p-elements in profinite groups

We investigate some properties of the p-elements of a profinite group G. We prove that if p is odd and the probability that a randomly chosen element of G is a p-element is positive, then G contains an open prosolvable subgroup. By contrast, there exist groups that are not virtually prosolvable but in which the probability that a randomly chosen element of G is a 2-element is arbitrarily close to 1. We prove also that if a profinite group G has the property that, for every p-element x, it is positive the probability that a randomly chosen element y of G generates with x a pro-p group, then G contains an open pro-p subgroup.