On inequalities for probabilities of joint occurrence of several events

Abstract

Various inequalities for probabilities of joint occurrence of several events are widely used in combinatorial analysis, probability theory and numerous applications. We derive a method for construction of upper and lower bounds for probabilities that r from n events occur. We use resolutions of those probabilities in sums of positive items and bound them separately. This yields new inequalities sharper than earlier ones which correspond to trivial resolutions. New inequalities are optimal. There exist examples which show that they may turn to equalities. Note that the method will give more sharp bounds when the number of moments increases. But this imply more difficult calculations as well. By the same method, similar bounds are obtained for conditional probabilities of corresponding events given a σ-field. Here, the accuracy of bounds also increases when one enlarges the numbers of moments under consideration. Taking an expectation from both parts of inequalities for conditional probabilities, one can derive sharper bounds for unconditional probabilities.