Boole's inequality

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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.^[1]

Formally, for a countable set of events A₁, A₂, A₃, ..., we have

${\displaystyle {\mathbb {P} }\left(\bigcup _{i=1}^{\infty }A_{i}\right)\leq \sum _{i=1}^{\infty }{\mathbb {P} }(A_{i}).}$

In measure-theoretic terms, Boole's inequality follows from the fact that a measure (and certainly any probability measure) is σ-sub-additive.