Inégalité de Boole

In probability theory, Boole's inequality, also known as the union bound, says that for any finite or countable set of events, the probability that at least one of the events happens is no greater than the sum of the probabilities of the individual events. This inequality provides an upper bound on the probability of occurrence of at least one of a countable number of events in terms of the individual probabilities of the events. Boole's inequality is named for its discoverer, George Boole. Formally, for a countable set of events A1, A2, A3, ..., we have In measure-theoretic terms, Boole's inequality follows from the fact that a measure (and certainly any probability measure) is σ-sub-additive. Boole's inequality may be proved for finite collections of events using the method of induction. For the case, it follows that For the case , we have Since and because the union operation is associative, we have Since by the first axiom of probability, we have and therefore For any events in in our probability space we have One of the axioms of a probability space is that if are disjoint subsets of the probability space then this is called countable additivity. If we modify the sets , so they become disjoint, we can show that by proving both directions of inclusion. Suppose . Then for some minimum such that . Therefore . So the first inclusion is true: . Next suppose that . It follows that for some . And so , and we have the other inclusion: . By construction of each , . For it is the case that So, we can conclude that the desired inequality is true: Boole's inequality may be generalized to find upper and lower bounds on the probability of finite unions of events. These bounds are known as Bonferroni inequalities, after Carlo Emilio Bonferroni; see . Let for all integers k in {1, ..., n}. When n is odd, the sequence of inequalities: and both hold. When n is even, then: and both hold. The chains of inequalities among the partial sums follow from the observation that the events Sk form a decreasing sequence of sets.