Probability is the mathematical framework for reasoning about chance and uncertainty. It assigns numbers between 0 and 1 to events, where 0 means impossible and 1 means certain. Though people had informally reasoned about chance for millennia, the rigorous mathematical theory began with a 17th-century gambling dispute and was fully formalised by Andrey Kolmogorov in 1933.
You know a coin flip has a "50-50" chance of landing heads. You know a die has a "1 in 6" chance of showing any particular number. Probability is the mathematics that makes these intuitions precise and lets us calculate the chances of much more complicated events.
In 1654, a French gambler named Chevalier de Méré asked mathematician Blaise Pascal a question about how to fairly divide the stakes of an interrupted gambling game. Pascal corresponded with Pierre de Fermat (yes, the same Fermat of Fermat's Last Theorem) to work out the answer — and in doing so, they laid the foundations of probability theory.
Probability underlies weather forecasts, insurance pricing, medical testing, game design, quality control in manufacturing, and the algorithms behind spam filters and recommendation systems. Almost every field that deals with uncertainty relies on probability theory.
A probability function P assigns a number to events in a sample space S, satisfying:
These three simple axioms, formalised by Andrey Kolmogorov, are enough to derive the entire mathematical theory of probability using measure theory.
P(A|B) — "the probability of A given that B has occurred" — is defined as P(A∩B)/P(B), provided P(B) > 0. This captures how new information updates our assessment of likelihood.
Events A and B are independent if knowing one occurred doesn't change the probability of the other: P(A|B) = P(A), equivalently P(A∩B) = P(A)·P(B). Two coin flips are independent; drawing cards without replacement generally is not.
Named after Thomas Bayes (1763, published posthumously), this formula lets you reverse conditional probabilities — critical in medical testing (updating disease probability given a test result), spam filtering, and modern machine learning.
A random variable assigns numerical values to outcomes of a random process. A probability distribution describes how likely each value is. The most famous distribution is the normal distribution (bell curve), which arises naturally whenever many small independent effects combine — described mathematically by the Central Limit Theorem.
Kolmogorov's 1933 monograph Grundbegriffe der Wahrscheinlichkeitsrechnung (Foundations of the Theory of Probability) placed probability on rigorous footing by defining it as a special case of measure theory: a probability space is a measure space (Ω, F, P) where Ω is the sample space, F is a σ-algebra of "measurable" events, and P is a measure with P(Ω) = 1. This unification allowed probability theory to draw on the full machinery of Lebesgue integration and functional analysis, resolving centuries of informal and sometimes inconsistent reasoning about probability in continuous settings (e.g., "the probability of picking an exact real number from an interval is 0, yet some number is picked").
Beyond the mathematical formalism, there's philosophical debate about what probability means. The frequentist interpretation defines probability as the long-run frequency of an event over many repeated trials. The Bayesian interpretation treats probability as a degree of belief or confidence, which can apply even to one-off events (like "the probability it will rain tomorrow" or "the probability a given scientific hypothesis is true") that can't naturally be repeated. Both interpretations use the same mathematical axioms but differ in how they're applied and understood — a debate still active in statistics and philosophy of science.
The Law of Large Numbers (Bernoulli, 1713, with rigorous versions by Chebyshev and Kolmogorov later) states that as you repeat a random experiment more and more times, the average of the results converges to the expected value. The Central Limit Theorem (de Moivre 1733 for the binomial case; general form by Lyapunov 1901 and Lindeberg 1922) states that the sum (or average) of many independent random variables tends toward a normal distribution, regardless of the original distribution of each variable — explaining why the bell curve appears so ubiquitously in nature and statistics.
Before Kolmogorov, extensive work on probability was done by Jacob Bernoulli (Ars Conjectandi, 1713, published posthumously, containing the Law of Large Numbers), Pierre-Simon Laplace (Théorie analytique des probabilités, 1812, developing much of classical probability theory), and others — but without a fully rigorous axiomatic foundation, leading to occasional paradoxes and inconsistencies in edge cases (particularly involving continuous probability distributions) that Kolmogorov's measure-theoretic approach ultimately resolved.