Exponential decay in the mapping class group

We show that the probability that a non-pseudo-Anosov element arises from a finitely supported random walk on a non-elementary subgroup of the mapping class group decays exponentially in the length of the random walk. More generally, we show that if R is a set of mapping class group elements with an upper bound on their translation lengths on the complex of curves, then the probability that a random walk lies in R decays exponentially in the length of the random walk.