Chance became rigorous when Kolmogorov wrote axioms
Gamblers asked first; physicists later found atoms behaving randomly. Probability theory turns that fog into a measure between zero and one on a sample space. Two classic theorems on averages and limiting shapes explain why chaos often settles into patterns.
Probability theory is the mathematical branch that handles chance rigorously through axioms, typically via a probability space: a sample space of outcomes, subsets called events, and a probability measure valued between zero and one. Interpretations of what probability means vary, yet the calculus stays formal. Core objects include discrete and continuous random variables, distributions, and stochastic processes that model single shocks or evolving uncertainty.
Perfect prediction of random events is impossible, yet long-run average laws and limiting-normal theorems describe how chance settles. As a foundation for statistics, the theory underwrites quantitative work wherever data matter. It also describes complex systems known only partly, as in statistical mechanics, and twentieth-century physics found probabilistic structure at atomic scales in quantum mechanics—though that field uses a different probability theory.
Roots reach Gerolamo Cardano's sixteenth-century gambling analysis and seventeenth-century work by Pierre de Fermat and Blaise Pascal, including the problem of points. Christiaan Huygens published on the subject in 1657; Pierre Laplace finished what is often called the classical definition in the nineteenth century. Early methods were combinatorial for discrete events before continuous variables forced analysis into the picture.
Andrey Kolmogorov fused Richard von Mises's sample-space idea with measure theory and issued his axiom system in 1933, still the dominant basis, though Bruno de Finetti preferred finite over countable additivity. A fair die's sample space has six faces; the event of an odd roll is the set one, three, five. Mutually exclusive pieces add: chances for one-or-six, three alone, and two-or-four sum to five sixths. Discrete theory covers dice, cards, walks, and coins; continuous theory needs cumulative distribution functions once classical counting breaks, as Bertrand's paradox shows.
Source: Probability theory
