Probability theory

Probability theory is the branch of mathematics concerned with probability. The modern mathematical theory of probability has its roots in attempts to analyze games of chance by Gerolamo Cardano in the sixteenth century, and by Pierre de Fermat and Blaise Pascal in the seventeenth century (for example the "problem of points").

These concepts have been given an axiomatic mathematical formalization in probability theory, which is used widely in such areas of study as mathematics, statistics, finance, gambling, science (in particular physics), artificial intelligence/machine learning, computer science, game theory, and philosophy to, for example, draw inferences about the expected frequency of events.

Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms.

In answering such questions, mathematicians interpret the probability values of probability theory.

Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space.

In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process (or “experiment”) consisting of states that occur randomly.

Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Kolmogorov combined the notion of sample space, introduced by Richard von Mises, and measure theory and presented his axiom system for probability theory in 1933.

In probability theory, the sample space (also called sample description space or possibility space ) of an experiment or random trial is the set of all possible outcomes or results of that experiment.

Any specified subset of these outcomes is called an event.

In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned.

Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes, which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion.

In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a family of random variables.

Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes, which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion.

The formal mathematical treatment of random variables is a topic in probability theory.

Two major results in probability theory describing such behaviour are the law of large numbers and the central limit theorem.

In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times.

Two major results in probability theory describing such behaviour are the law of large numbers and the central limit theorem.

In probability theory, the central limit theorem (CLT) establishes that, in some situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution (informally a "bell curve") even if the original variables themselves are not normally distributed.

Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics.

The approach is based on statistical methods, probability theory and the microscopic physical laws.

As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of data.

Inferences on mathematical statistics are made under the framework of probability theory, which deals with the analysis of random phenomena.

The earliest known forms of probability and statistics were developed by Arab mathematicians studying cryptography between the 8th and 13th centuries.

However, probability theory contains much that is mostly of mathematical interest and not directly relevant to statistics.

The modern mathematical theory of probability has its roots in attempts to analyze games of chance by Gerolamo Cardano in the sixteenth century, and by Pierre de Fermat and Blaise Pascal in the seventeenth century (for example the "problem of points").

The problem of points, also called the problem of division of the stakes, is a classical problem in probability theory.

The modern mathematical theory of probability has its roots in attempts to analyze games of chance by Gerolamo Cardano in the sixteenth century, and by Pierre de Fermat and Blaise Pascal in the seventeenth century (for example the "problem of points").

Pascal was an important mathematician, helping create two major new areas of research: he wrote a significant treatise on the subject of projective geometry at the age of 16, and later corresponded with Pierre de Fermat on probability theory, strongly influencing the development of modern economics and social science.

This culminated in modern probability theory, on foundations laid by Andrey Nikolaevich Kolmogorov.

Andrey Nikolaevich Kolmogorov (, 25 April 1903 – 20 October 1987) was a Soviet mathematician who made significant contributions to the mathematics of probability theory, topology, intuitionistic logic, turbulence, classical mechanics, algorithmic information theory and computational complexity.

Christiaan Huygens published a book on the subject in 1657 and in the 19th century, Pierre Laplace completed what is today considered the classic interpretation.

As a mathematician, Huygens was a pioneer on probability and wrote his first treatise on probability theory in 1657 with the work Van Rekeningh in Spelen van Gluck.

Initially, probability theory mainly considered events, and its methods were mainly combinatorial.

Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas.

Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Kolmogorov combined the notion of sample space, introduced by Richard von Mises, and measure theory and presented his axiom system for probability theory in 1933.

The Kolmogorov Axioms are the foundations of Probability Theory introduced by Andrey Kolmogorov in 1933.

Kolmogorov combined the notion of sample space, introduced by Richard von Mises, and measure theory and presented his axiom system for probability theory in 1933.

Richard Edler von Mises (19 April 1883 – 14 July 1953) was an Austrian scientist and mathematician who worked on solid mechanics, fluid mechanics, aerodynamics, aeronautics, statistics and probability theory.

Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes, which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion.

In probability theory and statistics, a probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment.

When doing calculations using the outcomes of an experiment, it is necessary that all those elementary events have a number assigned to them.

In probability theory, an elementary event (also called an atomic event or sample point) is an event which contains only a single outcome in the sample space.

Probability theory is the branch of mathematics concerned with probability.

Applied mathematics has significant overlap with the discipline of statistics, whose theory is formulated mathematically, especially with probability theory.

Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Kolmogorov combined the notion of sample space, introduced by Richard von Mises, and measure theory and presented his axiom system for probability theory in 1933. Given any set \Omega\, (also called ) and a σ-algebra on it, a measure P\, defined on is called a if

The main applications of measures are in the foundations of the Lebesgue integral, in Andrey Kolmogorov's axiomatisation of probability theory and in ergodic theory.

See Bertrand's paradox.

The Bertrand paradox is a problem within the classical interpretation of probability theory.

Given any set \Omega\, (also called ) and a σ-algebra on it, a measure P\, defined on is called a if

In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a collection Σ of subsets of X that