References:This question is inspired by a conjecture of Alon Amit that is solved by Miklós Abért, Nikolay Nikolov and Dan Segal in the following papers:(1) On the probability of satisfying a word in a group (2005).

(2) A characterization of finite soluble groups (2006).

They provided a probabilistic characterization for solvable groups as follows:

Theorem:A finite group $G$ is solvable if and only if there exists some positive constant $c>0$ (depending on $G$) such that, for every $r∈\mathbb{N}$ and every $w∈F_r$ (the free group on $r$ generators), the probability that $w(g_1,\cdots, g_r)=1$ is at least $c$, where the $g_i$ are uniformly, independently chosen elements of $G$.

This characterization interestingly relates the solvability of a group to the *ease of solving equations in that group*. (See Alon's explanations in his Quora post for more intuitive descriptions).

Now consider the notion of a hypoabelian group, that is a natural generalization of the notion of solvability by allowing the sequences of derivations of a group to be of a transfinite length. (See also Perfect Core).

Question:Is there any probabilistic characterization for hypoabelian groups similar to the above theorem for solvable groups? Is such a characterization related to the ordinal length of the sequence of derivations of the group? Do we need some extra set theoretic assumptions to get such a characterization for hypoabelian groups?