Part of a series on statistics |

Probability theory |
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The **Kolmogorov axioms** are the foundations of probability theory introduced by Russian mathematician 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 be a measure space with being the probability of some event E*,* and . Then is a probability space, with sample space , event space and probability measure .^{[1]}

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.

See also: Unitarity (physics) |

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

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.

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 .

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

since ,

by applying the third axiom to the left-hand side (note is disjoint with itself), and so

by subtracting from each side of the equation.

Given and are mutually exclusive and that :

*... (by axiom 3)*

and, ... *(by axiom 2)*

It immediately follows from the monotonicity property that

Given the complement rule and *axiom 1* :

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

That is, the probability that any event will *not* happen (or the event's complement) is 1 minus the probability that it will.

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.