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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. An alternative approach to formalising probability, favoured by some Bayesians, is given by Cox's theorem.
The assumptions as to setting up the axioms can be summarised as follows: Let (?, F, P) be a measure space with being the probability of some event E, and = 1. Then (?, F, P) is a probability space, with sample space ?, event space F and probability measure P.
The probability of an event is a non-negative real number:
where is the event space. It follows that is always finite, in contrast with more general measure theory. Theories which assign negative probability relax the first axiom.
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
From the Kolmogorov axioms, one can deduce other useful rules for studying probabilities. The proofs of these rules are a very insightful procedure that illustrates the power of the third axiom, and its interaction with the remaining two axioms. Four of the immediate corollaries and their proofs are shown below:
If A is a subset of, or equal to B, then the probability of A is less than, or equal to the probability of B.
In order to verify the monotonicity property, we set and , where and for . It is easy to see that the sets are pairwise disjoint and . Hence, we obtain from the third axiom that
Since, by the first axiom, the left-hand side of this equation is a series of non-negative numbers, and since it converges to which is finite, we obtain both and .
The probability of the empty set
In some cases, is not the only event with probability 0.
Proof of probability of the empty set
As shown in the previous proof, . However, this statement is seen by contradiction: if then the left hand side is not less than infinity;
If then we obtain a contradiction, because the sum does not exceed which is finite. Thus, . We have shown as a byproduct of the proof of monotonicity that .
The complement rule
Proof of the complement rule
Given and are mutually exclusive and that :
... (by axiom 3)
and, ... (by axiom 2)
The numeric bound
It immediately follows from the monotonicity property that
Proof of the numeric bound
Given the complement rule and axiom 1:
Another important property is:
This is called the addition law of probability, or the sum rule.
That is, the probability that AorB will happen is the sum of the probabilities that A will happen and that B will happen, minus the probability that both AandB will happen. The proof of this is as follows:
... (by Axiom 3)
and eliminating from both equations gives us the desired result.