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A Hypergeometric Distribution Problem

  1. Jan 25, 2008 #1
    1. The problem statement, all variables and given/known data
    Balls are randomly withdrawn, one at a time without replacement, from an urn that initially has N white and M black balls. Find the probability that n white balls are drawn before m black balls, n <= N, m <= M.

    2. Relevant equations
    A hypergeometric random variable with parameters n, N, m represents the number of white balls selected when n balls are randomly chosen from an urn that contains N balls, of which m are white. Its probability mass function is given by

    p(i) = C(m, i) * C(N - m, n - i) / C(N, n)

    3. The attempt at a solution
    How can n white balls be withdrawn before m black balls? It could happen in the following ways:

    - I draw out n balls and the all happen to be white,
    - I draw out n + 1 balls and all but one is black,
    - I draw out n + 2 balls and all but two are black, ...,
    - I draw out n + m - 1 balls and all but m - 1 are black

    The probability sought is the sum of the probabilities of each of the events described above right? If I draw k balls, n <= k < n + m, then the probability that n are white is C(N, n) * C(M, k - n) / C(N + M, k).

    Here's the books answer, which I disagree with:

    A total of n white balls will be withdrawn before a total of m black balls if and only if there are at least n white balls in the first n + m - 1 withdrawls. With X equal to the nuber of balls amount the first n + m - 1 withdrawn balls, then X is a hypergeometric random variable and

    [tex]P\{X \ge n\} = \sum_{i=n}^{n+m-1} \frac{\binom{N}{i} \binom{M}{n+m-1-i}}{\binom{N+M}{n+m-1}}[/tex]

    I don't understand why the answer is P{X >= n} since this is the probability that at least n white balls are selected from n + m - 1 withdrawn balls.
  2. jcsd
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