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Math Help - [SOLVED] Binomial Distribution Problem

  1. #1
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    Acolman, Mexico
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    [SOLVED] Binomial Distribution Problem

    Hello, I am lost with the following problem.

    Consider two players, player A and player B, in a game that involves tossing a (not necessarily fair) coin with p = P(H) such that A gains one dollar when a head (H) occurs and B gains one dollar if a T (tail) comes up. Assume that player A needs n while player B needs m more dollars to win the game (both n and m are non-negative integers)

    Let A be the event that n heads occur before m tails. A occurs if and only if there are at least n heads in the fi…rst n +m- 1 trials. Explain why this reasoning is correct. Then use the binomial distribution to write an expression for P(A) = P_{n,m}.


    I know it is a very long and wordy problem, any hint will be appreciated, Thanks.
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  2. #2
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    Quote Originally Posted by akolman View Post
    Hello, I am lost with the following problem.

    Consider two players, player A and player B, in a game that involves tossing a (not necessarily fair) coin with p = P(H) such that A gains one dollar when a head (H) occurs and B gains one dollar if a T (tail) comes up. Assume that player A needs n while player B needs m more dollars to win the game (both n and m are non-negative integers)

    Let A be the event that n heads occur before m tails. A occurs if and only if there are at least n heads in the fi…rst n +m- 1 trials. Explain why this reasoning is correct. Then use the binomial distribution to write an expression for P(A) = P_{n,m}.


    I know it is a very long and wordy problem, any hint will be appreciated, Thanks.
    The question is not correct because there are contradictions between the statements 1 and 2 ; and statements 2 and 3, as labeled below:

    Statement 1: Assume that player A needs n while player B needs m "more" dollars to win the game (both n and m are non-negative integers)

    Statement 2: A occurs if and only if there are at least n heads in the fi…rst n +m- 1 trials.

    Statement 3: Then use the binomial distribution to write an expression for P(A) = P_{n,m}.

    In Statement 1, the word "more" should have been struck out.
    P(A) in Statement 3 should have been P(A)=\binom{n+m-1}{n}p_h^np_t^{m-1}

    -------------------
    Assuming A needs to win n dollars, and B, m dollars, A will occur, if and only if there are n heads in the first n+m-1 trials.

    Example: Say n=3 dollars, m=2 dollars. For A to win there are n+m-1 trials, which have n+m-1=4 possible sample spaces {HHH},{HHTH},{HTHH},{THHH}

    P(A)=\binom{3+2-1}{3}p_h^3p_t^{3+2-1-3}=\binom{4}{3}p_h^3p_t^{1}

    Note: This is not a fair question because it does not tell us who will go first in the game. There is a big difference as to who will go first because the first to go will have a much better probability of winning. This you can test yourself by considering the expected value of each player. A flaw is found in this question in which it expected the solution to align with binomial distribution, for for which after a string of n number of heads appeared, B would bother to complete the remaining number trials.
    Last edited by novice; October 24th 2009 at 08:48 AM.
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