Conditional Probability 14 - Arizona State University garcia/3. probability trees to compute conditional

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  • 14.3Conditional Probability and Intersections of EventsObjectives1. Understand how to compute conditional probability.2. Calculate the probability of the intersection of two events.3. Use probability trees to compute conditional probabilities.4. Understand the difference between dependent and independent events.

    You know how to compute the probability of complements and unions of events. We willnow show you how to find the probability of intersections of events, but first you need tounderstand how the occurrence of one event can affect the probability of another event.

    Conditional ProbabilitySuppose that you and your friend Marcus cannot agree as to which video to rent and youdecide to settle the matter by rolling a pair of dice. You will each pick a number and thenroll the dice. The person whose total showing on the dice comes up first gets to pick the

    KEY POINT

    Conditional probability takesinto account that one eventoccurring may change theprobability of a second event.

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  • CHAPTER 14 y Probability682

    video. With your knowledge of probability, you know that the number you should pick is 7because it has the highest probability of appearing, namely .

    To illustrate the idea of conditional probability, lets now change this situation slightly.Assume that your friend Janelle will roll the dice before you and Marcus pick your num-bers. You are not allowed to see the dice, but Janelle will tell you something about the diceand then you will choose your number before looking at the dice.

    Suppose that Janelle tells you that the total showing is an even number. Would you stillchoose a 7? Of course not, because once you know the condition that the total is even, younow know that the probability of having a 7 is 0. A good way to think of this is that onceyou know that the total is even, you must exclude all pairs from the sample space such as(1, 4), (5, 6), and (4, 3) that give odd totals.

    In a similar way, suppose that you draw a card from a standard 52-card deck, put thatcard in your pocket, and then draw a second card. What is the probability that the secondcard is a king? How you answer this question depends on knowing what card is in yourpocket. If the card in your pocket is a king, then there are three kings remaining in the

    51 cards that are left, so the probability is . If the card in your pocket is not a king, then3

    51

    16

    the probability of the second card being a king is . Why? This discussion leads us to the

    formal definition of conditional probability.

    D E F I N I T I O N When we compute the probability of event F assuming that the event Ehas already occurred, we call this the conditional probability of F, given E. We denotethis probability as P(F E ). We read P(F E ) as the probability of F given that E hasoccurred, or in a quicker way, the probability of F given E.

    Do not let this new notation intimidate you. The notation P(F E ) simply means that you aregoing to compute a probability knowing that something else has already happened. Forexample, in our earlier discussion of Janelle, we said, The probability of having a total of 7knowing that the total is even is 0. We will restate this several times, each time increasingour use of symbols. So, we could say instead,

    If we now represent the event total is 7 by F and total is even by E, then we could writeour original statement as

    Similarly, lets return to the example of drawing two cards, and let A represent the eventthat we draw a king on our first card and put it in our pocket and, let B represent the event thatwe draw a king on our second card. Then we could write, The probability that we draw a

    second king given that the first card was a king is , as

    351

    P(B A) .

    First card was a king.

    Second card is a king.

    3

    51

    P(F E ) 0.

    Total is even.

    Total is 7.

    P(having a total of 7 given that the total is even) 0;

    or,

    P(having a total of 7 the total is even) 0.

    4

    51

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  • 14.3 y Conditional Probability and Intersections of Events

    The Venn diagram in Figure 14.12will help you remember how to com-pute conditional probability.

    We drew E with a heavy line inFigure 14.12 to emphasize that whenwe assume that E has occurred, we canthen think of the outcomes outside of Eas being discarded from the discussion.In computing conditional probability,you will find it useful to consider thesample space to be E and the event asbeing E F, rather than F. We will firststate a special rule for computing con-ditional probability when the outcomesare equally likely (all have the sameprobability of occurring). We will statethe more general conditional proba-bility rule later.

    S P E C I A L R U L E F O R C O M P U T I N G P ( F E ) B Y C O U N T I N G If E and F are

    events in a sample space with equally likely outcomes, then .

    EXAMPLE 1 Computing Conditional Probability by CountingAssume that we roll two dice and the total showing is greater than nine. What is the prob-ability that the total is odd?

    SOLUTION: This sample space has 36 equally likely outcomes. We will let G be the eventwe roll a total greater than nine and let O be the event the total is odd. Therefore,

    G = {(4, 6), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6)}.

    The set O consists of all pairs that give an odd total. Figure 14.13 shows you how to use thespecial rule for computing conditional probability to find P(O G).

    P(F E ) =n(E F )

    n(E )

    S

    E F

    We want the probability of theintersection outcomes, knowingthat were interested in onlyoutcomes for set E.

    FIGURE 14.12 To compute the probability of Fgiven E, we compare the outcomes in E F with theoutcomes in E.

    STotal greater

    than 9Odd total

    O G

    G O(5, 6)

    (6, 5)

    (5, 5)

    (6, 4)

    (4, 6)

    (6, 6)

    FIGURE 14.13 To compute P(O G), we comparethe number of outcomes in O G with the numberof outcomes in G.

    Therefore,

    Notice how the probability of rolling an odd total has changed from to when we know

    that the total showing is greater than nine.

    Now try Exercises 5 to 22. ]

    1

    3

    1

    2

    P(O G) =n(O G)

    n(G)=

    2

    6=

    1

    3.

    683

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  • CHAPTER 14 y Probability684

    PROBLEM SOLVING

    The Order PrincipleAs we have emphasized often, the order in which we do things is important in mathematics.In Example 1, P(O G) does not mean the same thing as P(G O). To understand the differ-ence, state what P(G O) represents and then compute P(G O).

    It is important to remember that the special rule for computing conditional probabilityonly works when the outcomes in the sample space are equally likely. Sometimes you maybe solving a problem where that is not the case; or, you may have a situation where it is notpossible to count the outcomes. In such cases, we need a rule for computing P(F E ) that isbased on probability rather than counting.

    G E N E R A L R U L E F O R C O M P U T I N G P ( F E ) If E and F are events in a sample

    space, then .

    We still can use Figure 14.12 to remember this rule; however, now instead of compar-ing the number of outcomes in E F with the number of outcomes in E, we compare theprobability of E F with the probability of E.

    EXAMPLE 2 Using the General Rule for Computing Conditional Probability

    The state bureau of labor statistics conducted a survey of college graduates comparingstarting salaries to majors. The survey results are listed in Table 14.6.

    P(F E ) =P (E F )

    P (E )

    If we select a graduate who was offered between $40,001 and $45,000, what is theprobability that the student has a degree in the health fields?

    SOLUTION: Each entry in Table 14.6 is the probability of an event. For example, the 8%that we highlighted is the probability of selecting a graduate in technology who earnsbetween $40,001 and $45,000, inclusive. The 14% that we highlighted tells us the proba-bility of selecting a graduate who majored in the health fields.

    Let R be the event the graduate received a starting salary between $40,001 and$45,000 and H be the event the student has a degree in the health fields. It is important indoing this problem that you identify clearly what you are given and what you must find. Weare given R and must find the probability of H, so we want to find P(H R), not P(R H).

    Because we want the probability of H given R, we can, in effect, ignore all the out-comes that do not correspond to a starting salary of $40,001 to $45,000. We darken thecolumns we want to ignore in Table 14.7.

    Major$30,000

    and Below$30,001

    to $35,000$35,001

    to $40,000$40,001

    to $45,000Above

    $45,000Totals

    (%)

    Liberal arts 6* 10 9 1 1 27

    Science 2 4 10 2 2 20

    Social sciences 3 6 7 1 1 18

    Health fields 1 1 8 3 1 14

    Technology 0 2 7 8 4 21

    Totals (%) 12 23 41 15 9 100

    TABLE 14.6 Survey comparing starting salaries to major in college.

    *These numbers are percentages.

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  • 14.3 y Conditional Probability and Intersections of Events

    Major$30,000 or Below

    $30,001 to $35,000

    $35,001 to $40,000

    $40,001 to $45,000

    Above $45,000

    Totals (%)

    Liberal arts 6 10 9 1 1 27

    Science 2 4 10 2 2 20

    Social sciences 3 6 7 1 1 18

    Health fields 1 1 8 3 1 14

    Technology 0 2 7 8 4 21

    Totals (%) 12 23 41 15 9 100

    TABLE 14.7 The columns we want to ignore are darkened