GIVEN: Positive integers $n,m,L$ and probabilities $p_1, p_2, \ldots, p_n$.

GOAL: Choose $L$ size-$m$ subsets $S_1, S_2, \ldots, S_L$ of $\{1,2,\ldots,n\}$ to maximize $\displaystyle \mathbb{E}[ \max_{1\leq \ell \leq L} |S \cap S_\ell| ]$, where the expectation is taken over (all $2^n$) $S \subseteq \{1,2,\ldots,n\}$ using the pdf $f$ defined as \[ f(S) = \left( \prod_{i \in S} p_i \right) \left( \prod_{i \not\in S} (1-p_i) \right)\ . \]

(Equivalently, $S$ is chosen by examining each $i \in \{1,2,\ldots,n\}$ independently, choosing $i$ to be in $S$ with probability $p_i$.)

Note: the problem is trivial unless $m < n$ and $L < \binom{n}{m}$.

A nice, closed-form expression for $\displaystyle \mathbb{E}[ \max_{1\leq \ell \leq L} |S \cap S_\ell| ]$ (in terms of $n,m,L$ and the $p_i$) would be a good start.