  It is known that every infinite index quasi-convex subgroup $H$ of a
non-elementary hyperbolic group $G$ is a free factor in a larger quasi-convex
subgroup of $G$. We give a probabilistic generalization of this result. That
is, we show that when $R$ is a subgroup generated by independent random walks
in $G$, then $\langle H, R\rangle\cong H\ast R$ with probability going to one
as the lengths of the random walks go to infinity and this subgroup is
quasi-convex in $G$. Moreover, our results hold for a large class of groups
acting on hyperbolic metric spaces and subgroups with quasi-convex orbits. In
particular, when $G$ is the mapping class group of a surface and $H$ is a
convex cocompact subgroup we show that $\langle H, R\rangle$ is convex
cocompact and isomorphic to $ H\ast R$.
