Commuting probability for the Sylow subgroups of a profinite group

Given two subgroups H, K of a compact group G, the probability that a random element of H commutes with a random element of K is denoted by Pr(H,K). We show that if G is a profinite group containing a Sylow 2-subgroup P, a Sylow 3-subgroup Q3 and a Sylow 5-subgroup Q5 such that Pr(P,Q3) and Pr(P,Q5) are both positive, then G is virtually prosoluble (Theorem 1). Furthermore, if G is a prosoluble group in which for every subset π⊆π(G) there is a Hall π-subgroup Hπ and a Hall π′-subgroup Hπ′ such that Pr(Hπ,Hπ′)>0, then G is virtually pronilpotent (Theorem 2).