Question

A bag contains 5 pair of socks, with each pair of different color. A random sample of 4 single socks is drawn from this bag. Any complete pairs in the sample is discarded and replaced by a new pair drawn from the bag. This process continues until the bag is empty OR 4 socks of different colors are held outside the bag. If the probability of the latter happening is m/n, where m, n are coprime natural numbers, then (m+n) is

Answer 1

8

Answer 2

Let's reinterpret the process. Each of the 5 pairs of socks is opened when its first sock is drawn and closed when its second sock appears. We want to find the probability that at some stage, the number of distinct open socks reaches 4.

The total number of Dyck paths of semilength 5 is the 5th Catalan number, which is C_5 = (1/6) * (10 choose 5) = 42.

We need to find the number of paths that never exceed 3 open socks. Using a reflection method, this number is 36.

The number of sequences in which the maximum open count is at least 4 is 42 - 36 = 6.

The required probability is 6/42 = 1/7.

Since the answer is in lowest terms as m/n = 1/7, we have m+n = 1+7 = 8.