# Probability

**probabilisticprobabilitieschanceprobableImprobableprobability theoryprobabilisticallyprobability calculusImprobabilitylikelihood**

Probability is a measure quantifying the likelihood that events will occur.wikipedia

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### Probability theory

**theory of probabilityprobabilityprobability theorist**

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. Like other theories, the theory of probability is a representation of its concepts in formal terms—that is, in terms that can be considered separately from their meaning.

Probability theory is the branch of mathematics concerned with probability.

### Frequentist probability

**frequentistFrequency probabilitystatistical probability**

Frequentist probability or frequentism is an interpretation of probability; it defines an event's probability as the limit of its relative frequency in many trials.

### Probability axioms

**axioms of probabilityKolmogorov axiomsaxioms**

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.

The assumptions as to setting up the axioms can be summarised as follows: Let (Ω, F, P) be a measure space with P being the probability of some event E, denoted P(E), and P(\Omega) = 1.

### Gambling

**gamblerbettinggaming**

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.

Since these high payoffs have very low probability, a house bias can quite easily be missed unless the devices are checked carefully.

### Propensity probability

**propensitypropensitiesobjective probability**

The propensity theory of probability is one interpretation of the concept of probability.

### Inductive reasoning

**inductioninductiveinductive logic**

In a sense, this differs much from the modern meaning of probability, which, in contrast, is a measure of the weight of empirical evidence, and is arrived at from inductive reasoning and statistical inference.

While the conclusion of a deductive argument is certain, the truth of the conclusion of an inductive argument may be probable, based upon the evidence given.

### Artificial intelligence

**AIA.I.artificially intelligent**

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.

By the late 1980s and 1990s, AI research had developed methods for dealing with uncertain or incomplete information, employing concepts from probability and economics.

### Odds

**4/15/13/1**

The sixteenth century Italian polymath Gerolamo Cardano demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes (which implies that the probability of an event is given by the ratio of favourable outcomes to the total number of possible outcomes ).

The odds for a possible event E are directly related to the (known or estimated) statistical probability of that event E. To express odds as a probability, or the other way around, requires a calculation.

### Gerolamo Cardano

**Girolamo CardanoCardanoJerome Cardan**

The sixteenth century Italian polymath Gerolamo Cardano demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes (which implies that the probability of an event is given by the ratio of favourable outcomes to the total number of possible outcomes ).

He was one of the most influential mathematicians of the Renaissance, and was one of the key figures in the foundation of probability and the earliest introducer of the binomial coefficients and the binomial theorem in the Western world.

### Ars Conjectandi

**1713**

Jakob Bernoulli's Ars Conjectandi (posthumous, 1713) and Abraham de Moivre's Doctrine of Chances (1718) treated the subject as a branch of mathematics.

Ars Conjectandi (Latin for "The Art of Conjecturing") is a book on combinatorics and mathematical probability written by Jacob Bernoulli and published in 1713, eight years after his death, by his nephew, Niklaus Bernoulli.

### Christiaan Huygens

**HuygensChristian HuygensChristiaan Huyghens**

Christiaan Huygens (1657) gave the earliest known scientific treatment of the subject.

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.

### Pierre de Fermat

**FermatPierre FermatFermat, Pierre de**

Aside from the elementary work by Cardano, the doctrine of probabilities dates to the correspondence of Pierre de Fermat and Blaise Pascal (1654).

He made notable contributions to analytic geometry, probability, and optics.

### Jacob Bernoulli

**Jakob BernoulliBernoulliJames Bernoulli**

Jakob Bernoulli's Ars Conjectandi (posthumous, 1713) and Abraham de Moivre's Doctrine of Chances (1718) treated the subject as a branch of mathematics.

However, his most important contribution was in the field of probability, where he derived the first version of the law of large numbers in his work Ars Conjectandi.

### Daniel Bernoulli

**BernoulliDanielBernoulli, Daniel**

Daniel Bernoulli (1778) introduced the principle of the maximum product of the probabilities of a system of concurrent errors.

He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics.

### Blaise Pascal

**PascalPascal, BlaisePascalian**

Aside from the elementary work by Cardano, the doctrine of probabilities dates to the correspondence of Pierre de Fermat and Blaise Pascal (1654).

In 1654, prompted by his friend the Chevalier de Méré, he corresponded with Pierre de Fermat on the subject of gambling problems, and from that collaboration was born the mathematical theory of probabilities.

### Italians

**ItalianItalian peopleItalian descent**

The sixteenth century Italian polymath Gerolamo Cardano demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes (which implies that the probability of an event is given by the ratio of favourable outcomes to the total number of possible outcomes ).

Gerolamo Cardano established the foundation of probability and introduced the binomial coefficients and the binomial theorem; he also invented several mechanical devices.

### Event (probability theory)

**eventeventsrandom event**

Probability is a measure quantifying the likelihood that events will occur. In Kolmogorov's formulation (see probability space), sets are interpreted as events and probability itself as a measure on a class of sets.

:This is especially common in formulas for a probability, such as

### Least squares

**least-squaresmethod of least squaresleast squares method**

Adrien-Marie Legendre (1805) developed the method of least squares, and introduced it in his Nouvelles méthodes pour la détermination des orbites des comètes (New Methods for Determining the Orbits of Comets).

### Richard Dedekind

**DedekindJulius Wilhelm Richard DedekindR. Dedekind**

In the nineteenth century authors on the general theory included Laplace, Sylvestre Lacroix (1816), Littrow (1833), Adolphe Quetelet (1853), Richard Dedekind (1860), Helmert (1872), Hermann Laurent (1873), Liagre, Didion, and Karl Pearson.

Dedekind returned to Göttingen to teach as a Privatdozent, giving courses on probability and geometry.

### Stochastic process

**stochastic processesstochasticrandom process**

Andrey Markov introduced the notion of Markov chains (1906), which played an important role in stochastic processes theory and its applications.

The study of stochastic processes uses mathematical knowledge and techniques from probability, calculus, linear algebra, set theory, and topology as well as branches of mathematical analysis such as real analysis, measure theory, Fourier analysis, and functional analysis.

### Theory

**theoreticaltheoriestheorist**

Like other theories, the theory of probability is a representation of its concepts in formal terms—that is, in terms that can be considered separately from their meaning.

Obvious examples include arithmetic (abstracting concepts of number), geometry (concepts of space), and probability (concepts of randomness and likelihood).

### Probability space

**probability measuresGaussian measureoutcomes**

In Kolmogorov's formulation (see probability space), sets are interpreted as events and probability itself as a measure on a class of sets.

### Conditional probability

**conditional probabilitiesconditionalconditioned**

Conditional probability is the probability of some event A, given the occurrence of some other event B.

In probability theory, conditional probability is a measure of the probability of an event occurring given that another event has (by assumption, presumption, assertion or evidence) occurred.

### Probability distribution

**distributioncontinuous probability distributiondiscrete probability distribution**

However, it is possible to define a conditional probability for some zero-probability events using a σ-algebra of such events (such as those arising from a continuous random variable).

In more technical terms, the probability distribution is a description of a random phenomenon in terms of the probabilities of events.

### Cache language model

The cache language model and other statistical language models that are used in natural language processing are also examples of applications of probability theory.

These occur in the natural language processing subfield of computer science and assign probabilities to given sequences of words by means of a probability distribution.