Axioms of Probability

In probability theory, axioms are fundamental principles that form the foundation of the mathematical framework for dealing with uncertainty and randomness. The axioms of probability, also known as Kolmogorov's axioms, were introduced by Russian mathematician Andrey Kolmogorov in the early 20th century. These axioms lay out the rules that any valid probability measure must satisfy. There are three axioms:

1. Non-Negativity Axiom: The probability of an event cannot be negative. For any event E, the probability P(E) is greater than or equal to 0: 0 ≤ P(E).

2. Additivity Axiom: The probability of the union of two mutually exclusive events is equal to the sum of their individual probabilities. For any two disjoint events E and F (meaning they have no outcomes in common), the probability of their union is given by: P(E ∪ F) = P(E) + P(F).

3. Normalization (Unit Measure) Axiom: The probability of the entire sample space is 1. The sample space is the set of all possible outcomes of an experiment. For any sample space S, the probability of S is 1: P(S) = 1.

By adhering to these axioms, probability theory ensures that the probabilities assigned to events are consistent and meaningful, facilitating the analysis and quantification of uncertainty in various fields, including statistics, science, finance, and decision-making.