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Math Help - Which outcome occurs first?

  1. #1
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    Which outcome occurs first?

    Two of the possible outcomes of an experiment are a and b, with probabilities p and q respectively and p+q\leq 1. If the experiment is repeated until either a or b occurs, what is the probability that b is the first to occur?

    My first thought was to work on a convolution of two geometric R.Vs. (ie number of trials since a occur, number of trials since b occurs) but these obviously are not independent since they cannot occur simultaneously.

    The solution says: "One can either workk on the joint distribution of the numbers of trials until the first a and b occur, or decompose in terms of the outcome of the first experiment." but I can't work out what it means.

    What is the joint distribution? The outcome of the first experiment can be a, b, or something else -- how do I decompose that?

    I'm feeling pretty stupid right now. Any help would be appreciated.
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  2. #2
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    Quote Originally Posted by harbottle View Post
    Two of the possible outcomes of an experiment are a and b, with probabilities p and q respectively and p+q\leq 1. If the experiment is repeated until either a or b occurs, what is the probability that b is the first to occur?

    My first thought was to work on a convolution of two geometric R.Vs. (ie number of trials since a occur, number of trials since b occurs) but these obviously are not independent since they cannot occur simultaneously.

    The solution says: "One can either workk on the joint distribution of the numbers of trials until the first a and b occur, or decompose in terms of the outcome of the first experiment." but I can't work out what it means.

    What is the joint distribution? The outcome of the first experiment can be a, b, or something else -- how do I decompose that?

    I'm feeling pretty stupid right now. Any help would be appreciated.
    Call the something else outcome C. To get the probability that A appears before B, you need to calculate the sum (using the sum of an infinite geometric series) of the probabilities of the final sequence of events:

    A
    CA
    CCA
    CCCA
    etc.

    I suggest you initially let r = 1 - (p + q) in the calculation.
    Last edited by mr fantastic; November 3rd 2010 at 03:14 AM. Reason: Fixed a typo.
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  3. #3
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    Thank you! That makes a lot of sense. The answer would be q/(1-(1-(p+q))) = q/(p+q).
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