Probability axioms

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]


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

First axiomEdit

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 axiomEdit

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 axiomEdit

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.


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 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.

Proof of monotonicity[5]Edit

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  :




Further consequencesEdit

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.

See alsoEdit


  1. ^ a b Kolmogorov, Andrey (1950) [1933]. Foundations of the theory of probability. New York, USA: Chelsea Publishing Company.
  2. ^ Aldous, David. "What is the significance of the Kolmogorov axioms?". David Aldous. Retrieved November 19, 2019.
  3. ^ Terenin Alexander; David Draper (2015). "Cox's Theorem and the Jaynesian Interpretation of Probability". arXiv:1507.06597. Bibcode:2015arXiv150706597T. Cite journal requires |journal= (help)
  4. ^ Hájek, Alan (August 28, 2019). "Interpretations of Probability". Stanford Encyclopedia of Philosophy. Retrieved November 17, 2019.
  5. ^ 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.
  6. ^ Gerard, David (December 9, 2017). "Proofs from axioms" (PDF). Retrieved November 20, 2019.
  7. ^ Jackson, Bill (2010). "Probability (Lecture Notes - Week 3)" (PDF). School of Mathematics, Queen Mary University of London. Retrieved November 20, 2019.

Further readingEdit