Random Elements in Profinite Groups

Let n be a positive integer and $F=\hat{F}_{n}$ the free profinite group of rank n. For each e-tuple (x₁,…,x _e ) in F^e we consider the closed (resp. normal closed) subgroup 〈x〉 (resp. [x]) generated by x₁,…,x _n in F. We investigate the probability that 〈x〉 (resp. [x]) is an open subgroup of F. Having done so, we strive to prove that with probability 1 each of the groups 〈x〉 and [x] are $\mathcal{C}$ -free of specific ranks.