Algorithms for analyzing the probability of M-out-of-N events

Simple algorithms are presented to compute both the exact probability of M or more out of N independent input events given unequal probabilities and, when there is uncertainty in the input event probabilities, the associated variance in the M-out-of-N probability. The performance of the M-out-of-N probability algorithm is on the order of N2 in time and N in space. The performance of the variance algorithm is N3 in time and N2 in space. The algorithms are not based on cut set methodology and, consequently, are not limited by the combinatorial explosion associated with cut set manipulation for the M-out-of-N gate. The algorithms are most useful when N exceeds the limitations of cut set manipulation techniques or the M-out-of-N probability is between 0·1 and 1, such that approximate quantification methods are inaccurate. In addition, the M-out-of-N probability is extended to permit the quantification of standard sensitivity and uncertainty importance measures for both individual input events and groups of input events. Example calculations illustrate the capabilities of the algorithms.