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] It is noteworthy that 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 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 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

References

  1. 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. 1. Terenin Alexander, 2. David Draper. "Cox's Theorem and the Jaynesian Interpretation of Probability" (PDF). arXiv. Retrieved November 19, 2019.
  4. Hájek, Alan (August 28, 2019). "Interpretations of Probability". Stanford Encyclopedia of Philosophy. Retrieved November 17, 2019.
  5. Ross, Sheldon M.,. A first course in probability (Ninth edition ed.). Upper Saddle River, New Jersey. pp. 27, 28. ISBN 978-0-321-79477-2. OCLC 827003384.CS1 maint: extra punctuation (link) CS1 maint: multiple names: authors list (link) CS1 maint: extra text (link)
  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 reading

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