Probability axioms

In Kolmogorov's probability theory, the probability P of some event E, denoted ${\displaystyle P(E)}$, is usually defined such that P satisfies the Kolmogorov axioms, named after the Russian mathematician Andrey Kolmogorov, which are described below.

These assumptions can be summarised as follows: 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.

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

where ${\displaystyle F}$ is the event space. In particular, ${\displaystyle P(E)}$ is always finite, in contrast with more general measure theory. Theories which assign negative probability relax the first axiom.

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