For any measure-preserving map \(T\) on a probability space \((X,\mu)\), and any measurable set \(A\) with \(\mu(A)\geq a\), it is shown that the average \(N^{-1}\sum_{j=0}^{N-1}\mu(A\cap T^{-j}A)\) is at least \(\sqrt{a^2+(1-a)^2}+a-1\). Examples are constructed to show this is sharp. The method of proof is combinatorial. By Rokhlin, it is enough to consider the case of \(X\) finite and \(T\) a permutation. The proof proceeds by careful counting arguments along the orbits of individual points. A similar result is found for related averages involving measurable functions and for nonconventional averages in systems generated by several transformations.