Edge-inclusion formula for a uniform spanning tree ($\mathbb{P}(e\in T)=c_e{R}_{\mathrm{eff}}(a,b)$)

For an edge $e=\{a,b\}$ of conductance $c_e$ in a finite electrical network, its inclusion probability in the conductance-weighted uniform spanning tree is $c_e{R}_{\mathrm{eff}}(a,b)$. The same formula holds for the wired limit using the limiting wired effective resistance.