Abstract 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.

中文翻译：

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用于系统和投资组合中相关故障的概率评估的组合分析

摘要 系统可靠性可用于评估单个结构和结构组合的性能。在许多情况下，人们想知道系统中 n 个组件中发生 m 次故障的概率，或者 n 个组件中至少发生 m 次故障的概率。示例包括单个基础设施或建筑物内的结构故障模式、投资组合内的建筑物和结构或基础设施系统的组件，例如交通、水、电力和通信。在不同组件中的故障事件在统计上独立且概率相同的情况下，组件故障的数量遵循二项式分布。然而，在大多数情况下，这些条件并不成立。在本文中，我们提出了一个组合公式，用于计算一般情况下 n 个系统组件中 m 个故障的概率，基于计算联合概率的包含-排除原则的扩展。结果已通过先前在组合数学的数学教科书中可用的现有数学方法进行了验证。当前的推导提出了一个封闭形式的会计方案，该方案用于解决组合中多个建筑故障的包含/排除问题，但也适用于结构系统中的多种故障模式。介绍了对建筑物组合的示例应用。结果已通过先前在组合数学的数学教科书中可用的现有数学方法进行了验证。当前的推导提出了一个封闭形式的会计方案，该方案用于解决组合中多个建筑故障的包含/排除问题，但也适用于结构系统中的多种故障模式。介绍了对建筑物组合的示例应用。结果已通过先前在组合数学的数学教科书中可用的现有数学方法进行了验证。当前的推导提出了一个封闭形式的会计方案，该方案是针对组合中多个建筑故障的包含/排除问题而导出的，但也适用于结构系统中的多种故障模式。介绍了对建筑物组合的示例应用。