  We prove that under two natural probabilistic models (studied by Cleary,
Elder, Rechnitzer and Taback), the probability that a random pair of elements
of Thompson's group $F$ generate the entire group is positive. We also prove
that for any $k$-generated subgroup $H$ of $F$ which contains a "natural" copy
of $F$, the probability of a random $(k+2)$-generated subgroup of $F$
coinciding with $H$ is positive.
