digplanet beta 1: Athena
Share digplanet:

Agriculture

Applied sciences

Arts

Belief

Business

Chronology

Culture

Education

Environment

Geography

Health

History

Humanities

Language

Law

Life

Mathematics

Nature

People

Politics

Science

Society

Technology

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[edit]

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[edit]

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[edit]

References[edit]

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


Original courtesy of Wikipedia: http://en.wikipedia.org/wiki/Boole's_inequality — Please support Wikipedia.
This page uses Creative Commons Licensed content from Wikipedia. A portion of the proceeds from advertising on Digplanet goes to supporting Wikipedia.
40 videos foundNext > 

Induction: Inequality Proofs

Proving inequalities with induction requires a good grasp of the 'flexible' nature of inequalities when compared to equations. Make sure that your logic is clear ...

Mod-01 Lec-04 Laws of Probability - I

Probability and Statistics by Dr.Somesh Kumar,Department of Mathematics,IIT Kharagpur. For more details on NPTEL visit http://nptel.iitm.ac.in.

Induction Inequality Proof Example 3: 5^n + 9 less than 6^n

Another viewer-submitted question. Inequality proofs seem particularly difficult when they involve powers of n, but they can be managed just like any other ...

Introduction to Inequality Proofs

I'm a high school mathematics and technology teacher from Sydney, Australia. My channel is all about learning - I love doing it, and I love helping others to do it ...

Proof of Jensen's Inequality

Proof of Jensen's Inequality for convex functions.

Inequalities Part One

How to solve an inequality in a linear equation- part one.. This is video 21 of 51 in the solving equation series. The next three series are "Word Problems", ...

Basic inequalities in probability theory

inequalities related to probabilities of unions and intersections.

Principles of Boolean Algebra

Principles of Boolean AlgebraAlgebra of logic is a mathematical device used to record, calculat, simplify, and transform logical propositions. Algebra of logic was ...

Between an Elusive Peace and a Haunting War

Mohamed ElBaradei, Director General Emeritus of the International Atomic Energy Agency, gave the Robert McNamara Lecture on War and Peace in the John ...

NOVA | Forgotten Genius

Watch a short trailer of the new NOVA "Forgotten Genius", which chronicles the amazing life story of Percy Julian an African American Scientist who many ...

40 videos foundNext > 

We're sorry, but there's no news about "Boole's inequality" right now.

Loading

Oops, we seem to be having trouble contacting Twitter

Support Wikipedia

A portion of the proceeds from advertising on Digplanet goes to supporting Wikipedia. Please add your support for Wikipedia!

Searchlight Group

Digplanet also receives support from Searchlight Group. Visit Searchlight