Suppose we have an Erdos random graph with $n$ vertices and $c n$ edges.

What can you say about the probability that the graph is connected?

(More importantly) If it is connected, what is the distribution on the number of bridges?

EDIT:

I am interested in asymptotics as $c$ is fixed but $n \rightarrow \infty$. That is, I know that the probability that the graph is connected is exponentially small in $n$, but I don't know what the exponent is.

As for the number of non-bridges, I would like some result
like the number of bridges for a random *connected* graph
is $> c' n$ where $c'$ is another constant, with probability approaching $1$.

sharp thresholdfor connectivity of $G(n,p)$. – Tony Huynh Jun 13 '11 at 16:04Journal of Mathematical Psychology, Volume 12, Issue 1, February 1975, Pages 90-98. – Joseph O'Rourke Jun 13 '11 at 16:43