  We prove that there exists a finitely generated group that satisfies a group
law with probability 1 but does not satisfy any group law. More precisely, we
construct a finitely generated group G in which the probability that a random
element chosen uniformly from a finite ball in its Cayley graph, or via any
non-degenerate random walk, satisfies the group law x^k=1 for some (fixed)
integer k, tends to 1. Yet, G contains a non-abelian free subgroup, and
therefore G does not satisfy any group law. In particular, this answers two
questions of Amir, Blachar, Gerasimova, and Kozma.
