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.