Probability, graphs and electrical networks

Explicit forms are presented for the probability that a random tree in a graph G contains any logically expressed set of edges, given that it contains any other logically expressed set of edges. Edge sets may be included or excluded, and may appear in a logical union or intersection. This probability is expressed in terms of various readily calculated or measured resistance values in a resistive electrical network based on G. Explicit expressions are presented for obtaining by inspection the resistance of the above type in a class of structures of considerable application, including, for example, the n-cube used in large-scale computational assemblies