  We investigate invariants for random elements of different hyperbolic groups.
We provide a method, using Cayley graphs of groups, to compute the probability
distribution of the minimal length of a random word, and explicitly compute the
drift in different cases, including the braid group $B_3$. We also compute in
this case the return probability. The action of these groups on the hyperbolic
plane is investigated, and the distribution of a geometric invariant, the
hyperbolic distance, is given. These two invariants are shown to be related by
a closed formula.
