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=                         Probability axioms                         =
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                             Introduction
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The Kolmogorov Axioms are the foundations of Probability Theory
introduced by Andrey Kolmogorov in 1933. These axioms remain central
and have direct contributions to mathematics, the physical sciences,
and real-world probability cases. It is noteworthy that an alternative
approach to formalising probability, favoured by some Bayesians, is
given by Cox's theorem.


                                Axioms
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The assumptions as to setting up the axioms can be summarised as
follows: Let (Ω, 'F', 'P') be a measure space with 'P' being the
probability of some event 'E', denoted P(E)',' and P(\Omega) = 1. Then
(Ω, 'F', 'P') is a probability space, with sample space Ω, event space
'F' and probability measure 'P'.


 {{Anchor|Non-negativity}}First axiom
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The probability of an event is a non-negative real number:
:P(E)\in\mathbb{R}, P(E)\geq 0 \qquad \forall E \in F

where F is the event space. It follows that P(E) is always finite, in
contrast with more general measure theory.  Theories which assign
negative probability relax the first axiom.


 {{Anchor|Unitarity|Normalization}}Second axiom
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This is the assumption of unit measure: that the probability that at
least one of the elementary events in the entire sample space will
occur is 1

: P(\Omega) = 1.


 {{Anchor|Sigma additivity|Finite additivity|Countable additivity|Finitely addit
ive}}Third axiom
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