Math

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A drawer has 10 pairs of gloves. If I grab 5 gloves at random, what is the probability that I pick at least one matched pair? What is the probability that I pick at least one right glove and one left glove?

  • Math -

    let's look at the prob that they are all different
    start by picking any glove, now you have 1
    there is 1 of the remaining 9 that will match
    we don't want that, so the prob that the 2nd is NOT a match is 8/9
    prob that the 2nd and third are NOT a match
    = (8/9)(7/8)
    prob that the 2nd, 3rd, 4th and 5th are NOT a match
    = (8/9)(7/8)(6/7)(5/5)
    so the prob that at least one match is found
    = 1 - (8/9)(7/8)(6/7)(5/5)
    = .....

    Prob(at least one right and one left glove)
    implies we don't want either all lefts or all rights
    there are 5 lefts and 5 rights
    prob(5 lefts) = (5/10)(4/9)(3/8)(2/7)(1/6)
    the same would be true for prob(5 right)

    so prob (all left or all right)
    = (5/10)(4/9)(3/8)(2/7)(1/6) + (5/10)(4/9)(3/8)(2/7)(1/6)
    = 2(5/10)(4/9)(3/8)(2/7)(1/6)

    so prob(at least one left one right)
    = 1 - 2(5/10)(4/9)(3/8)(2/7)(1/6)
    = ...

  • small typo - Math -

    in the first solution near the end

    = (8/9)(7/8)(6/7)(5/5)
    so the prob that at least one match is found
    = 1 - (8/9)(7/8)(6/7)(5/5)

    should have been:
    = (8/9)(7/8)(6/7)(5/6)
    so the prob that at least one match is found
    = 1 - (8/9)(7/8)(6/7)(5/6)

  • Math -

    Wouldnt it be from 20 gloves as we have 10 pairs.

    So for first glove we can pick any. [19 remaining]

    For second glove we have to select from 18 which are different - 18/19
    3rd glove - 16/18
    4th glove - 14/17
    5th glove - 12/16 ...

  • Math -

    Of course you are right, how silly of me, there are obviously 20 gloves.

    But.... why are you jumping from 18/19 to 16/18 etc
    the pattern still continues following my argument above.
    that is ...
    (18/19)(17/18)(16/17) ....

  • Math -

    Ohk so it will be -
    1st glove - 20
    2nd glove - 18/19
    3rd glove - 17/18
    4th glove - 16/17
    5th glove - 15/16

    Total probability = 1 - the probability of above things right ??

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