Probabilistic properties of profinite groups

Let C be a class of finite groups which is closed for subgroups, quotients and direct products. Given a profinite group G and an element x is an element of G, we denote by PC(x, G) the probability that x and a randomly chosen element of G generate a pro-C subgroup. We say that a profinite group G is C-positive if PC(x, G) > 0 for all x is an element of G. We establish several equivalent conditions for a profinite group to be C-positive when C is the class of finite soluble groups or of finite nilpotent groups. In particular, for the above classes, the profinite C-positive groups are virtually prosoluble (resp., virtually nilpotent). We also draw some consequences on the prosoluble (resp. pronilpotent) graph of a profinite group.