Random generation of Thompson group F

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