# Probability of a random collection of subsets being a cover

Consider the set $$[n]=\{1,2,\ldots,n\}$$. Suppose for each set $$A\subseteq [n]$$ I have a $$p_A \in [0,1]$$. I now create a random collection $$\mathcal{W}\subseteq\mathcal{P}([n])$$ of subsets of $$[n]$$ by including each $$A$$ with probability $$p_A$$, independently. What is the probability that their union covers $$[n]$$, that is, that $$\bigcup_{W\in\mathcal{W}} W = [n]$$

This seems like a problem that absolutely has been considered before -- it's easy to state and seems natural -- and the answer should "just" be some suitably symmetric multivariate polynomial. Unfortunately, I can't figure it out on my own, and my google-fu hasn't availed me either.

• You can easily generate some long formula (say, by inclusion-exclusion: $\sum_{B\subset[n]}(-1)^{|B|}\prod_{A\subset[n],A\cap B\ne\varnothing}(1-p_A)$) but why do you have any hope that it collapses to something nice? Jun 14 '19 at 1:20