This is something I just wondered about, and I don't know whether there is a closed-form answer or not. I've tried but without making progress, so any idea would be helpful.
Consider a complete graph with $2n$ nodes. Each edge $uv$ is assigned a direction $u\rightarrow v$ or $v\rightarrow u$, each with probability $1/2$. Let $k$ be the number of nodes whose out-degree is at least $n$ (i.e. it has an arrow to more than half of the remaining nodes.) What is the probability that $k=n$?