The **Kolmogorov axioms** are the foundations of probability theory introduced by Andrey Kolmogorov in 1933.^{[1]} These axioms remain central and have direct contributions to mathematics, the physical sciences, and real-world probability cases.^{[2]} An alternative approach to formalising probability, favoured by some Bayesians, is given by Cox's theorem.^{[3]}

## Axioms

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 *,* and = 1. Then (Ω, *F*, *P*) is a probability space, with sample space Ω, event space *F* and probability measure *P*.^{[1]}

### First axiom

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.

### Second 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

### Third axiom

This is the assumption of σ-additivity:

- Any countable sequence of disjoint sets (synonymous with
*mutually exclusive*events) satisfies

Some authors consider merely finitely additive probability spaces, in which case one just needs an algebra of sets, rather than a σ-algebra.^{[4]} Quasiprobability distributions in general relax the third axiom.

## Consequences

From the Kolmogorov axioms, one can deduce other useful rules for studying probabilities. The proofs^{[5]}^{[6]}^{[7]} of these rules are a very insightful procedure that illustrates the power the third axiom, and its interaction with the remaining two axioms. Four of the immediate corollaries and their proofs are shown below:

### Monotonicity

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.

*Proof of monotonicity*^{[5]}

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* :

## Further consequences

Another important property is:

This is called the addition law of probability, or the sum rule.
That is, the probability that *A* *or* *B* will happen is the sum of the probabilities that *A* will happen and that *B* will happen, minus the probability that both *A* *and* *B* will happen. The proof of this is as follows:

Firstly,

- ...
*(by Axiom 3)*

So,

- (by ).

Also,

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 *A ^{c}* 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 toss

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.

## See also

- Borel algebra
- σ-algebra
- Set theory
- Conditional probability
- Quasiprobability
- Fully probabilistic design

## References

- ^
^{a}^{b}Kolmogorov, Andrey (1950) [1933].*Foundations of the theory of probability*. New York, USA: Chelsea Publishing Company. **^**Aldous, David. "What is the significance of the Kolmogorov axioms?".*David Aldous*. Retrieved November 19, 2019.**^**Terenin Alexander; David Draper (2015). "Cox's Theorem and the Jaynesian Interpretation of Probability". arXiv:1507.06597. Bibcode:2015arXiv150706597T. Cite journal requires`|journal=`

(help)**^**Hájek, Alan (August 28, 2019). "Interpretations of Probability".*Stanford Encyclopedia of Philosophy*. Retrieved November 17, 2019.- ^
^{a}^{b}Ross, Sheldon M. (2014).*A first course in probability*(Ninth ed.). Upper Saddle River, New Jersey. pp. 27, 28. ISBN 978-0-321-79477-2. OCLC 827003384. **^**Gerard, David (December 9, 2017). "Proofs from axioms" (PDF). Retrieved November 20, 2019.**^**Jackson, Bill (2010). "Probability (Lecture Notes - Week 3)" (PDF).*School of Mathematics, Queen Mary University of London*. Retrieved November 20, 2019.

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## Further reading

- DeGroot, Morris H. (1975).
*Probability and Statistics*. Reading: Addison-Wesley. pp. 12–16. ISBN 0-201-01503-X. - McCord, James R.; Moroney, Richard M. (1964). "Axiomatic Probability".
*Introduction to Probability Theory*. New York: Macmillan. pp. 13–28. - Formal definition of probability in the Mizar system, and the list of theorems formally proved about it.