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7 August, 11:11

A closet contains n pairs of shoes. If 2r shoes are chosen at random, (where 2r < n), what is the probability that there will be

a) no complete pair

b) Exactly one complete pair

c) Exactly 2 complete pair

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Answers (1)
  1. 7 August, 14:31
    0
    We are choosing 2

    2

    r

    shoes. How many ways are there to avoid a pair? The pairs represented in our sample can be chosen in (2)

    (

    n

    2

    r

    )

    ways. From each chosen pair, we can choose the left shoe or the right shoe. There are 22

    2

    2

    r

    ways to do this. So of the (22)

    (

    2

    n

    2

    r

    )

    equally likely ways to choose 2

    2

    r

    shoes, (2) 22

    (

    n

    2

    r

    )

    2

    2

    r

    are "favourable."

    Another way: A perhaps more natural way to attack the problem is to imagine choosing the shoes one at a time. The probability that the second shoe chosen does not match the first is 2-22-1

    2

    n

    -

    2

    2

    n

    -

    1

    . Given that this has happened, the probability the next shoe does not match either of the first two is 2-42-2

    2

    n

    -

    4

    2

    n

    -

    2

    . Given that there is no match so far, the probability the next shoe does not match any of the first three is 2-62-3

    2

    n

    -

    6

    2

    n

    -

    3

    . Continue. We get a product, which looks a little nicer if we start it with the term 22

    2

    n

    2

    n

    . So an answer is

    22⋅2-22-1⋅2-42-2⋅2-62-3⋯2-4+22-2+1.

    2

    n

    2

    n



    2

    n

    -

    2

    2

    n

    -

    1



    2

    n

    -

    4

    2

    n

    -

    2



    2

    n

    -

    6

    2

    n

    -

    3



    2

    n

    -

    4

    r

    +

    2

    2

    n

    -

    2

    r

    +

    1

    .

    This can be expressed more compactly in various ways.
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