Bonferroni Inequalities

Let  be the probability that  is true, and  be the probability that at least one of , , ...,  is true. Then "the" Bonferroni inequality, also known as Boole's inequality, states that

where  denotes the union. If  and  are disjoint sets for all  and , then the inequality becomes an equality. A beautiful theorem that expresses the exact relationship between the probability of unions and probabilities of individual events is known as the inclusion-exclusion principle.

A slightly wider class of inequalities are also known as "Bonferroni inequalities."

REFERENCES:

Comtet, L. "Bonferroni Inequalities." §4.7 in Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, pp. 193-194, 1974.

Dohmen, K. Improved Bonferroni Inequalities with Applications: Inequalities and Identities of Inclusion-Exclusion Type. Berlin: Springer-Verlag, 2003.

Galambos, J. and Simonelli, I. Bonferroni-Type Inequalities with Applications. New York: Springer-Verlag, 1996.