In this paper, we construct a continuum family of nonisomorphic 3-generator groups in which the identity x^n=1 holds with probability 1, while failing to hold universally in each group. This resolves a recent question about the relationship between probabilistic and universal satisfaction of group identities. Our construction uses n -periodic products of cyclic groups of order n and two-generator relatively free groups satisfying identities of the form x^pn, y^pn^n=1. We prove that in each of these products, the probability of satisfying x^n=1 is equal to 1, despite the fact that the identity does not hold throughout any of these groups.
Atabekyan et al. (Wed,) studied this question.