Strengthened Bounds for the Probability of k-Out-Of-n Events

Abstract: Given a set of n random events in a probability space, represented by n Bernoulli variables (not necessarily independent,) we consider the probability that at least k out of n events occur. When partial distribution information, i.e., individual probabilities and all joint probabilities of up to m (m< n) events, are provided, only an upper or lower bound can be computed for this probability. Recently Prekopa and Gao (Discrete Appl. Math. 145 (2005) 444) proposed a polynomial-size linear program to obtain strong bounds for the probability of union of events, i.e., k=1. In this work, we propose inequalities that can be added to this linear program to strengthen the bounds. We also show that with a slight modification of the objective function this linear program and the inequalities can be used for the more general case where k is any positive integer less than or equal to n. We use the strengthened linear program to compute probability bounds for the examples used by Prekopa and Gao, and the comparison shows significant improvement in the bound quality.



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