Probability axioms

In probability theory, the probability P of some event E, denoted P(E), is usually defined in such a way that P satisfies the Kolmogorov axioms, named after Andrey Kolmogorov, which are described below.

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.

An alternative approach to formalising probability, favoured by some Bayesians, is given by Cox's theorem.

First axiom

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

where F is the event space and E is any event in F.

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.

and

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.

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.

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

See also

• Cox's theorem

Further reading

• Von Plato, Jan, 2005, "Grundbegriffe der Wahrscheinlichtkeitsrechnung" in Grattan-Guinness, I., ed., Landmark Writings in Western Mathematics. Elsevier: 960-69. (in English)
• Glenn Shafer; Vladimir Vovk, The origins and legacy of Kolmogorov’s Grundbegriffe, http://www.probabilityandfinance.com/articles/04.pdf 

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.