Let $V=\{v_1,\dots,v_n\}$ a set of $n$ vertices. Define $\mathcal{G}$ to be the set of all graphs on $V$. $|\mathcal{G}|=2^{\binom{n}{2}}$.
What is the probability that a random graph from $\mathcal{G}$ is connected?
Let $C(n)$ be the number of connected graphs on $n$ vertices, so the quantity we are looking for is just ${C(n)}\cdot {2^{-\binom{n}{2}}}$.
If $G$ is a connected graph on $n-1$ vertices and look on a new $n$'th vertex: it must have at least one edge to one of the $n-1$ vertices of $G$ in order that the new graph on $n$ vertices will be connected. Therefore $C(n)$ obeys the reccurence relation $$C(2)=1$$ $$C(n)=\sum_{i=1}^{n-1}\binom{n}{i}\cdot C(n-1)$$
but i'm not sure how to find a closed form for $C(n)$.
Your recursion is incorrect. It is also possible to create a connected graph on vertices v_1, v_2, \ldots, v_n by starting with a disconnected graph on n-1 vertices and adding some edges between v_n and the other vertices. I do not know that the exact expression for C(n) is known; however, as you're probably aware, Erdos and Renyi discuss such problems in their paper ``On the Evolution of Random Graphs". For the current problem they show, for example, that as n \rightarrow \infty, the probability that the graph is connected tends to 1. In fact, one can make the edge-probability much less than 1/2 and this result is still true.