  In the Cayley graph of the mapping class group of a closed surface, with
respect to any generating set, we look at a ball of large radius centered on
the identity vertex, and at the proportion among the vertices in this ball
representing pseudo-Anosov elements. A well-known conjecture states that this
proportion should tend to one as the radius tends to infinity. We prove that it
stays bounded away from zero. We also prove similar results for a large class
of subgroups of the mapping class group.
