Combinatorial analysis for probabilistic assessment of dependent failures in systems and portfolios

System reliability is usefully applied to assess the performance of individual structures and portfolios of structures. In many instances, one is interested in knowing the probability that m failures have occurred among n components in a system, or that at least m failures have occurred among n components. Examples include structural failure modes within a single infrastructure or building, buildings and structures within a portfolio, or components of infrastructure systems such as transportation, water, power and communications. In cases where failure events in different components are statistically independent and their probability is identical, the number of component failures follows the binomial distribution. However, in most situations, these conditions do not hold. In this paper, we present a combinatorial formulation for computing the probability of m failures out of n system components for the general case, based on an extension of the inclusion–exclusion principle for computing the probability of a union. The results have been verified by an existing mathematical approach previously available in a mathematical textbook on combinatorics. The current derivation presents a closed-form accounting scheme derived for the inclusion/exclusion problem of multiple building failures in a portfolio, but also applicable to multiple modes of failure in a structural system. An example application to a portfolio of buildings is presented.