Probability of (single) connecting paths in Erdos-Renyi graphs

Question

In an Erdos-Renyi graph with labeled vertices in $(1, ..., N)$, and for any pair of vertices $(r, s)$ with $r < s$ and a length $l$ in $(1, ..., s-r)$, I am looking for the probability of

1. there being at least one path of length $l$ joining $r$ and $s$
2. a path like in 1. being the only connecting path, knowing that it exists

Any ideas or pointers to the literature would be greatly appreciated!