Probability is the likelihood or chance that something is the case or will happen. Probability theory is used extensively in areas such as statistics, mathematics, science and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems. Image File history File links Question_book-3. ... A related article is titled uncertainty. ... This article is about the philosophical position. ... Agnosticism (Greek: α- a-, without + γνώσις gnōsis, knowledge; after Gnosticism) is the philosophical view that the truth value of certain claims — particularly metaphysical claims regarding theology, afterlife or the existence of God, gods, deities, or even ultimate reality — is unknown or, depending on the form of agnosticism, inherently unknowable due to... “Uncertain” redirects here. ... It has been suggested that this article or section be merged with estimation. ... For other uses, see Believe. ... Theory of knowledge redirects here: for other uses, see theory of knowledge (disambiguation) According to Plato, knowledge is a subset of that which is both true and believed Epistemology or theory of knowledge is the branch of philosophy that studies the nature, methods, limitations, and validity of knowledge and belief. ... A related article is titled uncertainty. ... This article is about the general notion of determinism in philosophy. ... Probability theory is the branch of mathematics concerned with analysis of random phenomena. ... This article is about the field of statistics. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... For other uses, see Philosophy (disambiguation). ...

Interpretations

Main article: Probability interpretations

The word probability does not have a consistent direct definition. Actually, there are two broad categories of probability interpretations: Frequentists talk about probabilities only when dealing with well defined random experiments. The relative frequency of occurrence of an experiment's outcome, when repeating the experiment, is a measure of the probability of that random event. Bayesians, however, assign probabilities to any statement whatsoever, even when no random process is involved, as a way to represent its subjective pausibility. The word probability has been used in a variety of ways since it was first coined in relation to games of chance. ... Statistical regularity has motivated the development of the relative frequency concept of probability. ... Random redirects here. ... Bayesian probability is an interpretation of probability suggested by Bayesian theory, which holds that the concept of probability can be defined as the degree to which a person believes a proposition. ...

History

Further information: Statistics

The scientific study of probability is a modern development. Gambling shows that there has been an interest in quantifying the ideas of probability for millennia, but exact mathematical descriptions of use in those problems only arose much later. This article is about the field of statistics. ... Gamble redirects here. ...

According to Richard Jeffrey, "Before the middle of the seventeenth century, the term 'probable' (Latin probabilis) meant approvable, and was applied in that sense, univocally, to opinion and to action. A probable action or opinion was one such as sensible people would undertake or hold, in the circumstances."^[1]

Aside from some elementary considerations made by Girolamo Cardano in the 16th century, the doctrine of probabilities dates to the correspondence of Pierre de Fermat and Blaise Pascal (1654). Christiaan Huygens (1657) gave the earliest known scientific treatment of the subject. Jakob Bernoulli's Ars Conjectandi (posthumous, 1713) and Abraham de Moivre's Doctrine of Chances (1718) treated the subject as a branch of mathematics. See Ian Hacking's The Emergence of Probability for a history of the early development of the very concept of mathematical probability. Gerolamo Cardano or Jerome Cardan (September 24, 1501 - September 21, 1576) was a celebrated Renaissance mathematician, physician, astrologer, and gambler. ... Pierre de Fermat Pierre de Fermat IPA: (August 17, 1601 – January 12, 1665) was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to modern calculus. ... Blaise Pascal (pronounced ), (June 20 [[1624 // ]] – August 19, 1662) was a French mathematician, physicist, and religious philosopher. ... Christiaan Huygens (pronounced in English (IPA): ; in Dutch: ) (April 14, 1629 – July 8, 1698), was a Dutch mathematician, astronomer and physicist; born in The Hague as the son of Constantijn Huygens. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Abraham de Moivre. ... The Doctrine of Chances is a book on probability theory by 18th-century French mathematician Abraham de Moivre, published in 1733. ... Ian Hacking, CC (born 1936 in Vancouver) is a philosopher, specializing in the philosophy of science. ...

The theory of errors may be traced back to Roger Cotes's Opera Miscellanea (posthumous, 1722), but a memoir prepared by Thomas Simpson in 1755 (printed 1756) first applied the theory to the discussion of errors of observation. The reprint (1757) of this memoir lays down the axioms that positive and negative errors are equally probable, and that there are certain assignable limits within which all errors may be supposed to fall; continuous errors are discussed and a probability curve is given. Roger Cotes (Burbage, Leicestershire July 10, 1682 – June 5, 1716 in Cambridge, Cambridgeshire) was an English mathematician. ... Thomas Simpson (August 20, 1710 – May 14, 1761) was a British mathematician, inventor and eponym of Simpsons rule to approximate definite integrals. ...

Pierre-Simon Laplace (1774) made the first attempt to deduce a rule for the combination of observations from the principles of the theory of probabilities. He represented the law of probability of errors by a curve y = φ(x), x being any error and y its probability, and laid down three properties of this curve: Pierre-Simon, marquis de Laplace (March 23, 1749 - March 5, 1827) was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy. ...

1. it is symmetric as to the y-axis;
2. the x-axis is an asymptote, the probability of the error being 0;
3. the area enclosed is 1, it being certain that an error exists.

He deduced a formula for the mean of three observations. He also gave (1781) a formula for the law of facility of error (a term due to Lagrange, 1774), but one which led to unmanageable equations. Daniel Bernoulli (1778) introduced the principle of the maximum product of the probabilities of a system of concurrent errors. For other uses, see Asymptote (disambiguation). ... Daniel Bernoulli Daniel Bernoulli (February 8, 1700 – March 17, 1782) was a Dutch-born mathematician who spent much of his life in Basel, Switzerland where he died. ...

The method of least squares is due to Adrien-Marie Legendre (1805), who introduced it in his Nouvelles méthodes pour la détermination des orbites des comètes (New Methods for Determining the Orbits of Comets). In ignorance of Legendre's contribution, an Irish-American writer, Robert Adrain, editor of "The Analyst" (1808), first deduced the law of facility of error, Least squares is a mathematical optimization technique that attempts to find a best fit to a set of data by attempting to minimize the sum of the squares of the differences (called residuals) between the fitted function and the data. ... Adrien-Marie Legendre (September 18, 1752 – January 10, 1833) was a French mathematician. ... Robert Adrain (September 30, 1775 - August 10, 1843) was a scientist and mathematician. ...

h being a constant depending on precision of observation, and c a scale factor ensuring that the area under the curve equals 1. He gave two proofs, the second being essentially the same as John Herschel's (1850). Gauss gave the first proof which seems to have been known in Europe (the third after Adrain's) in 1809. Further proofs were given by Laplace (1810, 1812), Gauss (1823), James Ivory (1825, 1826), Hagen (1837), Friedrich Bessel (1838), W. F. Donkin (1844, 1856), and Morgan Crofton (1870). Other contributors were Ellis (1844), De Morgan (1864), Glaisher (1872), and Giovanni Schiaparelli (1875). Peters's (1856) formula for r, the probable error of a single observation, is well known. John Herschel Sir John Frederick William Herschel (7 March 1792 – 11 May 1871) was an English mathematician and astronomer. ... Johann Carl Friedrich Gauss (pronounced ,  ; in German usually Gauß, Latin: ) (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... Sir James Ivory (1765 - September 21, 1842) was a Scottish mathematician. ... Friedrich Wilhelm Bessel (July 22, 1784 – March 17, 1846) was a German mathematician, astronomer, and systematizer of the Bessel functions (which, despite their name, were discovered by Daniel Bernoulli). ... Morgan Crofton (born 1826 in Dublin, Ireland, died in 1915 in Brighton, England) was a mathematician who contributed to the field of geometric probability theory. ... The tone or style of this article or section may not be appropriate for Wikipedia. ... James Whitbread Lee Glaisher (5 November 1848 - 7 December 1928) was a prolific British mathematician. ... Giovanni Virginio Schiaparelli (March 14, 1835 – July 4, 1910) was an Italian astronomer. ...

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. Augustus De Morgan and George Boole improved the exposition of the theory. Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... Pierre-Simon Laplace Pierre-Simon Laplace (March 23, 1749 – March 5, 1827) was a French mathematician and astronomer, the discoverer of the Laplace transform and Laplaces equation. ... Sylvestre François de Lacroix (April 28, 1765–May 24, 1843) was a French mathematician. ... Lambert Adolphe Jacques Quételet (February 22, 1796 – February 17, 1874) was a Belgian astronomer, mathematician, statistician and sociologist. ... Richard Dedekind Julius Wilhelm Richard Dedekind (October 6, 1831 – February 12, 1916) was a German mathematician who did important work in abstract algebra and the foundations of the real numbers. ... Paul Matthieu Hermann Laurent (2 September 1841 Echternach, Luxembourg - 19 February 1908 Paris, France) was a French mathematician. ... Karl Pearson FRS (March 27, 1857 – April 27, 1936) established the discipline of mathematical statistics. ... The tone or style of this article or section may not be appropriate for Wikipedia. ... Not to be confused with George Boolos. ...

On the geometric side (see integral geometry) contributors to The Educational Times were influential (Miller, Crofton, McColl, Wolstenholme, Watson, and Artemas Martin). In mathematics, the term integral geometry in is used in two ways, which, although related, imply different views of the content of the subject. ...

Mathematical treatment

In mathematics a probability of an event, A is represented by a real number in the range from 0 to 1 and written as P(A), p(A) or Pr(A). An impossible event has a probability of 0, and a certain event has a probability of 1. However, the converses are not always true: probability 0 events are not always impossible, nor probability 1 events certain. The rather subtle distinction between "certain" and "probability 1" is treated at greater length in the article on "almost surely". In probability theory, an event is a set of outcomes (a subset of the sample space) to which a probability is assigned. ... In probability theory, an event happens almost surely (a. ...

The opposite or complement of an event A is the event [not A] (that is, the event of A not occurring); its probability is given by P(not A) = 1 - P(A). As an example, the chance of not rolling a six on a six-sided die is 1 - (chance of rolling a six) = . See Complementary event for a more complete treatment. In probability theory, two events are called complementary if and only if precisely one of the possibilities must occur. ...

If two events, A and B are independent then the joint probability is Given two random variables X and Y, the joint probability distribution of X and Y is the probability distribution of X and Y together. ...

for example if two coins are flipped the chance of both being heads is .

If two events are mutually exclusive then the probability of either occurring is In logic, two mutually exclusive (or mutual exclusive according to some sources) propositions are propositions that logically cannot both be true. ...

For example, the chance of rolling a 1 or 2 on a six-sided die is .

If the events are not mutually exclusive then

.

For example, when drawing a single card at random from a regular deck of cards, the chance of getting a heart or a face card (J,Q,K) is , because of the 52 cards of a deck 13 are hearts, 12 are face cards, and 3 are both.

Conditional probability is the probability of some event A, given the occurrence of some other event B. Conditional probability is written P(A|B), and is read "the probability of A, given B". It is defined by This article defines some terms which characterize probability distributions of two or more variables. ...

If P(B) = 0 then is undefined. In mathematics, defined and undefined are used to explain whether or not expressions have meaningful, sensible, and unambiguous values. ...

Theory

Main article: Probability theory

Like other theories, the theory of probability is a representation of probabilistic concepts in formal terms—that is, in terms that can be considered separately from their meaning. These formal terms are manipulated by the rules of mathematics and logic, and any results are then interpreted or translated back into the problem domain. Probability theory is the branch of mathematics concerned with analysis of random phenomena. ... The word theory has a number of distinct meanings in different fields of knowledge, depending on their methodologies and the context of discussion. ... Probability theory is the branch of mathematics concerned with analysis of random phenomena. ...

There have been at least two successful attempts to formalize probability, namely the Kolmogorov formulation and the Cox formulation. In Kolmogorov's formulation (see probability space), sets are interpreted as events and probability itself as a measure on a class of sets. In Cox's theorem, probability is taken as a primitive (that is, not further analyzed) and the emphasis is on constructing a consistent assignment of probability values to propositions. In both cases, the laws of probability are the same, except for technical details. Andrey Kolmogorov Andrey Nikolaevich Kolmogorov (Андре́й Никола́евич Колмого́ров) (kahl-mah-GAW-raff) (April 25, 1903 in Tambov - October 20, 1987 in Moscow) was a Russian mathematician who made major advances in the fields of probability theory and topology. ... Richard Threlkeld Cox (1898 - May 2, 1991) was a professor of physics at Johns Hopkins University, known for Coxs theorem relating to the foundations of probability. ... In mathematics, the definition of the probability space is the foundation of probability theory. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In probability theory, an event is a set of outcomes (a subset of the sample space) to which a probability is assigned. ... In mathematics the concept of a measure generalizes notions such as length, area, and volume (but not all of its applications have to do with physical sizes). ... Coxs theorem, named after the physicist Richard Threlkeld Cox, is a derivation of the laws of probability theory from a certain set of postulates. ... In probability theory, the probability P of some event E, denoted , is defined in such a way that P satisfies the Kolmogorov axioms, named after Andrey Kolmogorov. ...

There are other methods for quantifying uncertainty, such as the Dempster-Shafer theory and possibility theory, but those are essentially different and not compatible with the laws of probability as they are usually understood. The Dempster-Shafer theory is a mathematical theory of evidence [SH76] based on belief functions and plausible reasoning, which is used to combine separate pieces of information (evidence) to calculate the probability of an event. ... Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. ...

Applications

Two major applications of probability theory in everyday life are in risk assessment and in trade on commodity markets. Governments typically apply probabilistic methods in environmental regulation where it is called "pathway analysis", often measuring well-being using methods that are stochastic in nature, and choosing projects to undertake based on statistical analyses of their probable effect on the population as a whole. It is not correct to say that statistics are involved in the modelling itself, as typically the assessments of risk are one-time and thus require more fundamental probability models, e.g. "the probability of another 9/11". A law of small numbers tends to apply to all such choices and perception of the effect of such choices, which makes probability measures a political matter. For the Parker Brothers board game, see Risk (game) For other uses, see Risk (disambiguation). ... Chicago Board of Trade Futures market Commodity markets are markets where raw or primary products are exchanged. ... Environmental law is a body of law which addresses the system of complex and interlocking rules which seeks to protect from destruction or development certain species or favored natural areas thought to be endangered by human encroachment. ... The well-being or quality of life of a population is an important concern in economics and political science. ... This article is about the field of statistics. ... For the Parker Brothers board game, see Risk (game) For other uses, see Risk (disambiguation). ... The law of small numbers may refer to the specific features of the Poisson distribution, as in the book The Law of Small Numbers by Ladislaus Bortkiewicz; or the tendency for an initial segment of data to show some bias that drops out later (one example in number theory being...

A good example is the effect of the perceived probability of any widespread Middle East conflict on oil prices - which have ripple effects in the economy as a whole. An assessment by a commodity trader that a war is more likely vs. less likely sends prices up or down, and signals other traders of that opinion. Accordingly, the probabilities are not assessed independently nor necessarily very rationally. The theory of behavioral finance emerged to describe the effect of such groupthink on pricing, on policy, and on peace and conflict. Economics Nobel Laureate Daniel Kahneman, was an important figure in the development of behavioral finance and economics and continues to write extensively in the field. ... Groupthink is a type of thought exhibited by group members who try to minimize conflict and reach consensus without critically testing, analyzing, and evaluating ideas. ...

It can reasonably be said that the discovery of rigorous methods to assess and combine probability assessments has had a profound effect on modern society. Accordingly, it may be of some importance to most citizens to understand how odds and probability assessments are made, and how they contribute to reputations and to decisions, especially in a democracy.

Another significant application of probability theory in everyday life is reliability. Many consumer products, such as automobiles and consumer electronics, utilize reliability theory in the design of the product in order to reduce the probability of failure. The probability of failure is also closely associated with the product's warranty. Reliability Theory of Aging and Longevity is a scientific approach aimed to gain theoretical insights into mechanisms of biological aging and species survival patterns by applying a general theory of systems failure, known as reliability theory. ... Car redirects here. ... Reliability theory developed apart from the mainstream of probability and statistics, and was used originally as a tool to help nineteenth century maritime insurance and life insurance companies compute profitable rates to charge their customers. ... In commercial and consumer transactions, a warranty is an obligation that an article or service sold is as factually stated or legally implied by the seller, and that often provides for a specific remedy such as repair or replacement in the event the article or service fails to meet the...

Relation to randomness

Main article: Randomness

In a deterministic universe, based on Newtonian concepts, there is no probability if all conditions are known. In the case of a roulette wheel, if the force of the hand and the period of that force are known, then the number on which the ball will stop would be a certainty. Of course, this also assumes knowledge of inertia and friction of the wheel, weight, smoothness and roundness of the ball, variations in hand speed during the turning and so forth. A probabilistic description can thus be more useful than Newtonian mechanics for analysing the pattern of outcomes of repeated rolls of roulette wheel. Physicists face the same situation in kinetic theory of gases, where the system, while deterministic in principle, is so complex (with the number of molecules typically the order of magnitude of Avogadro constant () that only statistical description of its properties is feasible. Random redirects here. ... This article is about the general notion of determinism in philosophy. ... It has been suggested that this article or section be merged with Classical mechanics. ... Kinetic theory or kinetic theory of gases attempts to explain macroscopic properties of gases, such as pressure, temperature, or volume, by considering their molecular composition and motion. ... The Avogadro constant (symbols: L, NA), also called the Avogadro number and, in German scientific literature, sometimes also known as the Loschmidt constant/number, is formally defined to be the number of entities in one mole,[1][2] that is the number of carbon-12 atoms in 12 grams (0. ...

A revolutionary discovery of 20th century physics was the random character of all physical processes that occur at microscopic scales and are governed by the laws of quantum mechanics. The wave function itself evolves deterministically as long as no observation is made, but, according to the prevailing Copenhagen interpretation, the randomness caused by the wave function collapsing when an observation is made, is fundamental. This means that probability theory is required to describe nature. Others never came to terms with the loss of determinism. Albert Einstein famously remarked in a letter to Max Born: Jedenfalls bin ich überzeugt, daß der Alte nicht würfelt. (I am convinced that God does not play dice). Although alternative viewpoints exist, such as that of quantum decoherence being the cause of an apparent random collapse, at present there is a firm consensus among the physicists that probability theory is necessary to describe quantum phenomena.^{[citation needed]} For a generally accessible and less technical introduction to the topic, see Introduction to quantum mechanics. ... A wave function is a mathematical tool that quantum mechanics uses to describe any physical system. ... Early twentieth century studies of the physics of very small-scale phenomena led to the Copenhagen interpretation. ... In quantum mechanics, quantum decoherence is the process by which quantum systems in complex environments exhibit classical behavior. ... Probability theory is the branch of mathematics concerned with analysis of random phenomena. ... “Einstein” redirects here. ... Max Born (December 11, 1882 – January 5, 1970) was a German physicist and mathematician. ... In quantum mechanics, quantum decoherence is the mechanism by which quantum systems interact with their environments to exhibit probabilistically additive behavior - a feature of classical physics - and give the appearance of wavefunction collapse. ...

See also

Decision theory is an area of study of discrete mathematics that models human decision-making in science, engineering and indeed all human social activities. ... It has been suggested that this article or section be merged with Equipossible. ... Fuzzy measure theory considers a number of special classes of measures, each of which is characterized by a special property. ... Game theory is a branch of applied mathematics that is often used in the context of economics. ... Not to be confused with information technology, information science, or informatics. ... // Probability The Doctrine of Chances Author: Abraham de Moivre Publication data: 1738 (2nd ed. ... In mathematics, a measure is a function that assigns a number, e. ... Probabilistic argumentation is a general theory of reasoning under uncertainty and ignorance. ... The aim of a probabilistic logic (or probability logic) is to combine the capacity of probability theory to handle uncertainty with the capacity of deductive logic to exploit structure. ... In probability theory, let S = {X1, ..., Xn}, with the Xi in {0,1,...,G-1}, be a set of random variables on the sample space Ω={0,1,...,G-1}n, a probability measure π is a random field if . There exist several types of random fields, such as Markov... In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ... This article is about the field of statistics. ... Please add any Wikipedia articles related to statistics that are not already on this list. ... In the mathematics of probability, a stochastic process is a random function. ... A single realization of a one-dimensional Wiener process A single realization of a three-dimensional Wiener process In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. ...

Footnotes

1. ^ Jeffrey, R.C., Probability and the Art of Judgment, Cambridge University Press. (1992). pp. 54-55 . ISBN 0-521-39459-7

Sources

• Olav Kallenberg, Probabilistic Symmetries and Invariance Principles. Springer -Verlag, New York (2005). 510 pp. ISBN 0-387-25115-4
• Kallenberg, O., Foundations of Modern Probability, 2nd ed. Springer Series in Statistics. (2002). 650 pp. ISBN 0-387-95313-2

...

Quotations

• Damon Runyon, "It may be that the race is not always to the swift, nor the battle to the strong - but that is the way to bet."
• Pierre-Simon Laplace "It is remarkable that a science which began with the consideration of games of chance should have become the most important object of human knowledge." Théorie Analytique des Probabilités, 1812.
• Richard von Mises "The unlimited extension of the validity of the exact sciences was a characteristic feature of the exaggerated rationalism of the eighteenth century" (in reference to Laplace). Probability, Statistics, and Truth, p 9. Dover edition, 1981 (republication of second English edition, 1957).

Damon Runyon Damon Runyon (October 4, 1884 – December 10, 1946) was a newspaperman and writer. ... Pierre-Simon, marquis de Laplace (March 23, 1749 - March 5, 1827) was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy. ... Richard von Mises. ...

External links

• Edwin Thompson Jaynes. Probability Theory: The Logic of Science. Preprint: Washington University, (1996). — HTML index with links to PostScript files and PDF
• Dictionary of the History of Ideas: Certainty in Seventeenth-Century Thought
• Dictionary of the History of Ideas: Certainty since the Seventeenth Century
• Figures from the History of Probability and Statistics (Univ. of Southampton)
• Probability and Statistics on the Earliest Uses Pages (Univ. of Southampton)
• Earliest Uses of Symbols in Probability and Statistics on Earliest Uses of Various Mathematical Symbols
• Analytical argumentations of probability and statistics.
• A tutorial on probability and Bayes’ theorem devised for first-year Oxford University students