Axiomatic definition of probability‌

The concept of axiomatic probability is based on the theory provided by Andrey Kolmogorov. The Kolmogorov axioms are the basis of axiomatic probability and are greatly concerned with real-world probability occurrence along with the usage in the field of Mathematics and Science.

Axiomatic Probability

The probability of an event is the possibility of occurrence of that particular event in comparison to the rest of the events. For example, while tossing a coin, the probability of getting heads is

Probability of a heads = (number of heads in a coin)/(total number of outcomes)

P(heads) = ½

In general, the probability of an event to occur out of the total number of given events is known as the probability of that particular event. The standard representation of probability of an event is P(E).

Conditions of probability

The probability of an event to occur can never be negative.
The probability of an event is always positive and greater than 0 i.e.
P(E) ≥ 0
The sum of probabilities of all the events is known as sample space. The sample space is always equal to unity.
P(E₁) + P(E₂) + …… + P(E_n) = 1

Here, P(E₁), P(E₂) and P(E_n) indicate the probability of event 1, probability of event 2 and so on for ‘n’ events.

Axioms of Kolmogorov

Axioms can be defined as statements or logics which are believed to be true and have no practical proof.

Kolmogorov axioms are the rules of probability that are beneficial in solving problems related to probability. The following are the three axioms provided by Kolmogorov.

Axiom 1: The probability of an event is not equal to 0

It states that the probability of an event is never equal to zero. It is always a positive non-zero real number.

P(E) ≠ 0

For every E ∈ R and P(E) ≥ 0.

Axiom 2: Assumption of unitary measure

This axiom is based on the assumption of unitary measures. This means that the probability of one of the elementary events which occur in the sample space will be equal to 1. Mathematically, it is represented as

P(Ω) = 1

Axiom 3: Additivity axiom

This axiom is also known as the axiom of additivity. It is applicable to disjoint sets or mutually exclusive sets of events of probability. According to this axiom, the following statement is satisfied by disjoint sets:

P(∪_i=1^∞ E_i) = ∑ _i=1^∞ . P(E_i)

which means that the union of the finite disjoint events of probability is equal to the summation of all the finite events of probability.