I would like to answer the following basic question:
Let $V$ be a collection of $n$ vertices and fix $x$ and $y$ in $V$. Let $G$ be a random graph on $n$ vertices and $M$ edges. What is the probability that $x$ and $y$ belong to the same connected component?
I've tried doing this by hand and I can not get anywhere with it. It seems like such a basic question to ask that I'm sure its been done in the literature somewhere but I can not find it. My thoughts now lead towards random graph theory and asymptotic probabilities.
I'd be grateful for any help you can provide.