Random-cluster single-edge conditional probability ($\varphi (\omega (e)=1\mid {\omega }_{E\setminus e})\in \{p,p/(p+q(1-p))\}$)

For an edge with endpoints already connected without that edge, including any boundary wiring, the conditional open probability is $p$. Otherwise opening the edge merges two components, so the conditional probability is $p/(p+q(1-p))$. For $q\ge 1$, this conditional probability increases when more other edges are open or more boundary vertices are identified. This is the local mechanism of boundary monotonicity of the random-cluster measure.