  We prove that generic elements of braid groups are pseudo-Anosov, in the
following sense: in the Cayley graph of the braid group with n $\ge$ 3 strands,
with respect to Garside's generating set, we prove that the proportion of
pseudo-Anosov braids in the ball of radius l tends to 1 exponentially quickly
as l tends to infinity. Moreover, with a similar notion of genericity, we prove
that for generic pairs of elements of the braid group, the conjugacy search
problem can be solved in quadratic time. The idea behind both results is that
generic braids can be conjugated "easily" into a rigid braid.
