Three Axioms of Kolmogorov’s

These axioms, as stated below, are a reduced version of those proposed by mathematician Andrey Kolmogorov in 1933. The core concepts of probability theory had previously been “thought to be somewhat unique,” therefore his goal was to place them in their “natural home, among the general notions of modern mathematics.” To this purpose, Kolmogorov provided a mathematical explanation of what a “random event” is (in terms of sets). We skipped over this part when expressing the axioms above, but it’s discussed in Kolmogorov’s original publication Foundations of the Theory of Probability.

Kolmogorov’s axioms placed probability in the context of measure theory. You are assigning a number to some form of mathematical object when you measure something (such as length, area, or volume) (a line segment, a 2D shape, or a 3D shape). Probability is a method of assigning a number to a mathematical object in a comparable way (collections of events). Because of Kolmogorov’s definition, the mathematical theory of measures could include probability theory as a specific case.

The axioms:

Pictures depicting the three Kolmogorov axioms of probability.

Probability theory is based on the three Kolmogorov axioms. Before we get into the axioms, we’ll go over some basic probability terminology.

First axiom:

Probabilities are real numbers that aren’t negative.

A probability must be a real number that is not negative.

The smallest probability of an event is zero, according to this principle. A probability theorem, on the other hand, does provide an upper bound.

Second axiom: 

At least one result in the sample space will occur with a probability of 1.

This axiom states that viewing an experiment will result in a given outcome.

Third axiom:

In a sample space with an unlimited number of disconnected events, the probability of the union of events is equal to the sum of all event probabilities.

This postulate establishes a link between a collection of disconnected events in a sample space and their individual probability. A probability theorem demonstrates how a finite set of discontinuous events can also be expressed as an infinite set.

Theorems of probability can then be derived using the axioms of probability.

Example of probability:

1] An experiment’s trial is a single instance of the experiment. A new experiment can be created by combining multiple repetitions of an experiment.

A trial is one instance of a twice-rolled dice experiment.

2] The result is a trial’s observed output.

In the twice-rolled dice experiment, 

Examples include rolling a 3 or a 1 and 5.

3] The sample space is the set of all possible experiment outcomes.

Example: Experiments using each side of the dice or arranged pairs of twice-rolled dice. 

4] In the sample space, an event A is a subset of outcomes.

Example: Rolling a side that is less than 4, rolling an even number, or rolling a 2 followed by a 3.

Set theory procedures are applicable to occurrences. The set of outcomes in A, B, or both is the union of events. The collection of outcomes in both A and B is the intersection of events.

The null event is defined as an occurrence with no outcomes. If events A and B have no common consequences, they are considered disconnected events (mutually exclusive).

A measure of probability P is a function that gives each measurable event a real number. The axioms of probability must be followed by a probability measure.

Conclusion:

Andrey Kolmogorov introduced the Kolmogorov axioms in 1933 as the foundations of probability theory. These axioms are still important, and they have direct applications in mathematics, physics, and real-world probability. The core concepts of probability theory had previously been “thought to be somewhat unique,” therefore his goal was to place them in their “natural home, among the general notions of modern mathematics.

Kolmogorov’s axioms placed probability in the context of measure theory.

Probability is a method of assigning a number to a mathematical object in a comparable way (collections of events). Because of Kolmogorov’s definition, the mathematical theory of measures could include probability theory as a specific case.

Probability theory is based on the three Kolmogorov axioms. First one is that the probability must be a real number that is not negative. The smallest probability of an event is zero, according to this principle.

The second one is: At least one result in the sample space will occur with a probability of 1.

And the third one is: In a sample space with an unlimited number of disconnected events, the probability of the union of events is equal to the sum of all event probabilities.

Set theory procedures are applicable to occurrences. The set of outcomes in A, B, or both is the union of events. The null event is defined as an occurrence with no outcomes.