A report on Probability

The probabilities of rolling several numbers using two dice.
Gerolamo Cardano (16th century)
Christiaan Huygens published one of the first books on probability (17th century)
Carl Friedrich Gauss

Branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true.

- Probability
The probabilities of rolling several numbers using two dice.

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The Poisson distribution, a discrete probability distribution.

Probability theory

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The Poisson distribution, a discrete probability distribution.
The normal distribution, a continuous probability distribution.

Probability theory is the branch of mathematics concerned with probability.

Painting of Pascal made by François II Quesnel for Gérard Edelinck in 1691

Blaise Pascal

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French mathematician, physicist, inventor, philosopher, writer, and Catholic theologian.

French mathematician, physicist, inventor, philosopher, writer, and Catholic theologian.

Painting of Pascal made by François II Quesnel for Gérard Edelinck in 1691
Pascal's birthplace.
An early Pascaline on display at the Musée des Arts et Métiers, Paris
Pascal's triangle. Each number is the sum of the two directly above it. The triangle demonstrates many mathematical properties in addition to showing binomial coefficients.
Pascal studying the cycloid, by Augustin Pajou, 1785, Louvre
An illustration of the (apocryphal) Pascal's barrel experiment
Puy de Dôme
Portrait of Pascal
Pascal
Man is only a reed...but he is a thinking reed.
Death mask of Blaise Pascal.
Pascal's epitaph in Saint-Étienne-du-Mont, where he was buried

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.

Silver didrachma from Crete depicting Talos, an ancient mythical automaton with artificial intelligence

Artificial intelligence

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Intelligence demonstrated by machines, as opposed to the natural intelligence displayed by animals including humans.

Intelligence demonstrated by machines, as opposed to the natural intelligence displayed by animals including humans.

Silver didrachma from Crete depicting Talos, an ancient mythical automaton with artificial intelligence
An ontology represents knowledge as a set of concepts within a domain and the relationships between those concepts.
A parse tree represents the syntactic structure of a sentence according to some formal grammar.
Feature detection (pictured: edge detection) helps AI compose informative abstract structures out of raw data.
Kismet, a robot with rudimentary social skills
A particle swarm seeking the global minimum
Expectation-maximization clustering of Old Faithful eruption data starts from a random guess but then successfully converges on an accurate clustering of the two physically distinct modes of eruption.
A neural network is an interconnected group of nodes, akin to the vast network of neurons in the human brain.
Representing images on multiple layers of abstraction in deep learning
For this project the AI had to learn the typical patterns in the colors and brushstrokes of Renaissance painter Raphael. The portrait shows the face of the actress Ornella Muti, "painted" by AI in the style of Raphael.
AI Patent families for functional application categories and sub categories. Computer vision represents 49 percent of patent families related to a functional application in 2016.
The word "robot" itself was coined by Karel Čapek in his 1921 play R.U.R., the title standing for "Rossum's Universal Robots"

To solve these problems, AI researchers have adapted and integrated a wide range of problem-solving techniques—including search and mathematical optimization, formal logic, artificial neural networks, and methods based on statistics, probability and economics.

John Venn, who provided a thorough exposition of frequentist probability in his book, The Logic of Chance (1866)

Frequentist probability

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John Venn, who provided a thorough exposition of frequentist probability in his book, The Logic of Chance (1866)

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 (the long-run probability).

A computer-simulated realization of a Wiener or Brownian motion process on the surface of a sphere. The Wiener process is widely considered the most studied and central stochastic process in probability theory.

Stochastic process

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Mathematical object usually defined as a family of random variables.

Mathematical object usually defined as a family of random variables.

A computer-simulated realization of a Wiener or Brownian motion process on the surface of a sphere. The Wiener process is widely considered the most studied and central stochastic process in probability theory.
A single computer-simulated sample function or realization, among other terms, of a three-dimensional Wiener or Brownian motion process for time 0 ≤ t ≤ 2. The index set of this stochastic process is the non-negative numbers, while its state space is three-dimensional Euclidean space.
Realizations of Wiener processes (or Brownian motion processes) with drift and without drift.
Mathematician Joseph Doob did early work on the theory of stochastic processes, making fundamental contributions, particularly in the theory of martingales. His book Stochastic Processes is considered highly influential in the field of probability theory.
Norbert Wiener gave the first mathematical proof of the existence of the Wiener process. This mathematical object had appeared previously in the work of Thorvald Thiele, Louis Bachelier, and Albert Einstein.

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.

The cover page of Ars Conjectandi

Ars Conjectandi

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The cover page of Ars Conjectandi
Christiaan Huygens published the first treaties on probability
Portrait of Jakob Bernoulli in 1687
Cutout of a page from Ars Conjectandi showing Bernoulli's formula for sum of integer powers. The last line gives his eponymous numbers.
Abraham de Moivre's work was built in part on Bernoulli's

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.

Pierre de Fermat

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French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality.

French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality.

Pierre de Fermat
The 1670 edition of Diophantus's Arithmetica includes Fermat's commentary, referred to as his "Last Theorem" (Observatio Domini Petri de Fermat), posthumously published by his son
alt=Plaque at the place of burial of Pierre de Fermat |Place of burial of Pierre de Fermat in Place Jean Jaurés, Castres. Translation of the plaque: in this place was buried on January 13, 1665, Pierre de Fermat, councillor at the Chambre de l'Édit (a court established by the Edict of Nantes) and mathematician of great renown, celebrated for his theorem,
Monument to Fermat in Beaumont-de-Lomagne in Tarn-et-Garonne, southern France
Bust in the Salle Henri-Martin in the Capitole de Toulouse
Holographic will handwritten by Fermat on 4 March 1660, now kept at the Departmental Archives of Haute-Garonne, in Toulouse

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

Bayesian probability

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Interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification of a personal belief.

Interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification of a personal belief.

It is associated with probabilities implied by the odds not being coherent.

Jacob Bernoulli

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One of the many prominent mathematicians in the Bernoulli family.

One of the many prominent mathematicians in the Bernoulli family.

Image from Acta Eruditorum (1682) wherein was published the critique of Bernoulli's Conamen novi systematis cometarum
Ars conjectandi, 1713 (Milano, Fondazione Mansutti).
Jacob Bernoulli's tombstone in Basel Münster
De gravitate aetheris, 1683
Opera, vol 1, 1744

He also discovered the fundamental mathematical constant e. 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.

A pseudorandomly generated bitmap.

Randomness

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Apparent or actual lack of pattern or predictability in events.

Apparent or actual lack of pattern or predictability in events.

A pseudorandomly generated bitmap.
Ancient fresco of dice players in Pompei.
The ball in a roulette can be used as a source of apparent randomness, because its behavior is very sensitive to the initial conditions.
In the Monty Hall problem, when the host reveals one door that contains a goat, this provides new information that needs to be factored into the calculation of probabilities.

Randomness applies to concepts of chance, probability, and information entropy.