Connections in randomly oriented graphs

Abstract

Given an undirected graph $G$, let us randomly orient $G$ by tossing independent (possibly biased) coins, one for each edge of $G$. Writing $a\rightarrow b$ for the event that there exists a directed path from a vertex $a$ to a vertex $b$ in such a random orientation, we prove that $\mathbb{P}(s\rightarrow a \cap s\rightarrow b) \ge \mathbb{P}(s\rightarrow a) \mathbb{P}(s\rightarrow b)$ for any three vertices $s$, $a$ and $b$ of $G$.