Given a graph $G=(V,E)$ where every edge $ij$ has a corresponding probability $p_{ij}$. We can then consider random subsets of the edges $C \in E$ for which the probability that an edge $ij$ is contained within $C$ is $p_{ij}$.
Now let us restrict to those $C$ that are edge covers. Given this condition, what is the probability that each edge is within $C$? Clearly $p_{ij}$ will now only be a lower bound, as edges may need to occur more often to ensure that $C$ is an edge cover. Is there a way by which the probabilities may be determined more exactly?