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 nonnegative 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 σadditivity:
Some authors consider merely finitely additive probability spaces, in which case one just needs an algebra of sets, rather than a σalgebra.
Consequences
From the Kolmogorov axioms, one can deduce other useful rules for calculating probabilities.
Monotonicity
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