Boole's inequality

Boole's inequality states that the probability of a union of events is at most the sum of their individual probabilities, regardless of dependence.

Union bound: A bound that limits the probability of a union by the sum of the probabilities of its events. This is the standard name for applying Boole's inequality to control a union.
Bonferroni inequalities: A sequence of upper and lower bounds for union probabilities obtained by truncating inclusion–exclusion. They sharpen Boole's bound by accounting for selected event intersections.
Probability space: A sample space equipped with a collection of events and a probability measure. The inequality applies to events within this mathematical structure.
George Boole: An English mathematician and logician whose work helped establish algebraic logic. The inequality bears his name, though its modern probability formulation is a general measure-theoretic fact.
Indicator function: A function that equals one when a condition holds and zero otherwise. Event indicators turn a union into a pointwise inequality whose expectation proves the bound.
Multiple comparisons problem: The inflation of false-positive risk when many statistical hypotheses are tested. Boole's inequality bounds the chance of at least one false rejection across tests.
Event (probability theory): A measurable subset of a probability space to which a probability is assigned. Boole's inequality compares the probability of a union of such sets with their separate probabilities.
Probability theory: The mathematical study of random events and their likelihoods. Boole's inequality is a foundational bound within this broader subject.
Subadditivity: A property in which a function of a union or sum is no greater than the sum of its separate values. Probability is subadditive, and Boole's inequality is its event-level expression.
Bonferroni correction: A procedure that controls family-wise error by dividing a significance level among multiple tests. It applies the union bound to keep the probability of any false rejection within a target.

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