The article begins by examining the fundamentals of traditional deterministic design philosophy. The initial section outlines the concepts of failure criteria and limit state functions, two traditional notions that are embedded in deterministic design philosophy. This is followed by a discussion regarding safety factors (a possible limit state function) and the common utilization of statistical concepts in deterministic engineering design approaches. Next, the fundamental aspects of a probabilistic failure analysis are explored, and it is shown that deterministic design concepts mentioned in the initial portion of the article are embedded in probabilistic design methods. For components fabricated from ceramic materials (and other similarly brittle materials) the probabilistic design approach yields the widely used Weibull analysis after suitable assumptions are incorporated. The authors point out that Weibull analysis provides the rare instance where closed form solutions are available for a probabilistic failure analysis. Since numerical methods are usually required to evaluate component reliabilities, a section on Monte Carlo methods is included to introduce the concept. The article concludes with a presentation of the technical aspects that support the numerical method known as fast probability integration (FPI). This includes a discussion of the Hasofer-Lind and Rackwitz-Fiessler approximations.