On the probability of nonexistence in binomial subsets

Let Γ be a hypergraph with vertex set Ω, let p : Ω → [0, 1], and let Ωp be a random set formed by including every ω ∈ Ω independently with probability p(ω). We investigate the general question of deriving fine (asymptotic) estimates for the probability that Ωp is an independent set in Γ, which is an omnipresent problem in probabilistic combinatorics. Our main result provides a sequence of lower and upper bounds on this quantity, each of which can be evaluated explicitly. Under certain natural conditions, we obtain an explicit closed formula that is asymptotic to this probability. We demonstrate the applicability of our results with two concrete examples: subgraph containment in random graphs and arithmetic progressions in random subsets of the integers.