- Probability axioms
In

probability theory , theprobability "P" of some event "E", denoted $P(E)$, is defined in such a way that "P" satisfies the**Kolmogorov axioms**, named afterAndrey Kolmogorov .These assumptions can be summarised as: Let (Ω, "F", "P") be a

measure space with "P"(Ω)=1. Then (Ω, "F", "P") is aprobability space , with sample space Ω, event space "F" and probability measure "P".**First axiom**The probability of an event is a non-negative real number::$P(E)geq\; 0\; qquad\; forall\; Ein\; F$

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.: $P(Omega)\; =\; 1.,$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 $E\_1,\; E\_2,\; ...$ satisfies $P(E\_1\; cup\; E\_2\; cup\; cdots)\; =\; sum\_i\; P(E\_i).$ Some authors consider merelyfinitely additive probability spaces, in which case one just needs analgebra of sets , rather than aσ-algebra .**Consequences**From the Kolmogorov axioms one can deduce other useful rules for calculating probabilities:

: $P(A\; cup\; B)\; =\; P(A)\; +\; P(B)\; -\; P(A\; cap\; B)$

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 theprobabilities that "A" will happen and that "B" will happen, minus theprobability that both "A" "and" "B" will happen. This can be extended to the

inclusion-exclusion principle .: $P(Omegasetminus\; E)\; =\; 1\; -\; P(E)$

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)

**External links*** [

*http://www.kolmogorov.com/ 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.

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