The sock matching problem

Abstract

When matching socks after doing the laundry, how many unmatched socks can appear in the process of drawing one sock at a time from the basket? By connecting the problem of sock matching to the Catalan numbers, we give the probability that k unmatched socks appear. We also show that, for each fixed k, this probability approaches 1 as the number of socks becomes large enough. The relation between the number of socks and the k for which a given probability is first reached is also discussed, but a complete answer is open. In any load of clothes to be washed by a college student, there are inevitably a variety of socks tossed in with all the other garments. By the time the clothes come out of the dryer, the socks have been thoroughly mixed in, hiding underneath shirts or in pant legs. The game of matching then begins: does the sock you just picked randomly out of the pile match any of the others you’ve already removed from the pile? How big is your stack of unmatched socks going to get? This creates a scenario in which there can be k unmatched socks out of n pairs. We wish to determine the likelihood of obtaining a maximum of k unmatched socks while folding a pile of laundry containing the n pairs of socks. We assume that each pair of socks is complete and unique, and that socks are drawn randomly, one at a time. 2. Background