Probability that a random graph on countably many vertices is connected

Question

Fix $0 < p < 1$ and let $G$ be a random graph on elements $\mathbb{N}$ where for $n,m \in \mathbb{N}$, the probability that there is an edge between $n$ and $m$ will $p$. What is the probability (in terms of $p$) that $G$ will be connected?

Answer 1

Hint: you can actually prove that between every pair of distinct vertices there is a path of length 2 (with probability 1).