# Probability that a random graph on countably many vertices is connected

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?

• See this question: math.stackexchange.com/questions/584228/… Edit: nevermind I misread the question I wasn't thinking about infinite graphs. – Ben Oct 28 '15 at 0:54
• Hint: you can actually prove that between every pair of distinct vertices there is (with probability 1) a path of length 2. – David Hackenger Oct 28 '15 at 1:01
• @DavidHackenger Well that's that. If you type that into an answer I'll accept it. – JustAskin Oct 28 '15 at 1:08