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Estimating probability of failureProbability of failure.The reliability index.Monte Carlo simulation.Uncertainty in Monte Carlo simulations.The uncertainty in the number of rare occurrence is the square root of the observed number.

1Limit state functionSimulation to predict the behavior of a physical system often try to answer what is the probability of a catastrophic event.We will use the term probability of failure.Following the textbook by Choi, Grandhi, and Canfield (CGC) we will denote the behavioral quantity that defines system failure by S(X), where X is a random vector of uncertainties in the simulation.The limit on S, or the resistance of the system is denoted as R(X). The limit state function is g=R-S.Failure occurs when g=R-S