It is well known that the set of hereditarily countable sets $H(\omega_1)$ —or, if you prefer, $H_{\omega_1}$— has cardinality $2^{\aleph_0}$, and I understand that every countable transitive model (ctm) of ZFC lives in this set. So, if there were a way to give $H(\omega_1)$ a sensible structure (separable metric, for instance) one might try to calculate the probability that a ctm satisfy CH (to name just one example).
I imagine that this idea has been explored and rejected quickly, but I couldn't find anything related to it.
A prior requirement, as I mentioned, would be to have some “definable” topology in the descriptive-set-theoretic sense (Polish, analytic, coanalytic, or the like).
Is there a sensible definable topology on $H(\omega_1)$?
About this, I found in the book Classification and Orbit Equivalence Relations by Hjorth that one can map $H(\omega_1)$ into the set of isomorphism classes of countable binary structures. Now, the countable binary structures form indeed a Polish space, so the odds are that this identification would give as something at least as complicated as $\boldsymbol{\Pi}_1^1(2^{\mathbb{N}\times\mathbb{N}})$ (since one has to say that the binary relation is well founded).