  We show that the probability that a finitely supported random walk on a
non-elementary subgroup of the the mapping class group gives a
non-pseudo-Anosov element 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.
