average probability for an edge be a in random spanning tree of a weighted graph

any cofactor of a Laplacian of a weighted graph will give the sum of all weighted spanning trees, lets denote it by $A$. The same can be calculated for spanning trees which avoid certain edge $e$, denote it by $A_e$. There, $P[e \in T]=1-\frac{A_e}{A}$, the probability that an edge $e$ being part of a random spanning tree. I would like to calculate the average probability of an edge $e$ being part of a random spanning tree of a weighted graph.
Would be glad for any help.

Thank you!

• mathoverflow.net/questions/100209/… is quite a similar question. – Brendan McKay Sep 14 '16 at 14:07
• I'm looking for an average probability. – Daniel Sep 14 '16 at 14:10
• Average over what? Do you mean average over $e$? – Brendan McKay Sep 14 '16 at 14:12
• Average over all $P_e$ – Daniel Sep 14 '16 at 14:16
• If $P_e$ is what you called $P[e\in T]$, that is just a number. What probability space do you want to average it over? – Brendan McKay Sep 14 '16 at 14:23