The Kolmogorov axioms are the foundations of
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
introduced by Russian mathematician
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.
Axioms
The assumptions as to setting up the axioms can be summarised as follows: Let
be a
measure space with
being the
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
of some
event
Event may refer to:
Gatherings of people
* Ceremony, an event of ritual significance, performed on a special occasion
* Convention (meeting), a gathering of individuals engaged in some common interest
* Event management, the organization of ev ...
E'','' and
. Then
is a
probability space, with sample space
, event space
and
probability measure .
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 event
In probability theory, an elementary event, also called an atomic event or sample point, is an event which contains only a single outcome in the sample space. Using set theory terminology, an elementary event is a singleton. Elementary events a ...
s 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.
Quasiprobability distribution
A quasiprobability distribution is a mathematical object similar to a probability distribution but which relaxes some of Kolmogorov's axioms of probability theory. Quasiprobabilities share several of general features with ordinary probabilities, ...
s in general relax the third axiom.
Consequences
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:
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''
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 set
:
In many cases,
is not the only event with probability 0.
''Proof of probability of the empty set''
Define
for
, then these are disjoint, and
, hence by the third axiom
; subtracting
(which is finite by the first axiom) yields
. From this together with the first axiom follows
, thus
.
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 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:
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
*
*
*
*
*
*
*
References
Further reading
*
*
Formal definitionof probability in the
Mizar system, and th
list of theoremsformally proved about it.
{{DEFAULTSORT:Probability Axioms
Probability theory
Mathematical axioms