  Consider the group of $n$ men and $n$ women, each with their own preference
list for a potential marriage partner. The stable marriage is a bipartite
matching such that no unmatched pair (man, woman) prefer each other to their
partners in the matching. Its non-bipartite version, with an even number $n$ of
members, is known as the stable roommates problem. Jose Alcalde introduced an
alternative notion of exchange-stable, one-sided, matching: no two members
prefer each other's partners to their own partners in the matching. Katarina
Cechl\'arov\'a and David Manlove showed that the e-stable matching decision
problem is $NP$-complete for both types of matchings. We prove that the
expected number of e-stable matchings is asymptotic to $\left(\frac{\pi
n}{2}\right)^{1/2}$ for two-sided case, and to $e^{1/2}$ for one-sided case.
However, the standard deviation of this number exceeds $1.13^n$, ($1.06^n$
resp.). As an obvious byproduct, there exist instances of preference lists with
at least $1.13^n$ ($1.06^n$ resp.) e-stable matchings. The probability that
there is no matching which is stable and e-stable is at least
$1-e^{-n^{1/6+o(1)}}$, ($1-O(2^{-n/2})$ resp.).
