  Let $S$ be a finite generating set of the mapping class group of a
finite-type hyperbolic surface. We show that mapping classes supported on a
fixed subsurface are not generic in the word metric with respect to $S$. We
also show that pseudo-Anosov mapping classes are generic in the word metric
with respect to $S'$, where $S'$ is $S$ plus a single mapping class. We also
observe the analogous results for well-behaved hierarchically hyperbolic groups
and groups quasi-isometric to them. This gives a version of quasi-isometry
invariant theory of counting group elements in groups.
