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.

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

$\backslash Pr\backslash leftA\_i\backslash right\backslash leq\; \backslash sum\_i\; \backslash Pr\backslash left*.$

Bonferroni inequalities

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

Define

$S\_1\; :=\; \backslash sum\_\{i=1\}^n\; \backslash Pr(A\_i),$

and for 2 < k ≤ n,

$S\_k\; :=\; \backslash sum\; \backslash Pr(A\_\{i\_1\}\backslash cap\; \backslash cdots\; \backslash cap\; A\_\{i\_k\}\; ),$

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

Then, for odd k ≥ 1,

$\backslash Pr\backslash left(\; \backslash bigcup\_\{i=1\}^n\; A\_i\; \backslash right)\; \backslash leq\; \backslash sum\_\{j=1\}^k\; (-1)^\{j+1\}\; S\_j,$

and for even k ≥ 2,

$\backslash Pr\backslash left(\; \backslash bigcup\_\{i=1\}^n\; A\_i\; \backslash right)\; \backslash geq\; \backslash sum\_\{j=1\}^k\; (-1)^\{j+1\}\; S\_j.$

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

See also

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Inclusion-exclusion principle

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Probability theory | Inequalities

Bonferroni-Ungleichung | Disuguaglianze di Boole e di Bonferroni