  We study the impact of certain identities and probabilistic identities on the
structure of finite groups. More specifically, let $w$ be a nontrivial word in
$d$ distinct variables and let $G$ be a finite group for which the word map
$w_G:G^d\rightarrow G$ has a fiber of size at least $\rho|G|^d$ for some fixed
$\rho>0$. We show that, for certain words $w$, this implies that $G$ has a
normal solvable subgroup of index bounded above in terms of $w$ and $\rho$. We
also show that, for a larger family of words $w$, this implies that the
nonsolvable length of $G$ is bounded above in terms of $w$ and $\rho$, thus
providing evidence in favor of a conjecture of Larsen. Along the way we obtain
results of some independent interest, showing roughly that most elements of
large finite permutation groups have large support.
