  Let $\Sigma$ be a compact orientable surface of finite type with at least one
boundary component. Let $\Gamma \leq \textup{Mod}(\Sigma)$ be a non virtually
solvable subgroup. We answer a question of Lubotzky by showing that there
exists a finite dimensional homological representation $\rho$ of
$\textup{Mod}(\Sigma)$ such that $\rho(\Gamma)$ is not virtually solvable. We
then apply results of Lubotzky and Meiri to show that for any random walk on
such a group the probability of landing on a power, or on an element with
topological entropy $0$ both decrease exponentially in the length of the walk.
