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Math Help - probability problem about the probability that Team A will win this game

  1. #1
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    Smile probability problem about the probability that Team A will win this game

    Hi,

    A very simple prob question. Hope sb could help:

    Two basketball teams: team A makes it shots 48% of the time and team B makes its shots 52% of the time. The outcome of the each shot is independent of all other shots and that the team who makes the most shots during the game wins. If each team takes 100 shots in a game, what is the probability that team A wins this game?

    Please explain why. Thanks in advance.
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by notatall View Post
    Hi,

    A very simple prob question. Hope sb could help:

    Two basketball teams: team A makes it shots 48% of the time and team B makes its shots 52% of the time. The outcome of the each shot is independent of all other shots and that the team who makes the most shots during the game wins. If each team takes 100 shots in a game, what is the probability that team A wins this game?

    Please explain why. Thanks in advance.
    Let X be the random variable that represents the number of shots made by team A. Then X has a binomial distribution, in particular X\sim\mathcal{B}(100,.48).

    What we're after here is P(X\geq51). But that's the same as saying 1-P(X\leq 50).

    Now recall that if X\sim\mathcal{B}(n,p) then \displaystyle P(X\leq x) = \sum\limits_{k=0}^x {n\choose k}p^k(1-p)^{n-k}

    Now, use this formula to get the answer.

    Does this make sense? Can you finish this problem?
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  3. #3
    MHF Contributor chisigma's Avatar
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    For both teams the probability of 'k over n' shots is...

    \displaystyle P_{A} (k)= \binom{n}{k} p_{a}^{k}\ (1-p_{a})^{n-k}

    \displaystyle P_{B} (k)= \binom{n}{k} p_{b}^{k}\ (1-p_{b})^{n-k}

    If we set n=100, p_{a}=.48, p_{b}=.52 the probability that A wins is...

    \displaystyle P_{A wins}= \sum_{i=1}^{100} P_{A} (i)\ \sum_{k=0}^{i-1} P_{B} (k)

    Kind regards

    \chi \sigma
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  4. #4
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    Yes. It makes perfect sense to me. Thanks a lot.
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  5. #5
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    Thanks I will compare your solution with Chris's method and let you know if they agree.
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  6. #6
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    Hi Chris,

    Thanks for your answer. But I think the way you compute the probability that team A wins has problem. Since the team which makes the most shots wins, you have to consider the number of shots that team B makes when team A makes certain number of shots. Once team B makes less shots than team A, team A wins the game. So I guess chisigma's answer makes more sense. But thanks anyway
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  7. #7
    MHF Contributor matheagle's Avatar
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    I was trying to clean up the sum, but it's a mess.

    \sum_{i=1}^{100}{100\choose i} (.48)^i(.52)^{100-i} \sum_{k=0}^{i-1}{100\choose k}(.52)^k(.48)^{100-k}
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  8. #8
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    Hi,

    I have done this problem in the way you suggested and the probability that team A wins is 0.2620. I am wondering if there is an easy way to get the answer since

    \displaystyle P_{A wins}= \sum_{i=1}^{100} P_{A} (i)\ \sum_{k=0}^{i-1} P_{B} (k)

    is hard to compute. I used Matlab to get the answer.

    Thanks, it really helps
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  9. #9
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    Thanks I think Chrisigma is correct and I have to use Matlab to actually get the final answer.
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  10. #10
    MHF Contributor chisigma's Avatar
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    The computation of...

    \displaystyle P_{Awins} = \sum_{i=1}^{n} \binom{n}{i}\ p_{a}^{i}\ (1-p_{a})^{n-i}\ \sum_{k=0}^{i-1} \binom{n}{k}\ p_{b}^{k}\ (1-p_{b})^{n-k} (1)

    ... requires only a minimal knowledge of computer programming. Setting p_{a}= .48 and p_{b}= .52 I obtained...

    n=10 \rightarrow P_{Awins} = .342510814481

    n=100 \rightarrow P_{Awins} = .261961725783

    n=1000 \rightarrow P_{Awins} =.0349809721915

    If necessary greater value of n can be examined...

    Kind regards

    \chi \sigma
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  11. #11
    MHF Contributor matheagle's Avatar
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    fine, I have less than mininal knowledge of computing, lol
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