8. Let $n$ be a positive integer. Randomly select non-empty subsets $A, B$ of $\{1,2, \cdots, n\}$. Then the probability that $A \cap B$ is not the empty set is $\qquad$
8. $\frac{4^{n}-3^{n}}{\left(2^{n}-1\right)^{2}}$.

Since $\{1,2, \cdots, n\}$ has a total of $2^{n}$ subsets, among which, $\mathrm{C}_{n}^{k}$ subsets $X$ satisfy $|X|=k, 2^{n-k}$ subsets $Y$ satisfy $X \cap Y=\varnothing(1 \leqslant k \leqslant n)$.
Therefore, the required probability is
$$
\frac{1}{\left(2^{n}-1\right)^{2}} \sum_{k=1}^{n} \mathrm{C}_{n}^{k}\left(2^{n}-2^{n-k}\right)=\frac{4^{n}-3^{n}}{\left(2^{n}-1\right)^{2}}
$$