We now describe the two different formats for the representation of F considered in this paper. In the first format F is represented by m failure sets F1, F2, . . . , Fm. Each failure set is specified by an n-tuple in the input. The set of failure states is then F=m,t=1Fi. We develop a formula for the probability of a union of events. A Monte Carlo algorithm which estimates the failure probability of an n-component system is developed based on this formula. One trial of the algorithm outputs an unbiased estimator of Pr[F]. Let Y be the average of the estimators produced by many trials of the algorithm. We show that, when the algorithm is run for an amount of time proportional to nm, Y is provably close to Pr[F] with high probability.
The second format for the representation of F can be described as follows. A network is an undirected graph G, where the edges in the graph correspond to the components in the system. Let G be a planar network and let x1, . . . , xK be K specified nodes in G. For the planar K-terminal problem, the network is in a failing state if there is no path of working edges between some pair of specified nodes. We develop a Monte Carlo algorithm to estimate Pr[F] for the planar K-terminal problem. The algorithm works especially well when the edge failure probabilities are small. In this case the algorithm produces an estimator Y which is provably close to Pr[F] with high probability in time polynomial in the size of the graph. This compares very favorably with the execution times of other methods used for solving this problem.