Incoming-edge coupling of site and bond percolation

Suppose every vertex of a directed graph has at most $\Delta$ incoming bonds. In bond percolation with parameter $q$, open a site whenever at least one incoming bond is open. Incoming bond sets have different heads and hence are disjoint, so these site indicators are independent random variables. Supplement their probabilities to $r=1-(1-q{)}^{\Delta }$. Every vertex reached by a nontrivial open directed path is now open. Conditioning the root to be open gives ${\theta }_s(r)\ge r{\theta }_b(q)$, since a simple directed path from the root never uses a bond entering the root. Thus $p_H^s\le 1-(1-p_H^b{)}^{\Delta }$.