Residual properties of free groups and probabilistic methods

Let w be a non-trivial word in two variables. We prove that the probability that two randomly chosen elements x, y of a nonabelian finite simple group S satisfy w(x, y) = 1 tends to 0 as |S| → ∞. As a consequence of this result, we obtain a new short proof of a well-known conjecture of Magnus concerning free groups. We also use it to prove an analogue of the Tits alternative: If a linear group Γ is not virtually solvable then its profinite completion Γ̂ has a "virtually dense" free subgroup of finite rank.