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Based on a black box model of a complex system, and on intervals and probabilities describing the known information about the inputs, we want to estimate the system's reliability. This problem is motivated by a number of problem areas, most specifically in engineering reliability analysis under conditions of poor measurement and high complexity of system models. Using the results of tests performed on the system's computer model, we can estimate the lower and upper bounds of the probability that the system is in a desirable state. This is equivalent to using Monte-Carlo sampling to estimate cumulative belief and plausibility values of functionally propagated finite random intervals. In this paper, we prove that these estimates are correct in the sense that under reasonable assumptions, these estimates converge to the actual probability bounds. [ABSTRACT FROM AUTHOR]