  Let $G \curvearrowright X$ be a nonelementary action by isometries of a
hyperbolic group $G$ on a hyperbolic metric space $X$. We show that the set of
elements of $G$ which act as loxodromic isometries of $X$ is generic. That is,
for any finite generating set of $G$, the proportion of $X$--loxodromics in the
ball of radius $n$ about the identity in $G$ approaches $1$ as $n \to \infty$.
We also establish several results about the behavior in $X$ of the images of
typical geodesic rays in $G$; for example, we prove that they make linear
progress in $X$ and converge to the Gromov boundary $\partial X$. Our
techniques make use of the automatic structure of $G$, Patterson--Sullivan
measure on $\partial G$, and the ergodic theory of random walks for groups
acting on hyperbolic spaces. We discuss various applications, in particular to
Mod(S), Out($F_N$), and right--angled Artin groups.
