Probability of randomly designated subsets cover the universe

Question

Let $U=\{1,2,\ldots,n\}$ and $S \subseteq \mathscr{P}(U)$. Let $T$ be a subset of $S$, randomly constructed selecting independently each element of $S$ with probability $p$.

Is there a polynomial time algorithm that computes:

$$\mathbb{Pr}\left[ U =\bigcup_{X \in T} X \right]$$

Or is there any equivalent famous problem?