Suppose $G$ is a random simple graph, that has $n$ vertices, and edges, that are present independently with probability $p$. What is the probability of $G$ being connected?
It is quite easy to calculate the probability for small $n$ (as for small $n$ we can classify all connected graphs with $n$ vertices). Thus for $n = 1$ the probability is $1$, for $n = 2$ it is $p$, and for $n = 3$ it is $3p^2 - 2p^3$. However, I do not know, how to calculate it for arbitrary $n$.
Any help will be appreciated.