Commuting probability for the Sylow subgroups of a finite group

For subsets X, Y of a finite group G, let Pr(X, Y) denote the probability that two random elements x is an element of X and y is an element of Y commute.Suppose that G is a finite group in which for any distinct primes p, q is an element of pi(G) there is a Sylow p-subgroup P and a Sylow q-subgroup Q of G such that Pr(P, Q) >= & varepsilon;. We show that F2(G) has & varepsilon;-bounded index in G.If G is a finite soluble group in which for any prime p is an element of pi(G) there is a Sylow p-subgroup P and a Hall p '-subgroup H such that Pr(P, H) >= & varepsilon;, then F(G) has & varepsilon;-bounded index in G.Moreover, we establish criteria for nilpotency and solubility of G.