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Boole's inequality

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

\Pr\left[\bigcup_{i} A_i\right] \leq \sum_i \Pr\left[A_i\right].

Bonferroni inequalities

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 := \sum_{i=1}^n \Pr(A_i),
S_2 := \sum_{i<j} \Pr(A_i \cap A_j),

and for 2 < k ≤ n,

S_k := \sum \Pr(A_{i_1}\cap \cdots \cap A_{i_k} ),

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

Then, for odd k ≥ 1,

\Pr\left( \bigcup_{i=1}^n A_i \right) \leq \sum_{j=1}^k (-1)^{j+1} S_j,

and for even k ≥ 2,

\Pr\left( \bigcup_{i=1}^n A_i \right) \geq \sum_{j=1}^k (-1)^{j+1} S_j.

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

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