Expected rank and randomness in rooted graphs

For a rooted graph G, let EV"b(G;p) be the expected number of vertices reachable from the root when each edge has an independent probability p of operating successfully. We determine the expected value of EV"b(G;p) for random trees, and include a connection to unrooted trees. We also consider rooted digraphs, computing the expected value of a random orientation of a rooted graph G in terms of EV"b(G;p). We consider optimal location of the root vertex for the class of grid graphs, and we also briefly discuss a polynomial that incorporates vertex failure.