Let $T$ be a subset of $\{1,2, \ldots, n\}$ of size $\frac{n}{2}$, sampled uniformly at random from all subsets of size $\frac{n}{2}$ of the same set $\{1,2, \ldots, n\}$. Let $S \subseteq\{1,2, \ldots, n\}$ be an arbitrary subset of size $\frac{n}{2}$. What is the probability that $|S \cap T|=\frac{n}{4}$.
