Let n = b₁ + ... + b_k = b₁' + ... + b_k' be a pair of compositions of n into k positive parts. We say this pair is {\em irreducible} if there is no positive jk for which b₁ + ... + b_j = b₁' + ... + b_j'. The probability that a random pair of compositions of n is irreducible is shown to be asymptotic to 8/n. This problem leads to a problem in probability theory. Two players move along a game board by rolling a die, and we ask when the two players will first coincide. A natural extension is to show that the probability of a first return to the origin at time n for any mean-zero variance V random walk is asymptotic to \sqrt{V/(2\pi)} n^-3/2. We prove this via two methods, one analytic and one probabilistic.