Probability of randomly designated subsets cover the universe

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?

There is no polynomial-time algorithm that computes an exact solution (i.e. the probability expressed as a ratio of two nonnegative integers $$a/b$$) to your problem unless $$\textbf{P} = \textbf{NP}$$. I will show that such an exact solution would provide a polynomial time algorithm for set-cover.

Let $$(U, S \subseteq \mathscr{P}(U), k)$$ be an instance of set-cover. Let $$m = |S|$$.

Let $$\Gamma = \left\{ K \in \mathscr{P}(S) : U =\bigcup_{X \in K} X \right\}$$

For each $$h \in \mathbb{N}$$ let:

$$\nu_h = |\{K \in \Gamma : |K|=h\}|$$

Observe that our instance of set-cover is satisfiable if and only if $$\nu_k \neq 0$$.

Now, let $$T$$ be a set, constructed as in the problem statement. $$T$$ covers $$U$$ if and only if $$T \in \Gamma$$. Therefore:

$$\mathbb{Pr}\left[ U =\bigcup_{X \in T} X \right] = \mathbb{Pr}[T \in \Gamma] = \sum_{X\in\Gamma}p^{|X|}(1-p)^{m-|X|} = \sum_{h=0}^m \nu_h p^h(1-p)^{m-h}$$

The rightmost expression is a polynomial over the ring $$\mathbb{Q}[p]$$, which means that its coefficients can be found by computing its value at $$m-1$$ points and applying an algorithm such as the finite differences method.