In probability theory, the probability P of some event E, denoted P(E), is defined in such a way that P satisfies the Kolmogorov axioms, named after Andrey Kolmogorov. Probability theory is the branch of mathematics concerned with analysis of random phenomena. ... Probability is the likelihood that something is the case or will happen. ... In probability theory, an event is a set of outcomes (a subset of the sample space) to which a probability is assigned. ... Andrey Nikolaevich Kolmogorov (Russian: Андре́й Никола́евич Колмого́ров) (April 25, 1903 - October 20, 1987) was a Soviet mathematician who made major advances in different academic fields (among them probability theory, topology, intuitionistic logic, turbulence, classical mechanics and computational complexity). ...

These assumptions can be summarised as: Let (Ω, F, P) be a measure space with P(Ω)=1. Then (Ω, F, P) is a probability space, with sample space Ω, event space F and probability measure P. In mathematics, a measure is a function that assigns a number, e. ...

First axiom

The probability of an event is a non-negative real number:

where F is the event space.

Second axiom

This is the assumption of unit measure: that the probability that some elementary event in the entire sample space will occur is 1. More specifically, there are no elementary events outside the sample space.

This is often overlooked in some mistaken probability calculations; if you cannot precisely define the whole sample space, then the probability of any subset cannot be defined either.

Third axiom

This is the assumption of σ-additivity:

Any countable sequence of pairwise disjoint events E₁,E₂,... satisfies

Some authors consider merely finitely-additive probability spaces, in which case one just needs an algebra of sets, rather than a σ-algebra. In mathematics the term countable set is used to describe the size of a set, e. ... The algebra of sets develops and describes the basic properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality (mathematics) and set inclusion. ... In mathematics, a σ-algebra (pronounced sigma-algebra) or σ-field over a set X is a collection Σ of subsets of X that is closed under countable set operations, meaning that the union or the intersection of countably many members of the algebra is also a member. ...

Consequences

From the Kolmogorov axioms one can deduce other useful rules for calculating probabilities:

This is called the addition law of probability, or the sum rule. That is, the probability that A or B will happen is the sum of the probabilities that A will happen and that B will happen, minus the probability that both A and B will happen. This can be extended to the inclusion-exclusion principle. In combinatorial mathematics, the inclusion-exclusion principle (also known as the sieve principle) states that if A1, ..., An are finite sets, then where |A| denotes the cardinality of the set A. For example, taking n = 2, we get a special case of double counting: in words, we can count the...

That is, the probability that any event will not happen is 1 minus the probability that it will.

See also

• Cox's theorem

Coxs theorem, named after the physicist Richard Threlkeld Cox, is a derivation of the laws of probability theory from a certain set of postulates. ...

Further reading

• Von Plato, Jan, 2005, "Grundbegriffe der Wahrscheinlichtkeitsrechnung" in Grattan-Guinness, I., ed., Landmark Writings in Western Mathematics. Elsevier: 960-69. (in English)

Ivor Grattan-Guinness (Born 23 June 1941, in Bakewell, England) is a prolific historian of mathematics and logic, at Middlesex University. ...

External links

• The Legacy of Andrei Nikolaevich Kolmogorov Curriculum Vitae and Biography. Kolmogorov School. Ph.D. students and descendants of A.N. Kolmogorov. A.N. Kolmogorov works, books, papers, articles. Photographs and Portraits of A.N. Kolmogorov.