Let $G=(V,E)$ be a finite simple 2-connected graph. Let $B\subseteq E$ be a set of bad edges of size $k:=|B|$. For each edge $e\in E$ we toss a fair coin ($p=1/2$) and if the outcome is head, we hit the edge $e$ and we let it die (sorry about that). At the end of the process, we consider the set of survived edges $S$ and we look at the graph $G[S]$ consisting of survived edges only. Let $C_1, \ldots, C_n$ be the connected components of $G[S]$ and let $e=(u,v)$ be a dead edge: we say that $e$ died with honor if there exists just a single component $C_i$ s.t. $u,v\in C_i$, that is $e$ dies with honor if it doesn't connect two different components of the survived edges graph.
What can we say about the probability $P_H$ of the event $H=$"all the bad edges $B$ are dead and at least one died with honor" ? Can we lower bound somehow this probability as a function of $k$ ? Can we find structural hypothesis over $G$ or particular distributions for random $G$ in order to do that ?