By a random $n\times n$ bipartite graph, I mean a random bipartite graph on two vertex classes of size $n$, with the edges added independently, each with probability $p$.
I want to find the probability that such a graph contains an isolated vertex.
Let $X$ and $Y$ be the vertex classes. I can calculate the probability that $X$ contains an isolated vertex by considering one vertex first and using the fact that vertices in $X$ are independent.
But I don't know how to calculate the probability that $X\cup Y$ contains an isolated vertex. Can someone help? Thanks!