Bonferroni inequalities

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. Boole's inequality is named after 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 ${\displaystyle n=1}$ case, it follows that

For the case ${\displaystyle n}$, we have

Since ${\displaystyle \mathbb {P} (A\cup B)=\mathbb {P} (A)+\mathbb {P} (B)-\mathbb {P} (A\cap B),}$ and because the union operation is associative, we have

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