One can pose this question in more generality (for dense free subgroups in semisimple Lie groups), but anyway:
It is known to some that there are elements $x,y \in SO(3)$, which generate a dense, free subgroup. Let $F_n$ be the set of words of length $\leq n$. Let $\mu_n$ denote the probability measure that is uniform on $F_n$.
Question: Does $\mu_n$ converge to the Haar measure? If so, how fast?