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.