Probability theory

A probability model begins with a process whose result is uncertain. Its sample space is the set of possible outcomes, and an event is a collection of outcomes of interest. A probability assigns each event a value from 0 to 1: 0 represents an impossible event in the model and 1 a certain one. When a finite set of outcomes is assumed to be equally likely, an event's probability can be calculated as the fraction of outcomes that belong to it. That shortcut depends on the equal-likelihood assumption; it is not a universal definition of probability.

In the standard mathematical framework, probabilities are non-negative, the probability of the full sample space is 1, and probabilities add across mutually exclusive events. These principles support rules for unions, complements, and intersections. Conditional probability describes the chance of an event when some other event is known to have occurred; for events A and B with P(B) greater than zero, it is P(A given B) = P(A and B) / P(B). Two events are independent when learning that one occurred does not change the probability assigned to the other; in the standard formulation, P(A and B) then equals P(A)P(B).

A random variable assigns a numerical value to each outcome, allowing questions about quantities such as the number of successes in repeated trials or a measurement produced by a random process. Its probability distribution describes how probability is allocated among possible values or ranges of values. The expected value is a probability-weighted average and, in repeated-trial settings, represents a long-run average—not a promise about the result of a single trial. Variance and standard deviation describe aspects of the distribution's spread.

Probability can be interpreted in more than one way. A frequentist approach relates it to the relative frequency expected over repeated trials, while Bayesian approaches represent degrees of belief that are updated in light of evidence. The formal rules help calculate consequences once probabilities and a model have been supplied, but do not by themselves determine the correct initial probabilities. Applications include statistical inference, scientific modeling, risk analysis, and decision-making. Conclusions therefore depend on the quality of the evidence and whether the model's assumptions fit the situation.

Mathematical probability grew in seventeenth-century Europe from analysis of games of chance, including correspondence between Blaise Pascal and Pierre de Fermat. Its modern axiomatic formulation came later: Andrey Kolmogorov's 1933 work set out a framework in which probabilities obey explicit mathematical rules. That framework made it possible to treat probability rigorously across finite and more complex sample spaces.