Probability axioms

The probability axioms do not specify or assume any particular interpretation of probability, but may be motivated by starting from a philosophical definition of probability and arguing that the axioms are satisfied by this definition. For example,

• Cox's theorem derives the laws of probability based on a "logical" definition of probability as the likelihood or credibility of arbitrary logical propositions.
• The Dutch book arguments show that rational agents must make bets which are in proportion with a subjective measure of the probability of events.

The third axiom, σ-additivity, is relatively modern, and originates with Lebesgue's measure theory. Some authors replace this with the strictly weaker axiom of finite additivity, which is sufficient to deal with some applications.

Kolmogorov axioms

In order to state the Kolmogorov axioms, the following pieces of data must be specified:

• The sample space, $\mathrm{\Omega }$, which is the set whose members comprise all possible outcomes or elementary events.
• The space of all events, each of which is some set of outcomes (i.e., some subset of $\mathrm{\Omega }$). The event space, $F$, must be a σ-algebra on $\mathrm{\Omega }$.
• The probability measure $P$ which assigns to each event ${\displaystyle E\in F}$ its probability, ${\displaystyle P(E)}$.

Taken together, these assumptions mean that ${\displaystyle (\mathrm{\Omega },F,P)}$ is a measure space. It is additionally assumed that ${\displaystyle P(\mathrm{\Omega })=1}$, making this triple a probability space.

First axiom

The probability of an event is a non-negative real number. This assumption is implied by the fact that ${\displaystyle P}$ is a measure on ${\displaystyle F}$.

${\displaystyle P(E)\ge 0\mathrm{\forall }E\in F}$

Theories which assign negative probability relax the first axiom.

Second axiom

This is the assumption of unit measure: that the probability that one of the elementary events in the entire sample space will occur is 1.$${\displaystyle P(\mathrm{\Omega })=1}$$From this axiom it follows that ${\displaystyle P(E)}$ is always finite, in contrast with more general measure theory.

Third axiom

This is the assumption of σ-additivity: Any countable sequence of disjoint events (synonymous with mutually exclusive events) ${\displaystyle E_1,E_2,\dots }$ satisfies

${\displaystyle P\left(\bigcup _{i=1}^{\mathrm{\infty }}{E}_i\right)=\sum _{i=1}^{\mathrm{\infty }}P(E_i).}$

This property again is implied by the fact that ${\displaystyle P}$ is a measure. Note that, by taking ${\displaystyle E_1=\mathrm{\Omega }}$ and ${\displaystyle E_i=\mathrm{\varnothing }}$ for all ${\displaystyle i>1}$, one deduces that ${\displaystyle P(\mathrm{\varnothing })=0}$. This in turn shows that σ-additivity implies finite additivity.

Some authors consider merely finitely additive probability spaces, in which case one just needs an algebra of sets, rather than a σ-algebra. Quasiprobability distributions in general relax the third axiom.