  Word maps provide a wealth of information about finite groups. We examine the
connection between the probability distribution induced by a word map and the
underlying structure of a finite group. We show that a finite group is
nilpotent if and only if every surjective word map has fibers of uniform size.
Moreover, we show that probability distributions themselves are sufficient to
identify nilpotent groups, and these same distributions can be used to
determine abelian groups up to isomorphism. In addition we answer a question of
Amit and Vishne.
