Bernoulli Scheme with Closure

A problem of the following type is investigated: Given a finite set $E$ in which a class of subsets has been identified, what is the probability that a set of elements sampled from $E$ by the Bernoulli scheme will belong to that class? Inequalities analogous to those obtained earlier [M. V. Lomonosov and V. P. Polesskii, Probl. Peredachi Inf., 1971, vol. 7, no. 4, pp. 78–81; 1972, vol. 8, no. 2, pp. 47–53] for the special case of the connectivity probability of a random graph are proved with certain assumptions on the identified class of subsets. Concurrently, all the proofs are simplified, and some of the results are strengthened.