Summary: Various applied problems require the analysis of systems with randomly failing elements. A typical such instance occurs in assessing the reliability of a transportation or communication network, in which the individual links fail independently with known probabilities. One is then interested in the probability that goods or information will be successfully transported from some source location to some terminal location. Exact calculation of this probability is known to be difficult, so this paper develops a method for obtaining lower and upper bounds on the required value. In fact, our procedure yields a sequence of matched lower and upper bounds, which can be monitored as the algorithm progresses. Rather than simply producing a numerical value, the method produces a functional (symbolic) form for the answer, especially useful for subsequent sensitivity analyses.