What is the expected maximum matching size of a bipartite graph $(A\cup B, V)$ where $\lvert A\rvert = n$ and $\lvert B\rvert = n$ and the probability of a edge existing between $A$ and $B$ is a fixed $p$. If a general formula doesn't exist how about for $p = 0.5$ or some other $p$.
There is not likely to be any exact formula, but we can investigate the asymptotic behavior. If $p$ is a little bit larger than $(\log n)/n$, then with high probability there exists a maximum matching, so the expected size of the maximum matching is close to $n$. See https://math.stackexchange.com/q/1267387/14578.
So the only interesting case is where $p < (\log n)/n$. I don't have an estimate for that case, but perhaps someone else will be able to handle that case.