Random Generation of Thompsons group $F$

We prove that under two natural probabilistic models (studied by Cleary, Elder, Rechnitzer and Taback), the probability that a random pair of elements of Thompsons 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.