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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.
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