Probability Sock Problem

Question

A drawer contains n different and distinguishable pairs of socks (a total of 2n socks). A person randomly selects 2r of the socks, where 2r < n.

a) What is the probability of having no matches?

b) What is the probability of having exactly 1 match?

Answer 1

A. How many ways can we pick 2r socks out of 2n socks and not get any matches?

nC2r × 2^r

How many ways can we pick 2r socks out of 2n socks?

2nC2r

So, ( nC2r × 2^r ) / ( 2nC2r ).

B. Pick exactly one matching pair means picking r-1 non-matching pairs and one matching pair.

How many ways can I pick r-1 non-matching pairs and one matching pair?

( nCr-1 ) × (r-1) × 2^r-2

Divide this by the above denominator and you are done!