I have a number of discrete finite sets, $A_0$ through $A_n$. I do not actually know their contents, but I know the size of each set and the size of the intersection between $A_0$ and each of the other sets.
If necessary, I can also know the size of the "universe" - ie: how many elements exist in total (it's finite, though large), though I'd prefer to not use this information if possible. Let's call that value $N$.
Given the above, and the set of sets $A_1$ through $A_n$ that some element $x$ is a member of, how can we estimate the probability that $x$ is a member of $A_0$? Please state additional assumptions if you need to make any. For example, it may be reasonable to assume that the individual sets $A_1$ through $A_n$ are independent (ie: membership or lack therof in one of those sets does not affect the probability membership in any of the other sets, except $A_0$).