TITLE: Expectation of the cardinality of the intersection of subsets
QUESTION [6 upvotes]: Let A be a set of n distinct elements, and let A' and A'' be independent permutations of A, where |A'| = |A''| = k.   What is the expectation for |A' ∩ A''| for any given k <= n?

REPLY [11 votes]: By linearity of expectation, the expectation value is $n$ times the probability that any given element of $A$ is in both sets. The probability for that is $(k/n)^2$, so the desired expectation value is $k^2/n$.