Probability axioms

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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.

Contents

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. In particular, P(E) is always finite, in contrast with more general measure theory.

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:

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.

Monotonicity

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