Kolmogorov Axioms

Axioms introduced by Andrey Kolmogorov in 1933 which act as the foundations of probability theory

Axioms

Let () be a probability space. That is, is the sample space, is the event space, and is a probability function.

1. Probability is a Non-Negative Real Number

Axiom

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

(see universally quantified)

It follows that is always finite.

2. Unit Measure

Axiom

The probability that at least one of the events in the entire sample space will occur is 1:

3. Assumption of Sigma-Additivity

See sigma-additive

Axiom

Any countable sequence of disjoint (mutually exclusive events) satisfies:

As a result, we can also assume finite additivity.

Intuition

All this axiom says is, for example, if we have a bag with red and blue balls, the probability of drawing a blue or red ball is the same as the probability of drawing a blue ball plus the probability of drawing a red ball

Consequences

Monotonicity

See monotonicity

Proposition

If all the events of are also in event , then the probability of is less than or equal to the probability of .

That is,

(see subset)

Probability of the Empty Set

Proposition

Complement Rule

$

complement

Complementary Events

$

complement

Complement

Definition - Without explicit mention of the universe

is the set of elements not in

Definition - With explicit mention of the universe

is the set of all elements in but not in

Equivalent to

Alternate notation: , ,

Axiom - Absolute Complement

Law

De Morgan's laws:

in fact, this is how De Morgan's Laws are originally defined, and the complement of a boolean is actually defined by sets too.

Definition

A complement of an event is the opposite event, like the complement. The complement of event is written as .


and