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. Boole's inequality is named after George Boole.

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

${\mathbb P}\biggl(\bigcup_{i} A_i\biggr) \le \sum_i {\mathbb P}(A_i).$

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

Proof

Boole's inequality may be proved for finite collections of events using the method of induction.

For the $n=1$ case, it follows that

$\mathbb P(A_1) \le \mathbb P(A_1).$

For the case $n$, we have

${\mathbb P}\biggl(\bigcup_{i=1}^{n} A_i\biggr) \le \sum_{i=1}^{n} {\mathbb P}(A_i).$

Since $\mathbb P(A \cup B) = \mathbb P(A) + \mathbb P(B) - \mathbb P(A \cap B),$ and because the union operation is associative, we have

${\mathbb P}\biggl(\bigcup_{i=1}^{n+1} A_i\biggr) = {\mathbb P}\biggl(\bigcup_{i=1}^n A_i\biggr) + \mathbb P(A_{n+1}) - {\mathbb P}\biggl(\bigcup_{i=1}^n A_i \cap A_{n+1}\biggr).$

Since

${\mathbb P}\biggl(\bigcup_{i=1}^n A_i \cap A_{n+1}\biggr) \ge 0,$

by the first axiom of probability, we have

${\mathbb P}\biggl(\bigcup_{i=1}^{n+1} A_i\biggr) \le {\mathbb P}\biggl(\bigcup_{i=1}^n A_i\biggr) + \mathbb P(A_{n+1})$,

and therefore

${\mathbb P}\biggl(\bigcup_{i=1}^{n+1} A_i\biggr) \le \sum_{i=1}^{n} {\mathbb P}(A_i) + \mathbb P(A_{n+1}) = \sum_{i=1}^{n+1} {\mathbb P}(A_i)$.

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, see Bonferroni (1936).

Define

$S_1 := \sum_{i=1}^n {\mathbb P}(A_i),$

and

$S_2 := \sum_{1\le i<j\le n} {\mathbb P}(A_i \cap A_j),$

as well as

$S_k := \sum_{1\le i_1<\cdots<i_k\le n} {\mathbb P}(A_{i_1}\cap \cdots \cap A_{i_k} )$

for all integers k in {3, ..., n}.

Then, for odd k in {1, ..., n},

${\mathbb P}\biggl( \bigcup_{i=1}^n A_i \biggr) \le \sum_{j=1}^k (-1)^{j-1} S_j,$

and for even k in {2, ..., n},

${\mathbb P}\biggl( \bigcup_{i=1}^n A_i \biggr) \ge \sum_{j=1}^k (-1)^{j-1} S_j.$

Boole's inequality is recovered by setting k = 1. When k = n, then equality holds and the resulting identity is the inclusion–exclusion principle.

See also

• Diluted inclusion–exclusion principle
• Schuette–Nesbitt formula
• Boole–Fréchet inequalities

References

This article incorporates material from Bonferroni inequalities on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.