Probabilistically nilpotent groups

We show that, for a finitely generated residually finite group $\mathrm{\Gamma }$, the word $[x_1,\dots ,x_k]$ is a probabilistic identity of $\mathrm{\Gamma }$ if and only if $\mathrm{\Gamma }$ is virtually nilpotent of class less than $k$. Related results, generalizations and problems are also discussed.