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Math Help - Negative binomial problem?

  1. #1
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    Negative binomial problem?

    Suppose that two teams are playing a series of games, each of which is independently won by team A with probability p and by team B with probability 1-p. The winner of the series is the first team to win i games.

    If i=4, find the probability that a total of 7 games are played. Also show that this probability is maximized when p=0.5


    I thought that this is a pretty straight forward negative binomial problem, but I don't seem to be getting the correct answer. Here's my work:

    pr(X=7)= {7-1}\choose{4-1} (0.5^{4})(0.5^{3})=0.15625

    The problem is that the probability of 7 games being played doesn't seem to be maximized when p=0.5.

    For example, if I set p=0.6 then I get:

    pr(X=7)= {7-1}\choose{4-1} (0.6^{4})(0.4^{3})=0.16589

    I assume that I'm reading the question wrong, using the wrong distribution, committing a math error, or some combination of the three. Any help is appreciated.
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  2. #2
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    Re: Negative binomial problem?

    hello there
    actually every function of the form
    m*p^4*(1-p)^3
    where m is an integer has amaximum at
    x=4/7
    now in ur case m is abolinomial coefficient so it must be an integer
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  3. #3
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    Re: Negative binomial problem?

    Thank you for replying, but I don't understand what you mean.

    Is this not a negative binomial problem? Is my math wrong?
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  4. #4
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    Re: Negative binomial problem?

    I figured out the problem. I wasn't taking into account that either team can win the series. Therefore it's:

    {6}\choose{3} p^{4}(1-p)^{3}+ {6}\choose{3} p^{3}(1-p)^{4}

    Where
    p is the probability of team A winning a game

    and

    (1-p) is the probability of team B winning a game
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