  In the free group $F_k$, an element is said to be primitive if it belongs to
a free generating set. In this paper, we describe what a generic primitive
element looks like. We prove that up to conjugation, a random primitive word of
length $N$ contains one of the letters exactly once asymptotically almost
surely (as $N \to \infty$).
  This also solves a question from the list `Open problems in combinatorial
group theory' [Baumslag-Myasnikov-Shpilrain 02']. Let $p_{k,N}$ be the number
of primitive words of length $N$ in $F_k$. We show that for $k \ge 3$, the
exponential growth rate of $p_{k,N}$ is $2k-3$. Our proof also works for giving
the exact growth rate of the larger class of elements belonging to a proper
free factor.
