Bounds for the probability of union of events following monotonic distribution

Sharp bounds for the probability of occurrence of at least one out of n events are found when the probability distribution of the occurrences is supposed to follow monotonic distribution and first m(m≤n−1) binomial moments are known. The bounds are found explicitly for two different cases of increasing and decreasing distributions using operations research techniques. Inverses are found for two different classes of matrices and the bounds are obtained using those inverses.