In probability theory, Boole's inequality, named after George Boole, (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.

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.

[ Bonferroni inequalities

Boole's inequality may be generalised to find upper and lower bounds on the probability of finite unions of events. These bounds are known as Bonferroni inequalities, after Carlo Emilio Bonferroni.

Define

and for 2 < k ≤ n,

where the summation is taken over all k-tuples of integers .

Then, for odd k ≥ 1,

and for even k ≥ 2,

Boole's inequality is recovered by setting k = 1.

When k = n, then equality holds and the resulting identity is the inclusion-exclusion principle.

[ References

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