Boole's inequality

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


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

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

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 &le; 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 &ge; 1,
 * $$\Pr\left( \bigcup_{i=1}^n A_i \right) \leq \sum_{j=1}^k (-1)^{j+1} S_j,$$

and for even k &ge; 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.