \text{rolling a 17}\}.</mrow>
            </md>
        We will designate the probability of an event <m>A</m> by <m>P(A)</m>.</p>

        <exercise>
            <statement>
                <p>Find <m>P(A)</m>, <m>P(B)</m>, and <m>P(C)</m>.</p>
             </statement>
         </exercise>


        <p>Sample spaces may either be finite or infinite.  For example, if we toss a single die, there are only 6 possible outcomes.  We must roll a 1, 2, 3, 4, 5, or 6.  Of course, there are many more possible events for rolling a single die.  We could have the event of rolling an even number or the event of rolling a 2 or a 5.  However, there are still only a finite number of events.</p>

        <exercise>
            <statement>
                <p>Suppose we roll a single 6-sided die.  Suppose  <m>A = \{ \text{rolling a 3} \}</m> and <m>B = \{\text{rolling an even number}\}.</m>
                    <ol>

                        <li><p>What is the event <m>A \cup B</m>?</p></li>

                        <li><p>What is the event <m>A \cap B</m>?</p></li>

                        <li><p>List all possible events for rolling a single die.</p></li>

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

         <paragraphs>
            <title>Probability with Equally Likely Outcomes</title>

            <p>Let <m>S</m> be a sample space with a finite number of outcomes.  If all outcomes in <m>S</m> are equally likely, then the probability of an event <m>A</m> is
                <me>P(A) = \frac{\text{number of outcomes in }A}{\text{number of outcomes in }S}.</me></p>
        </paragraphs>

        <exercise>
            <statement>
                <p>Which of the following are equally likely events?
                    <ol>

                        <li><p>You go to MATH S<ndash />304.  You are on time or late.</p></li>

                        <li><p>You guess on a true-false question.  Your answer is right or wrong.</p></li>

                        <li><p>You shot a basketball free throw.  You either make or miss the shot.</p></li>

                        <li><p>A baby is born.  It is either a boy or a girl.</p></li>

                        <li><p>You catch a flight from Boston to Boise.  The plane is on time or late.</p></li>

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

         <exercise>
            <statement>
                <p>You roll a 6-sided die six times.  Which is more likely?
                    <ol>

                        <li><p>1 2 3 4 5 6</p></li>

                        <li><p>1 1 1 1 1 1</p></li>

                        <li><p>Both events are equally likely.</p></li>

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

        <exercise>
            <statement>
                <p>Jane, one of your students, says, <q>Out of Canada, Mexico, and the U.S., the probability of being born in Canada is 1/3, since there are three countries.</q>  Is Jane correct?  If not, what would you tell her?</p>
             </statement>
         </exercise>

        <exercise>
            <statement>
                <p>An experiment consists of selecting the last digit of a telephone number.  Assume that each of the ten digits is equally likely to appear as the last digit.
                    <ol>

                        <li><p>What is the sample space?</p></li>

                        <li><p>What is the event consisting of outcomes that the digit is less than 5?</p></li>

                        <li><p>What is the event consisting of outcomes where the digit is odd?</p></li>

                        <li><p>What is the event consisting of outcomes that this digit is not 2?</p></li>

                        <li><p>Find the probability of each of the events in (b) through (d).</p></li>

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

        <exercise>
            <statement>
                <p>A box contains three red balls, five black balls, and four white balls.  Suppose that one ball is drawn at random.  Find the probability of each of the following events.
                    <ol>

                        <li><p>A black ball is drawn.</p></li>

                        <li><p>A black or a white ball is drawn.</p></li>

                        <li><p>Neither a red ball or a white ball is drawn.</p></li>

                        <li><p>A red ball is not drawn.</p></li>

                        <li><p>A black ball and a white ball are drawn.</p></li>

                        <li><p>A black ball or a red ball or a white ball is drawn.</p></li>

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

        <exercise>
            <statement>
                <p>A pair of fair dice is rolled and we observe the sum of the two dice.
                    <ol>

                        <li><p>What is the sample space?</p></li>

                        <li><p>What is the probability of rolling a 7?</p></li>

                        <li><p>What is the probability of rolling a 7 or an 11?</p></li>

                        <li><p>What is the probability of rolling a 2, 3, or 12?</p></li>

                        <li><p>What is the probability of rolling a 4, 5, 6, 8, 9, or 10?</p></li>

                        <li><p>What is the probability of rolling a 1?</p></li>

                        <li><p>What is the probability of rolling a sum less than 6?</p></li>

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

         <p>So far all of our sample spaces have been finite.  It is possible to have an infinite sample space.</p>

        <exercise>
            <statement>
                <p>Suppose that an object is dropped at random on the area shown below.
                    <ol>

                        <li><p>What is the sample space?</p></li>

                        <li><p>What is the probability that the object will land in the circle?</p></li>

                        <li><p>What is different about this exercise?</p></li>

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

        <figure xml:id="figure-prob-intro-1">

            <image width="80%" xml:id="prob-intro-1">
                <latex-image-code><![CDATA[\begin{tikzpicture}[scale=1.5]
                        \draw (0,0) -- (0, 2) -- (3, 2) -- (3,0) -- (0, 0);
                        \draw (1,1) -- (2,1);
                        \draw (1.5,1) circle (0.5);
                        \node [above] at (1.5, 2) {3 ft};
                        \node [above] at (1.5, 1) {1 ft};
                        \node [left] at (0, 1) {2 ft};
                        \end{tikzpicture}]]>
                </latex-image-code>                
            </image>
        </figure>

        <exercise>
            <statement>
                <p>Suppose that we spin the following spinner with the first spin giving the numerator of a fraction and the second spin giving the denominator of a fraction.  What is the probability that the fraction will be greater than <m>1/2</m>?</p>
             </statement>
        </exercise>


        <figure xml:id="figure-prob-intro-2">

            <image width="80%" xml:id="prob-intro-2">
                <latex-image-code><![CDATA[\begin{tikzpicture}[scale=0.9]
                    \draw (0,0) circle (3);
                    \filldraw[fill=green!20!white, draw=black]
                    (0,0) -- (0,3) arc (90:150:3) -- cycle;
                    \filldraw[fill=red!20!white, draw=black]
                    (0,0) -- (150:3) arc (150:210:3) -- cycle;
                    \filldraw[fill=blue!20!white, draw=black]
                    (0,0) -- (210:3) arc (210:270:3) -- cycle;
                    \filldraw[fill=yellow!20!white, draw=black]
                    (0,0) -- (270:3) arc (270:330:3) -- cycle;
                    \filldraw[fill=cyan!20!white, draw=black]
                    (0,0) -- (30:3) arc (30:90:3) -- cycle;
                    \node at (0: 1.5) {1};
                    \node at (60: 1.5) {2};
                    \node at (120: 1.5) {3};
                    \node at (180: 1.5) {4};
                    \node at (240: 1.5) {5};
                    \node at (300: 1.5) {6};
                    \draw[->,black,very thick] (120:1) -- (300:1);
                    \end{tikzpicture}]]>
                </latex-image-code>                
            </image>
        </figure>



    </section>

    <section xml:id="section-prob-rules">
        <title>Probability Rules</title>


 

        <subsection>
            <title>Probability Axioms</title>

            <p>We have the following rules that govern probability.
                <ol>
                    <li><p>For any event <m>A</m>, <m>P(A) \geq 0</m>.</p></li>

                    <li><p><m>P(S) = 1</m></p></li>

                    <li><p>We say two events <m>A</m> and <m>B</m> are <term>disjoint</term> if <m>A \cap B = \emptyset</m>.  In this case,
                        <me>P(A \text{ or } B) = P(A) + P(B).</me></p></li>

                </ol>
            We sometimes say that disjoint events are <term>mutually exclusive</term>.</p>
        </subsection>

        <exercise>
            <statement>
                <p>Suppose that you roll a pair of dice.  List at least five pairs of events that are disjoint.</p>
             </statement>
         </exercise>

        <exercise>
            <statement>
                <p>Assume that we have a pair of dice and that it is equally likely that a one through six will occur on each die for any particular roll.
                    <ol>
                        <li><p>What is the probability of rolling a 3 or a 5 if you roll only one die?</p></li>

                        <li><p>What is the probability of rolling a 3 or a 5 if you roll a pair of dice?</p></li>

                        <li><p>What is the probability of rolling a 3 or a 7 if you roll a pair of dice?</p></li>

                        <li><p>What is the probability of rolling a 3 or a 7 if you roll only one die?</p></li>

                        <li><p>What is the probability of not rolling a 3 or an odd number if you roll only one die?</p></li>

                        <li><p>What is the probability of not rolling a 3 or an odd number if you roll a pair of dice?</p></li>

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

        <p>To understand the structure of any proposed mathematical object, we should engage in formulating and proving theorems.  Suppose that we have a sample space <m>S</m>.  If <m>A</m> and <m>B</m> are events in <m>S</m>, prove each of the following theorems.</p>

        <exercise>
            <statement>
                <p>If <m>A \subseteq B</m>, then <m>P(A) \leq P(B)</m>.</p>
            </statement>
        </exercise>

        <exercise>
            <statement>
                <p>If <m>A \subseteq B</m>, then <m>P(B) - P(A) = P(B - A)</m>.</p>
            </statement>
        </exercise>

        <exercise>
            <statement>
                <p>For every event <m>A</m>, <m>0 \leq P(A) \leq 1</m>.</p>
            </statement>
        </exercise>

        <exercise>
            <statement>
                <p><m>P(\emptyset) = 0</m>.</p>
            </statement>
        </exercise>

        <exercise>
            <statement>
                <p>If <m>A'</m> is the complement of an event <m>A</m>, then
                    <me>P(A') = 1 - P(A).</me></p>
            </statement>
        </exercise>

        <exercise>
            <statement>
                <p>For any two events <m>A</m> and <m>B</m> (not necessarily disjoint),
                    <me>P(A \cup B ) = P(A) + P(B) - P(A \cap B).</me></p>
             </statement>
         </exercise>

        <exercise>
            <statement>
                <p>For any two events <m>A</m> and <m>B</m>,
                    <me>P(A) = P(A \cap B) + P(A \cap B').</me></p>
             </statement>
        </exercise>

        <p>Several of these axioms and theorems can be generalized to <m>n</m> events. For example, if <m>A_1, A_2, \ldots, A_n</m> are disjoint sets, then
            <m>P(A_1 \cup A_2 \cup \cdots \cup A_n) = P(A_1) + P(A_2) + \cdots + P(A_n).</m></p>

                <exercise>
            <statement>
                <p>Suppose that you flip a coin once.  Name two events, $A$ and $B$, that are mutually exclusive.</p>
             </statement>
         </exercise>

        <exercise>
            <statement>
                <p>The probability that a person has black hair is 22\%, and the probability that a person has brown eyes is 72\%.  Nick, who is one of your students, says that the probability that a person has either black hair or brown eyes is 94\%.  Is Nick correct?  If not, what would you say to the student?</p>
             </statement>
         </exercise>

        <exercise>
            <statement>
                <p>At a certain hospital, 40 patients have lung cancer, 30 patients smoke, and 25 have lung cancer and smoke.  Suppose the hospital has 200 patients.  If a patient chosen at random is known to smoke, what is the probability that the patient has lung cancer?</p>
             </statement>
         </exercise>

     </section>

</chapter>