Probabilities for Intersecting Systems and Random Subsets of Finite Sets

Let F sub k be a family of subsets of {1,2,...,n}, each two of which have at least k elements in common, and let S be a random subset (sample) of {1,2,...,n} obtained by choosing each i = n independently with probability p sub i. Assuming that 1>p sub 1 p sub 1 >= p sub 2 >=...>= p sub n > 0, we investigate the problem of determining an F sub k that maximizes the probability that at least one of the sets in F sub k will be included in S.