Let n=b1​+...+bk​=b1′​+⋅+bk′​ be a pair of compositions of
n into k positive parts. We say this pair is {\em irreducible} if there is
no positive j<k for which b1​+...bj​=b1′​+...bj′​. 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 V/(2π)​n−3/2. We prove
this via two methods, one analytic and one probabilistic