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Math Help - Poisson Approximation Problem

  1. #1
    Yan
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    Poisson Approximation Problem

    A box contains 1000 balls, of which 2 are black and the rest are white.

    a) which of the following is most likely to occur in 1000 draws with replacement from the box?

    -fewer than 2 black balls
    -exactly 2 black balls
    -more than 2 black balls

    b) if two series of 1000 draws are made at random from this box, what, approximately, is the chance that they produce the same number of black balls?
    Last edited by CaptainBlack; October 22nd 2008 at 01:14 AM. Reason: Doing what the OP should have done
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Yan View Post
    A box contains 1000 balls, of which 2 are black and the rest are white.

    a) which of the following is most likely to occur in 1000 draws with replacement from the box?

    b) if two series of 1000 draws are made at random from this box, what, approximately, is the chance that they produce the same number of black balls?
    Question a) is incomplete.

    The number of black balls in a sample of 1000 draws with replacement has a binomial distribution B(1000,1/500) , which may be approximated by a Poisson distribution P(2) .

    The answer to b) is the sum:

    p\approx \sum_{i=0}^{\infty} [p(i,2)]^2

    where p(i,\lambda) denotes the Poisson probability of outcome i, with mean \lambda.

    The sum will for all practical purposes have converged by the time you reach i=5 or so.

    CB
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  3. #3
    Grand Panjandrum
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    Original question updated for the poster.

    CB
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