A sharp upper probability bound for the occurrence of at least m out of n events

Abstract

Consider a fixed set of n events. Let S1 denote the sum of the probabilities of occurrence of each event and S2 the sum of the probabilities of occurrence of each of the pairs of events. Using a dual linear programming method a sharp upper bound is derived for the probability of occurrence of at least m out of the n events when S1 and S2 are given. The bound improves one of the inequalities obtained by Sathe et al. (1980).

On the relationship between the discrete and continuous bounding moment problems and their numerical solutions

“…Prékopa (1988Prékopa ( , 1990a and Samuels and Studden (1989), based primarily on Prékopa's works, have discovered that the sharp Bonferroni bounds of Dawson and Sankoff (1967), Sobel and Uppuluri (1972), and others can be interpreted as optimum values of moment problems. Results in this respect, for special cases, have earlier been obtained by Kwerel (1975a, b), Kounias and Marin (1976), Platz (1985) and a few others. Prékopa also developed general linear programming methodology for the discrete moment problems that included methods to obtain bounding formulas for cases where the number of moments is small and algorithms for the solution of problems.…”

“…This is because the Vandermonde systems are ill-conditioned (Bender et al 2002;Björck and Pereyra 1970;Markov 2001). Kounias and Marin (1976), Kwerel (1975a, b) and Platz (1985) exploited the special structure of problem (4) to derive special, low order probability bounds, using binomial moments. In Prékopa (1988Prékopa ( , 1990a the author gave characterizations for the dual feasible bases and showed how the bounds that have already been known in the literature, can be obtained by the use of them, if the number of moments is small.…”