probability

Probability is a branch of mathematics that deals with the analysis of random events and the likelihood of their occurrence. It provides a quantitative framework for assessing uncertainty and predicting outcomes in systems influenced by chance. The probability of an event is expressed as a value between 0 and 1, where 0 indicates impossibility, 1 signifies certainty, and intermediate values reflect varying degrees of likelihood.

The concept of probability can be traced back to the 17th century, with contributions from mathematicians such as Blaise Pascal and Pierre de Fermat. It has since evolved into a fundamental component of statistics, gambling theory, actuarial science, and various scientific disciplines. In probability theory, events are classified as independent or dependent, and models often include probability distributions that describe how random variables behave. Common distributions include the normal, binomial, and Poisson distributions, each applicable under specific conditions.

Probability is formally defined through axioms established by Andrey Kolmogorov in 1933, which provide a rigorous mathematical foundation. These axioms underpin methods for calculating probabilities, such as classical, empirical, and subjective approaches. Classical probability involves equally likely outcomes, empirical probability is based on observed data, and subjective probability reflects personal belief in an event's likelihood.

Applications of probability are widespread, including risk assessment, decision-making under uncertainty, machine learning, and simulations. It enables individuals and organizations to make informed decisions by quantifying uncertainty and evaluating potential outcomes systematically. As a critical tool for understanding complex systems involving randomness, probability continues to influence various fields of science and technology.