So, I'm trying to show that a random graph is almost surely connected.
I want to know if my intuition is correct, and if so, how to formalize that intuition into a proof. If a graph $G=(V,E)$ has $|V|=n$, and it is disconnected, then it has at least 2 components. Let us arbitrarily divide the vertices into 2 subsets $U, W \subseteq V$ such that $|U|=k$ and $|W|=n-k$ such that $U$ and $W$ are connected. Then for $G$ to be disconnected, no vertex in $U$ can be connected to any other vertex in $W$. Therefore the probability that none of the vertices between $U$ and $W$ are connected is $(\frac{1}{2})^{k(n-k)}$, and since $k(n-k) \leq \frac{n^2}{4}$, $P \leq (\frac{1}{2})^{\frac{n^2}{4}})$, and when $n$ is large, this probability tends to 0, making the graph almost always connected for a random graph.
If this intuition is correct, how do I formalize it for all possible configurations of $U$ and $W$? Any help would be thoroughly appreciated!