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 (Ω, F, P) be a measure space with being the probability of some event E, and . Then (Ω, F, P) is a probability space, with sample space Ω, event space F and probability measure P.
The probability of an event is a non-negative real number:
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. Quasiprobability distributions in general relax the third axiom.
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 . From the properties of the empty set ( ), 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 setEdit
In some cases, is not the only event with probability 0.
Proof of probability of the empty setEdit
As shown in the previous proof, . This statement can be proved by contradiction: if then the left hand side is infinite;
If we have a contradiction, because the left hand side is infinite while must be finite (from the first axiom). Thus, . We have shown as a byproduct of the proof of monotonicity that .
The complement ruleEdit
Proof of the complement ruleEdit
Given and are mutually exclusive and that :
... (by axiom 3)
and, ... (by axiom 2)
The numeric boundEdit
It immediately follows from the monotonicity property that
Proof of the numeric boundEdit
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 an event in A or B will happen is the sum of the probability of an event in A and the probability of an event in B, minus the probability of an event that is in both A and B. The proof of this is as follows:
- ... (by Axiom 3)
- (by ).
and eliminating from both equations gives us the desired result.
An extension of the addition law to any number of sets is the inclusion–exclusion principle.
Setting B to the complement Ac of A in the addition law gives
That is, the probability that any event will not happen (or the event's complement) is 1 minus the probability that it will.
Simple example: coin tossEdit
Consider a single coin-toss, and assume that the coin will either land heads (H) or tails (T) (but not both). No assumption is made as to whether the coin is fair.
We may define:
Kolmogorov's axioms imply that:
The probability of neither heads nor tails, is 0.
The probability of either heads or tails, is 1.
The sum of the probability of heads and the probability of tails, is 1.
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- Formal definition of probability in the Mizar system, and the list of theorems formally proved about it.