Boole's inequality

Statement

Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $A_1, A_2, A_3, \dots$ be a finite or infinite sequence of events. Then $$P\left(\bigcup_{i} A_i\right) \le \sum_{i} P(A_i).$$

Remarks

The inequality is also called the union bound. It needs no hypothesis on the events: they need not be disjoint or independent. Equality holds when the events are pairwise disjoint. The Bonferroni inequalities refine it with the terms of the inclusion–exclusion principle. For example, $P\left(\bigcup_{i=1}^n A_i\right) \ge \sum_i P(A_i) - \sum_{i < j} P(A_i \cap A_j)$.

History

The inequality is named after George Boole.

Explanations

L1 human

The chance that at least one of several bad things happens is at most the sum of their chances. If each of $10$ parts fails with probability $0.01$, then some part fails with probability at most $0.1$.