SIMON-MARAIS 2022 A2

Let $n$ be a positive integer, and let $S$ be a set with $2^n$ elements. Let $A_1, A_2, \ldots, A_n$ be randomly and independently chosen subsets of $S$, where each possible subset of $S$ is chosen with equal probability. Let $P_n$ be the probability that

$$A_1 \cup A_2 \cup \cdots \cup A_n = S \quad \text{and} \quad A_1 \cap A_2 \cap \cdots \cap A_n = \emptyset.$$

Prove that $\lim_{n \to \infty} P_n = \frac{1}{e^2}$.