# Probability of having a good set by choosing independently from Universe

Let $$S$$, $$T$$ be two disjoint subsets of a universe $$U$$ such that $$|S| = |T| = n$$. Suppose we select a random subset $$R\subseteq U$$ by independently sampling each element of $$U$$ with probability $$p$$; that means, for each element $$i$$ of $$U$$ independently we include $$i$$ in $$R$$ with probability $$p$$. We say that the random subset $$R$$ is good if the following two conditions hold: $$R\cap S = \emptyset$$ and $$R\cap T = \emptyset$$. Show that for $$p=1/n$$, the probability that $$R$$ is good is larger than some positive constant.