It's well known that for any graph $G = (V,E)$ that if $G$ is not connected, then its compliment $\overline{G}$ is connected. So, it's impossible to have both $G$ and $\overline{G}$ be disconnected. However, it's entirely possible for both $G$ and $\overline{G}$ to be connected. Is anything known about the probability that a random graph has both $G$ and $\overline{G}$ connected? For simplicity say $G(n,1/2)$?