On estimates for probabilities of combinations of events, the Jordan formula and the Bonferroni inequalities

Abstract

We discuss the method which allows us to derive optimal upper and lower bounds for probabilities and conditional probabilities given a σ-field for various combinations of events. We define the optimality as the possibility of bounds to be equalities for some sets of events. We derive new generalizations of the Jordan formula and the Bonferroni inequalities. Conditional variants of these results are discussed as well.