I'm trying to prove that for a random graph on $n$ vertices with edge-probability $p \in (0, 1)$ is almost surely connected as $n$ grows large. I've tried making an argument using the probability of having a component of size $< n/2$ or a path not existing between some vertices $u, v$ and using the union bound, but the probabilities don't quite work out. Could someone give me a hint in the right direction?
Edit: The model in question is starting with $n$ vertices, and having each edge appear independently with probability $p$.