# Probability theory

**theory of probabilityprobabilityprobability theoristprobabilisticprobabilistprobabilistic argumentmathematical probabilityprobabilistslaws of probabilitymeasure-theoretic probability theory**

Probability theory is the branch of mathematics concerned with probability.wikipedia

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### Probability

**probabilisticprobabilitieschance**

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.

### Probability interpretations

**philosophy of probabilityinterpretation of probabilityinterpretations of probability**

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.

### Probability space

**probability measuresGaussian measureoutcomes**

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.

### Sample space

**event spacespacerepresented by points**

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.

### Event (probability theory)

**eventeventsrandom event**

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.

### Stochastic process

**stochastic processesstochasticrandom process**

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.

### Random variable

**random variablesrandom variationrandom**

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.

### Law of large numbers

**strong law of large numbersweak law of large numbersBernoulli's Golden Theorem**

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.

### Central limit theorem

**Lyapunov's central limit theoremlimit theoremscentral limit**

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.

### Statistical mechanics

**statistical thermodynamicsstatistical mechanicalnon-equilibrium statistical mechanics**

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.

### Statistics

**statisticalstatistical analysisstatistician**

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.

### Probability and statistics

**probability, statistics**

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.

### Problem of points

**divide the stakes fairlygambling problemproblem of the points**

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.

### Blaise Pascal

**PascalPascal, BlaisePascalian**

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.

### Andrey Kolmogorov

**KolmogorovAndrei KolmogorovA. N. Kolmogorov**

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

**HuygensChristian HuygensChristiaan Huyghens**

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.

### Combinatorics

**combinatorialcombinatorial mathematicscombinatorial analysis**

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.

### Probability axioms

**axioms of probabilityKolmogorov axiomsaxioms**

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.

### Richard von Mises

**von MisesRichard Edler von Misesvon Mises, Richard**

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.

### Probability distribution

**distributioncontinuous probability distributiondiscrete probability distribution**

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.

### Elementary event

**elementaryelementary eventssample points**

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.

### Mathematics

**mathematicalmathmathematician**

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.

### Measure (mathematics)

**measuremeasure theorymeasurable**

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.

### Bertrand paradox (probability)

**Bertrand's paradoxBertrand paradoxBertrand's paradox (probability)**

See Bertrand's paradox.

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

### Sigma-algebra

**σ-algebraσ-algebrasigma algebra**

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