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Math Help - A hard probability problem

  1. #1
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    A hard probability problem

    I'm stuck at this impossible problem... i'd aprecciate if you help me

    A box contains n socks, such that:

    - b are white
    - r are red
    - a are blue

    (So, n = b + r + a)
    Consider an experiment that consists of taking a sock from the box (without replacement), until you have a pair of the same colour. Find the following probabilities:


    a) The pair (of the same colour) is obtained at the k-th extraction n = b + r + a
    b) The obtained pair is white
    c) Knowing that the pair will be red: the pair is obtained at the third extraction.
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  2. #2
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    Re: A hard probability problem

    This does sound tough, what have you done? Try to create a tree diagram with the first couple of extrations, this may help you find a pattern.
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  3. #3
    Moo
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    Re: A hard probability problem

    Hello,

    In my opinion, it's possible to simplify a bit the problem. For the first question, the event will occur after (or at) the 2nd draw and before (or at) the 4th draw. So you just have to compute 3 probabilities, and they're not that complicated (thanks to the symmetry in the variables).
    But I'm not sure if this is a conventional way, if this is how you'd be asked to solve the problem.
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  4. #4
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    Re: A hard probability problem

    I've done many things but nothing interesting...
    I've been trying to separate part (a) by colours, but i'm really stuck... don't know how to begin solving a problem like this...

    About the simplification... hmmm
    Do you mean i have to solve for k=0, k=1, k=2, k=3 etc.
    And try to see a recurrence?

    Help
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  5. #5
    Moo
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    Re: A hard probability problem

    No, I mean try to think about it. How many draws do you have to make in order to have a pair ? You can't have a pair at the first draw since you need 2 socks for it. So it's at least at the second draw. Then what is the biggest number of draws in order to get a pair ? Suppose you've got no luck and you keep picking socks of different colours : on the first 3 draws, you have a different colour. Then on the 4th draw, whatever colour you get, it'll form a pair with one of the previous ones !
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  6. #6
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    Re: A hard probability problem

    I envision a tree:
    Code:
                                +-----------------------+-----------------------+
                                |                       |                       |
    k=1                       white                    red                     blue
                                |
             +------------------+------------------+
             |                  |                  |
    k=2    white               red               blue
        (pair white)            |
                      +---------+-----------+
                      |         |           |
    k=3             white      red        blue
                (pair white)(pair red)      |
                                  +---------+-----------+
                                  |         |           |
    k=4                         white      red        blue
                            (pair white)(pair red)(pair blue)
    For each node you can set up the conditional chance.
    For instance:
    P(white in 1st draw)=b/n
    P(white in 2nd draw | white in 1st draw)=(b-1)/(n-1)
    P(white in 3rd draw | white in 1st draw, red in 2nd draw)=(b-1)/(n-2)


    Now for instance the chance for a pair of white socks in the second draw is [product rule]:
    P(white in 2nd draw, white in 1st draw) = P(white in 2nd draw | white in 1st draw) P(white in 1st draw)=(b-1)/(n-1) * b/n

    The chance to finish with a pair of white socks is [sum rule]:
    P(pair of white socks) = sum P(each possible way to finish with 2 white socks)
    Last edited by ILikeSerena; December 29th 2011 at 04:18 AM.
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