On upper and lower bounds for probabilities of combinations of events

TL;DR

This paper develops new bounds for the probabilities of combinations of events, including conditions and expectations, applicable in general measurable spaces, enhancing understanding of probabilistic event interactions.

Contribution

The paper introduces novel upper and lower bounds for event probability combinations, applicable to general measurable spaces and measures with sign, with discussions on equality conditions and variants for conditional probabilities.

Findings

New bounds for probabilities of event combinations

Applicable to measures with sign and general measurable spaces

Enhanced bounds for conditional probabilities

Abstract

We derive new upper and lower bounds for probabilities that $r$ or at least $r$ from $n$ events occur. These bounds can turn to equalities. The method is discussed as well. It works for measurable space and measures with sign, too. We also discuss variants of the results for conditional probability of above events given $\sigma$-field. Taking expectations from both parts of inequalities for conditional probabilities can yield better bounds for unconditional ones.