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Probabilities: Estimating the probability of overlapping

  1. Jan 16, 2007 #1
    Hi all
    I recently ran into this problem:
    I have two bins. Each bin contains N numbered balls, from 1 to N.
    For both the bins, the probability of the ball numbered k to be
    selected equals to P(ball-k-selected)=k/SUM(1:N) (in other versions
    this can be any given probability distribution)

    Simple case:
    Having selected 1 ball from the first bin, and 1 ball from
    the second bin, i want to find the probability of the ball
    having the same number.

    If i am correct, the probability for this is SUM(k=1:N) (P(ball-k-selected)^2).

    Complex case:
    Having selected m balls from the first bin, and m balls from
    the second bin, i need the probability of holding at least
    one pair of balls with the same number at the end of the
    experiment.

    Assumption: The selection is without replacement. However, for
    simplicity we can assume that the probability of a ball to be selected
    remains stable during the experiment, and is given by
    P(ball-k-selected)=k/SUM(1:N)

    If something is not clear, please let me know.

    Thanks in advance for any contributions!
     
  2. jcsd
  3. Jan 17, 2007 #2

    EnumaElish

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    Science Advisor
    Homework Helper

    Denote integers with lowercase and sets with UPPERCASE. The set of balls in bin i is Ni.

    There are C(n, m) = n!/(m!(n-m)!) combinations (subsets) that you can draw from either bin.

    Let S1 be a subset of m balls from the 1st bin. Let S2 be the corresponding subset from the 2nd bin. How many subsets of m balls can you form out of the set of the remaining balls in the 2nd bin, S2' = N2\S2? The answer is s2' = |S2'| = C(n-m, m). That's the answer to the question, "for a given S1, how many disjoint subsets of the same size are there?" Since there are C(n,m) ways to construct S1, there are C(n,m)C(n-m,m) ways to construct two disjoint subsets, each with m elements.

    Now you need to calculate the probability of obtaining these disjoint subsets.
     
    Last edited: Jan 18, 2007
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