We prove that if G is a sufficiently large finite almost simple group of Lie type, then given a fixed nontrivial element x in G and a coset of G modulo its socle, the probability that x and a random element of the coset generate a subgroup containing the socle is uniformly bounded away from 0 (and goes to 1 if the field size goes to infinity). This is new even if G is simple. Together with results of Lucchini and Burness--Guralnick--Harper, this proves a conjecture of Lucchini and has an application to profinite groups. A key step in the proof is the determination of the limits for the proportion of elements in a classical group which fix no subspace of any bounded dimension.