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Probability axioms

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Foundations of probability theory

Probability theory
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The standardprobability axioms are the foundations ofprobability theory introduced by Russian mathematicianAndrey Kolmogorov in 1933.^[1] Theseaxioms remain central and have direct contributions to mathematics, the physical sciences, and real-world probability cases.^[2]

There are several other (equivalent) approaches to formalising probability.Bayesians will often motivate the Kolmogorov axioms by invokingCox's theorem or theDutch book arguments instead.^[3]^[4]

Kolmogorov axioms

The assumptions as to setting up the axioms can be summarised as follows: Let $(\Omega ,F,P)$ be ameasure space such that $P(E)$ is theprobability of someevent $E {\displaystyle E}$ , and $P(\Omega )=1$ . Then $(\Omega ,F,P)$ is aprobability space, with sample space $\Omega$ , event space $F {\displaystyle F}$ andprobability measure $P {\displaystyle P}$ .^[1]

First axiom

The probability of an event is a non-negative real number:

P(E)\in \mathbb {R} ,P(E)\geq 0\qquad \forall E\in F

where $F {\displaystyle F}$ is the event space. It follows (when combined with the second axiom) that $P(E)$ is always finite, in contrast with more generalmeasure theory. Theories which assignnegative probability relax the first axiom.

Second axiom

This is the assumption ofunit measure: that the probability that at least one of theelementary events in the entire sample space will occur is 1.

P(\Omega )=1

Third axiom

This is the assumption ofσ-additivity:

Anycountable sequence ofdisjoint sets (synonymous withmutually exclusive events)

E_{1},E_{2},\ldots

satisfies

P\left(\bigcup _{i=1}^{\infty }E_{i}\right)=\sum _{i=1}^{\infty }P(E_{i}).

Some authors consider merelyfinitely additive probability spaces, in which case one just needs analgebra of sets, rather than aσ-algebra.^[5]Quasiprobability distributions in general relax the third axiom.

Consequences

From the Kolmogorov axioms, one can deduce other useful rules for studying probabilities. The proofs^[6]^[7]^[8] of these rules are a very insightful procedure that illustrates the power of the third axiom, and its interaction with the prior two axioms. Four of the immediate corollaries and their proofs are shown below:

Monotonicity

\quad {\text{if}}\quad A\subseteq B\quad {\text{then}}\quad P(A)\leq P(B).

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

Source:^[6]

In order to verify the monotonicity property, we set $E_{1}=A$ and $E_{2}=B\setminus A$ , where $A\subseteq B$ and $E_{i}=\varnothing$ for $i\geq 3$ . From the properties of theempty set ( $\varnothing$ ), it is easy to see that the sets $E_{i}$ are pairwise disjoint and $E_{1}\cup E_{2}\cup \cdots =B$ . Hence, we obtain from the third axiom that

P(A)+P(B\setminus A)+\sum _{i=3}^{\infty }P(E_{i})=P(B).

Since, by the first axiom, the left-hand side of this equation is a series of non-negative numbers, and since it converges to $P(B)$ which is finite, we obtain both $P(A)\leq P(B)$ and $P(\varnothing )=0$ .

The probability of the empty set

P(\varnothing )=0.

In many cases, $\varnothing$ is not the only event with probability 0.

Proof of the probability of the empty set

$P(\varnothing \cup \varnothing )=P(\varnothing )$ since $\varnothing \cup \varnothing =\varnothing$ ,

$P(\varnothing )+P(\varnothing )=P(\varnothing )$ by applying the third axiom to the left-hand side (note $\varnothing$ is disjoint with itself), and so

$P(\varnothing )=0$ by subtracting $P(\varnothing )$ from each side of the equation.

The complement rule

$P\left(A^{\complement }\right)=P(\Omega -A)=1-P(A)$

Proof of the complement rule

Given $A {\displaystyle A}$ and $A^{\complement }$ are mutually exclusive and that $A\cup A^{\complement }=\Omega$ :

$P(A\cup A^{\complement })=P(A)+P(A^{\complement })$ ... (by axiom 3)

and, $P(A\cup A^{\complement })=P(\Omega )=1$ ...(by axiom 2)

$\Rightarrow P(A)+P(A^{\complement })=1$

$\therefore P(A^{\complement })=1-P(A)$

The numeric bound

It immediately follows from the monotonicity property that

0\leq P(E)\leq 1\qquad \forall E\in F.

Proof of the numeric bound

Given the complement rule $P(E^{c})=1-P(E)$ andaxiom 1 $P(E^{c})\geq 0$ :

$\Rightarrow 1\geq P(E)$

$\therefore 0\leq P(E)\leq 1$

Further consequences

Another important property is:

P(A\cup B)=P(A)+P(B)-P(A\cap B).

This is called the addition law of probability, or the sum rule.That is, the probability that an event inAorB will happen is the sum of the probability of an event inA and the probability of an event inB, minus the probability of an event that is in bothAandB. The proof of this is as follows:

Firstly,

P(A\cup B)=P(A)+P(B\setminus A)

.(by Axiom 3)

So,

P(A\cup B)=P(A)+P(B\setminus (A\cap B))

(by

B\setminus A=B\setminus (A\cap B)

).

Also,

P(B)=P(B\setminus (A\cap B))+P(A\cap B)

and eliminating $P(B\setminus (A\cap B))$ from both equations gives us the desired result.

An extension of the addition law to any number of sets is theinclusion–exclusion principle.

SettingB to the complementA^c ofA in the addition law gives

P\left(A^{c}\right)=P(\Omega \setminus A)=1-P(A)

That is, the probability that any event willnot happen (or the event'scomplement) 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 or as to whether or not any bias depends on how the coin is tossed.^[9]

We may define:

\Omega =\{H,T\}

F=\{\varnothing ,\{H\},\{T\},\{H,T\}\}

Kolmogorov's axioms imply that:

P(\varnothing )=0

The probability ofneither headsnor tails, is 0.

P(\{H,T\}^{c})=0

The probability ofeither headsor tails, is 1.

P(\{H\})+P(\{T\})=1

The sum of the probability of heads and the probability of tails, is 1.

See also

Borel algebra – Class of mathematical setsPages displaying short descriptions of redirect targets
Conditional probability – Probability of an event occurring, given that another event has already occurred
Fully probabilistic design
Intuitive statistics
Quasiprobability – Concept in statisticsPages displaying short descriptions of redirect targets
Set theory – Branch of mathematics that studies sets
σ-algebra – Algebraic structure of set algebraPages displaying short descriptions of redirect targets

References

^^a ^bKolmogorov, Andrey (1950) [1933].Foundations of the theory of probability. New York, US: Chelsea Publishing Company.
^Aldous, David."What is the significance of the Kolmogorov axioms?".David Aldous. RetrievedNovember 19, 2019.
^Cox, R. T. (1946). "Probability, Frequency and Reasonable Expectation".American Journal of Physics.14 (1):1–10.Bibcode:1946AmJPh..14....1C.doi:10.1119/1.1990764.
^Cox, R. T. (1961).The Algebra of Probable Inference. Baltimore, MD: Johns Hopkins University Press.
^Hájek, Alan (August 28, 2019)."Interpretations of Probability".Stanford Encyclopedia of Philosophy. RetrievedNovember 17, 2019.
^^a ^bRoss, 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.{{cite book}}: CS1 maint: location missing publisher (link)
^Gerard, David (December 9, 2017)."Proofs from axioms"(PDF). RetrievedNovember 20, 2019.
^Jackson, Bill (2010)."Probability (Lecture Notes - Week 3)"(PDF).School of Mathematics, Queen Mary University of London. RetrievedNovember 20, 2019.
^Diaconis, Persi; Holmes, Susan; Montgomery, Richard (2007)."Dynamical Bias in the Coin Toss"(PDF).SIAM Review.49 (211–235):211–235.Bibcode:2007SIAMR..49..211D.doi:10.1137/S0036144504446436. Retrieved5 January 2024.

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 theMizar system, and thelist of theorems formally proved about it.

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