Any profinite group can be viewed as a probabilistic space. This approach was explored intensively during the last years. On one hand, the people have been interested in properties of this probabilistic space like to be PFG (positively finite generated). A profinite group is called PFG if for some $k$, random $k$ elements of the group generate it with positive probability. On the other hand, the probabilistic aspect of profinite groups is used in the solution of different kind of problems. In the first direction we present new characterizations of PFG profinite groups which permit us to prove that an open subgroup of a PFG profinite group is also PFG. In the second direction we solve a problem posed by A. Mann, showing that there exist a constant $c$ such that the number of finite groups of order $n$ which can be defined by $r$ relations is at most $n^{cr}$.