Consider a uniform random tournament with $n$ vertices. (Between any two vertices $x,y$, with probability $0.5$ draw an edge from $x$ to $y$; otherwise draw an edge from $y$ to $x$.) Let $p(n)$ denote the probability that there is only one vertex with the maximum out-degree. What is $\lim_{n\rightarrow\infty}p(n)$?

I would be surprised if this statement hasn't been written and proved anywhere, but I can't find it. Does anyone know of a reference, or a result that implies it?