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

${\mathbb {P} }{\biggl (}\bigcup _{i}A_{i}{\biggr )}\leq \sum _{i}{\mathbb {P} }(A_{i}).$

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

Proof

Proof using induction

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

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

$\mathbb {P} (A_{1})\leq \mathbb {P} (A_{1}).$

For the case $n$, we have

${\displaystyle {\mathbb {P} }{\biggl (}\bigcup _{i=1}^{n}A_{i}{\biggr )}\leq \sum _{i=1}^{n}{\mathbb {P} }(A_{i}).}$

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

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

Since

${\displaystyle {\mathbb {P} }{\biggl (}\bigcup _{i=1}^{n}A_{i}\cap A_{n+1}{\biggr )}\geq 0,}$

by the first axiom of probability, we have

${\displaystyle {\mathbb {P} }{\biggl (}\bigcup _{i=1}^{n+1}A_{i}{\biggr )}\leq {\mathbb {P} }{\biggl (}\bigcup _{i=1}^{n}A_{i}{\biggr )}+\mathbb {P} (A_{n+1})}$,

and therefore

${\displaystyle {\mathbb {P} }{\biggl (}\bigcup _{i=1}^{n+1}A_{i}{\biggr )}\leq \sum _{i=1}^{n}{\mathbb {P} }(A_{i})+\mathbb {P} (A_{n+1})=\sum _{i=1}^{n+1}{\mathbb {P} }(A_{i})}$.

Proof without using induction

For any events in ${\displaystyle A_{1}...A_{i}}$in our probability space we have

${\displaystyle \mathbb {P} (\bigcup _{i}A_{i})\leq \sum _{i}\mathbb {P} (A_{i})}$

One of the axioms of a probability space is that if ${\displaystyle B_{1}...B_{i}}$ are disjoint subsets of the probability space then

${\displaystyle \mathbb {P} (\bigcup _{i}B_{i})=\sum _{i}\mathbb {P} (B_{i})}$

this is called countable additivity.

If ${\displaystyle B\subset A}$ then ${\displaystyle \mathbb {P} (B)\leq \mathbb {P} (A)}$

Indeed, from the axioms of a probability distribution,

${\displaystyle \mathbb {P} (A)=\mathbb {P} (B)+\mathbb {P} (A-B)}$

Note that both terms on the right are nonnegative.

Now we have to modify the sets ${\displaystyle A_{i}}$, so they become disjoint.

${\displaystyle B_{i}=A_{i}-\bigcup _{j=1}^{i-1}A_{j}}$

So if ${\displaystyle B_{i}\subset A_{i}}$, then we know

${\displaystyle \bigcup _{i=1}^{\infty }B_{i}=\bigcup _{i=1}^{\infty }A_{i}}$

Therefore can we make following equation

${\displaystyle \mathbb {P} (\bigcup _{i}A_{i})=\mathbb {P} (\bigcup _{i}B_{i})=\sum _{i}\mathbb {P} (B_{i})\leq \sum _{i}\mathbb {P} (A_{i})}$

Bonferroni inequalities

Boole's inequality may be generalised to find upper and lower bounds on the probability of finite unions of events.^[1] 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

$S_{2}:=\sum _{1\leq i<j\leq n}{\mathbb {P} }(A_{i}\cap A_{j}),$

as well as

$S_{k}:=\sum _{1\leq i_{1}<\cdots <i_{k}\leq n}{\mathbb {P} }(A_{i_{1}}\cap \cdots \cap A_{i_{k}})$

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

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

${\mathbb {P} }{\biggl (}\bigcup _{i=1}^{n}A_{i}{\biggr )}\leq \sum _{j=1}^{k}(-1)^{j-1}S_{j},$

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

${\mathbb {P} }{\biggl (}\bigcup _{i=1}^{n}A_{i}{\biggr )}\geq \sum _{j=1}^{k}(-1)^{j-1}S_{j}.$

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

1. Casella, George; Berger, Roger L. (2002). Statistical Inference. Duxbury. pp. 11–13. ISBN 0-534-24312-6.

• Bonferroni, Carlo E. (1936), "Teoria statistica delle classi e calcolo delle probabilità", Pubbl. d. R. Ist. Super. di Sci. Econom. e Commerciali di Firenze (in Italian), 8: 1–62, Zbl 0016.41103
• Dohmen, Klaus (2003), Improved Bonferroni Inequalities via Abstract Tubes. Inequalities and Identities of Inclusion–Exclusion Type, Lecture Notes in Mathematics, 1826, Berlin: Springer-Verlag, pp. viii+113, ISBN 3-540-20025-8, MR 2019293, Zbl 1026.05009
• Galambos, János; Simonelli, Italo (1996), Bonferroni-Type Inequalities with Applications, Probability and Its Applications, New York: Springer-Verlag, pp. x+269, ISBN 0-387-94776-0, MR 1402242, Zbl 0869.60014
• Galambos, János (1977), "Bonferroni inequalities", Annals of Probability, 5 (4): 577–581, doi:10.1214/aop/1176995765, JSTOR 2243081, MR 0448478, Zbl 0369.60018
• Galambos, János (2001), "Bonferroni inequalities", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

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