  Given a finite generating set $S$, let us endow the mapping class group of a
closed hyperbolic surface with the word metric for $S$. We discuss the
following question: does the proportion of non-pseudo-Anosov mapping classes in
the ball of radius $R$ decrease to 0 as $R$ increases? We show that any finite
subset $S'$ of the mapping class group is contained in a finite generating set
$S$ such that this proportion decreases exponentially. Our strategy applies to
weakly hyperbolic groups and does not refer to the automatic structure of the
group.
