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Probability is the branch of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
concerning numerical descriptions of how likely an
event Event may refer to: Gatherings of people * Ceremony, an event of ritual significance, performed on a special occasion * Convention (meeting), a gathering of individuals engaged in some common interest * Event management, the organization of eve ...
is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty."Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th Ed, (2009), .William Feller, ''An Introduction to Probability Theory and Its Applications'', (Vol 1), 3rd Ed, (1968), Wiley, . The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%). These concepts have been given an
axiomatic An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
mathematical formalization in
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
, which is used widely in areas of study such as
statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indus ...
,
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
,
science Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earliest archeological evidenc ...
,
finance Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline ...
,
gambling Gambling (also known as betting or gaming) is the wagering of something of value ("the stakes") on a random event with the intent of winning something else of value, where instances of strategy are discounted. Gambling thus requires three el ...
,
artificial intelligence Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machines, as opposed to intelligence displayed by animals and humans. Example tasks in which this is done include speec ...
,
machine learning Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence. Machine ...
,
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (inclu ...
,
game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
, and
philosophy Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. ...
to, for example, draw inferences about the expected frequency of events. Probability theory is also used to describe the underlying mechanics and regularities of
complex systems A complex system is a system composed of many components which may interact with each other. Examples of complex systems are Earth's global climate, organisms, the human brain, infrastructure such as power grid, transportation or communication ...
.

# Interpretations

When dealing with
experiment An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into cause-and-effect by demonstrating what outcome occurs whe ...
s that are
random In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual ra ...
and
well-defined In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined or ''ambiguous''. A func ...
in a purely theoretical setting (like tossing a coin), probabilities can be numerically described by the number of desired outcomes, divided by the total number of all outcomes. For example, tossing a coin twice will yield "head-head", "head-tail", "tail-head", and "tail-tail" outcomes. The probability of getting an outcome of "head-head" is 1 out of 4 outcomes, or, in numerical terms, 1/4, 0.25 or 25%. However, when it comes to practical application, there are two major competing categories of probability interpretations, whose adherents hold different views about the fundamental nature of probability: *
Objectivists Objectivism is a philosophical system developed by Russian-American writer and philosopher Ayn Rand. She described it as "the concept of man as a heroic being, with his own happiness as the moral purpose of his life, with productive achievement ...
assign numbers to describe some objective or physical state of affairs. The most popular version of objective probability is
frequentist 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 (the long-run probability). Probabilities can be found (in principle) by a re ...
, which claims that the probability of a random event denotes the ''relative frequency of occurrence'' of an experiment's outcome when the experiment is repeated indefinitely. This interpretation considers probability to be the relative frequency "in the long run" of outcomes. A modification of this is
propensity probability The propensity theory of probability is a probability interpretation in which the probability is thought of as a physical propensity, disposition, or tendency of a given type of situation to yield an outcome of a certain kind, or to yield a long ...
, which interprets probability as the tendency of some experiment to yield a certain outcome, even if it is performed only once. * Subjectivists assign numbers per subjective probability, that is, as a degree of belief. The degree of belief has been interpreted as "the price at which you would buy or sell a bet that pays 1 unit of utility if E, 0 if not E", although that interpretation is not universally agreed upon. The most popular version of subjective probability is
Bayesian probability Bayesian probability is an 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 ...
, which includes expert knowledge as well as experimental data to produce probabilities. The expert knowledge is represented by some (subjective)
prior probability distribution In Bayesian statistical inference, a prior probability distribution, often simply called the prior, of an uncertain quantity is the probability distribution that would express one's beliefs about this quantity before some evidence is taken in ...
. These data are incorporated in a
likelihood function The likelihood function (often simply called the likelihood) represents the probability of random variable realizations conditional on particular values of the statistical parameters. Thus, when evaluated on a given sample, the likelihood funct ...
. The product of the prior and the likelihood, when normalized, results in a
posterior probability distribution The posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood via an application of Bayes' rule. From an epistemological perspective, the posterio ...
that incorporates all the information known to date. By
Aumann's agreement theorem Aumann's agreement theorem was stated and proved by Robert Aumann in a paper titled "Agreeing to Disagree", which introduced the set theoretic description of common knowledge. The theorem concerns agents who share a common prior and update their ...
, Bayesian agents whose prior beliefs are similar will end up with similar posterior beliefs. However, sufficiently different priors can lead to different conclusions, regardless of how much information the agents share.

# Etymology

The word ''probability'' derives from the Latin , which can also mean " probity", a measure of the
authority In the fields of sociology and political science, authority is the legitimate power of a person or group over other people. In a civil state, ''authority'' is practiced in ways such a judicial branch or an executive branch of government.''Th ...
of a
witness In law, a witness is someone who has knowledge about a matter, whether they have sensed it or are testifying on another witnesses' behalf. In law a witness is someone who, either voluntarily or under compulsion, provides testimonial evidence, e ...
in a
legal case A legal case is in a general sense a dispute between opposing parties which may be resolved by a court, or by some equivalent legal process. A legal case is typically based on either civil or criminal law. In most legal cases there are one or mo ...
in
Europe Europe is a large peninsula conventionally considered a continent in its own right because of its great physical size and the weight of its history and traditions. Europe is also considered a subcontinent of Eurasia and it is located entire ...
, and often correlated with the witness's
nobility Nobility is a social class found in many societies that have an aristocracy. It is normally ranked immediately below royalty. Nobility has often been an estate of the realm with many exclusive functions and characteristics. The characteris ...
. In a sense, this differs much from the modern meaning of ''probability'', which in contrast is a measure of the weight of
empirical evidence Empirical evidence for a proposition is evidence, i.e. what supports or counters this proposition, that is constituted by or accessible to sense experience or experimental procedure. Empirical evidence is of central importance to the sciences a ...
, and is arrived at from
inductive reasoning Inductive reasoning is a method of reasoning in which a general principle is derived from a body of observations. It consists of making broad generalizations based on specific observations. Inductive reasoning is distinct from ''deductive'' r ...
and
statistical inference Statistical inference is the process of using data analysis to infer properties of an underlying distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers properti ...
. Hacking, I. (2006) ''The Emergence of Probability: A Philosophical Study of Early Ideas about Probability, Induction and Statistical Inference'', Cambridge University Press,

# History

The scientific study of probability is a modern development of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
. Gambling shows that there has been an interest in quantifying the ideas of probability for millennia, but exact mathematical descriptions arose much later. There are reasons for the slow development of the mathematics of probability. Whereas games of chance provided the impetus for the mathematical study of probability, fundamental issues are still obscured by the superstitions of gamblers. According to
Richard Jeffrey Richard Carl Jeffrey (August 5, 1926 – November 9, 2002) was an American philosopher, logician, and probability theorist. He is best known for developing and championing the philosophy of radical probabilism and the associated heuristic of ...
, "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."Jeffrey, R.C., ''Probability and the Art of Judgment,'' Cambridge University Press. (1992). pp. 54–55 . However, in legal contexts especially, 'probable' could also apply to propositions for which there was good evidence.Franklin, J. (2001) ''The Science of Conjecture: Evidence and Probability Before Pascal,'' Johns Hopkins University Press. (pp. 22, 113, 127) The sixteenth-century
Italian Italian(s) may refer to: * Anything of, from, or related to the people of Italy over the centuries ** Italians, an ethnic group or simply a citizen of the Italian Republic or Italian Kingdom ** Italian language, a Romance language *** Regional It ...
polymath
Gerolamo Cardano Gerolamo Cardano (; also Girolamo or Geronimo; french: link=no, Jérôme Cardan; la, Hieronymus Cardanus; 24 September 1501– 21 September 1576) was an Italian polymath, whose interests and proficiencies ranged through those of mathematician, ...
demonstrated the efficacy of defining
odds Odds provide a measure of the likelihood of a particular outcome. They are calculated as the ratio of the number of events that produce that outcome to the number that do not. Odds are commonly used in gambling and statistics. Odds also have ...
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). Aside from the elementary work by Cardano, the doctrine of probabilities dates to the correspondence of
Pierre de Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...
and
Blaise Pascal Blaise Pascal ( , , ; ; 19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, philosopher, and Catholic writer. He was a child prodigy who was educated by his father, a tax collector in Rouen. Pascal's earliest ...
(1654).
Christiaan Huygens Christiaan Huygens, Lord of Zeelhem, ( , , ; also spelled Huyghens; la, Hugenius; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor, who is regarded as one of the greatest scientists ...
(1657) gave the earliest known scientific treatment of the subject.
Jakob Bernoulli Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family. He was an early proponent of Leibnizian calculus and sided with Gottfried Wilhelm Leibniz during the L ...
'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. The seminal work consolidated, apa ...
'' (posthumous, 1713) and Abraham de Moivre's '' Doctrine of Chances'' (1718) treated the subject as a branch of mathematics. See
Ian Hacking Ian MacDougall Hacking (born February 18, 1936) is a Canadian philosopher specializing in the philosophy of science. Throughout his career, he has won numerous awards, such as the Killam Prize for the Humanities and the Balzan Prize, and been ...
's ''The Emergence of Probability'' and James Franklin's ''The Science of Conjecture'' for histories of the early development of the very concept of mathematical probability. The theory of errors may be traced back to
Roger Cotes Roger Cotes (10 July 1682 – 5 June 1716) was an English mathematician, known for working closely with Isaac Newton by proofreading the second edition of his famous book, the '' Principia'', before publication. He also invented the quadratur ...
'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 certain assignable limits define the range of all errors. Simpson also discusses continuous errors and describes a probability curve. The first two laws of error that were proposed both originated with
Pierre-Simon Laplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
. The first law was published in 1774, and stated that the frequency of an error could be expressed as an exponential function of the numerical magnitude of the error—disregarding sign. The second law of error was proposed in 1778 by Laplace, and stated that the frequency of the error is an exponential function of the square of the error.Wilson EB (1923) "First and second laws of error".
Journal of the American Statistical Association The ''Journal of the American Statistical Association (JASA)'' is the primary journal published by the American Statistical Association, the main professional body for statisticians in the United States. It is published four times a year in March, ...
, 18, 143
The second law of error is called the normal distribution or the Gauss law. "It is difficult historically to attribute that law to Gauss, who in spite of his well-known precocity had probably not made this discovery before he was two years old."
Daniel Bernoulli Daniel Bernoulli FRS (; – 27 March 1782) was a Swiss mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family from Basel. He is particularly remembered for his applications of mathematics to mec ...
(1778) introduced the principle of the maximum product of the probabilities of a system of concurrent errors.
Adrien-Marie Legendre Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are nam ...
(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''). 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, :$\phi\left(x\right) = ce^,$ where $h$ is a constant depending on precision of observation, and $c$ is 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 Sir John Frederick William Herschel, 1st Baronet (; 7 March 1792 – 11 May 1871) was an English polymath active as a mathematician, astronomer, chemist, inventor, experimental photographer who invented the blueprint and did botanical ...
's (1850).
Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
gave the first proof that 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 Friedrich Wilhelm Bessel (; 22 July 1784 – 17 March 1846) was a German astronomer, mathematician, physicist, and geodesist. He was the first astronomer who determined reliable values for the distance from the sun to another star by the method ...
(1838), W.F. Donkin (1844, 1856), and Morgan Crofton (1870). Other contributors were
Ellis Ellis is a surname of Welsh and English origin. Retrieved 21 January 2014 An independent French origin of the surname is said to derive from the phrase fleur-de-lis. Surname A * Abe Ellis (Stargate), a fictional character in the TV series ''S ...
(1844), De Morgan (1864), Glaisher (1872), and
Giovanni Schiaparelli Giovanni Virginio Schiaparelli ( , also , ; 14 March 1835 – 4 July 1910) was an Italian astronomer and science historian. Biography He studied at the University of Turin, graduating in 1854, and later did research at Berlin Observator ...
(1875). Peters's (1856) formula for ''r'', the probable error of a single observation, is well known. In the nineteenth century, authors on the general theory included
Laplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
,
Sylvestre Lacroix Sylvestre can refer to: People Surname Given name Middle name * Carlos Sylvestre Begnis (1903–1980), Argentine medical doctor and politician * Philippe Sylvestre Dufour (1622–1687), French Protestant apothecary, banker, collector, ...
(1816), Littrow (1833),
Adolphe Quetelet Lambert Adolphe Jacques Quetelet FRSF or FRSE (; 22 February 1796 – 17 February 1874) was a Belgian astronomer, mathematician, statistician and sociologist who founded and directed the Brussels Observatory and was influential in introd ...
(1853),
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His ...
(1860), Helmert (1872), Hermann Laurent (1873), Liagre, Didion and
Karl Pearson Karl Pearson (; born Carl Pearson; 27 March 1857 – 27 April 1936) was an English mathematician and biostatistician. He has been credited with establishing the discipline of mathematical statistics. He founded the world's first university s ...
. Augustus De Morgan and
George Boole George Boole (; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher, and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in I ...
improved the exposition of the theory. In 1906, Andrey Markov introduced the notion of
Markov chains A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...
, which played an important role in
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
es theory and its applications. The modern theory of probability based on the
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simi ...
was developed by
Andrey Kolmogorov Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
in 1931. On the geometric side, contributors to ''The Educational Times'' included Miller, Crofton, McColl, Wolstenholme, Watson, and Artemas Martin. See integral geometry for more information.

# Theory

Like other
theories A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be ...
, 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. These formal terms are manipulated by the rules of mathematics and logic, and any results are interpreted or translated back into the problem domain. There have been at least two successful attempts to formalize probability, namely the
Kolmogorov Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
formulation and the
Cox Cox may refer to: * Cox (surname), including people with the name Companies * Cox Enterprises, a media and communications company ** Cox Communications, cable provider ** Cox Media Group, a company that owns television and radio stations ** ...
probability space In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
), sets are interpreted as
events Event may refer to: Gatherings of people * Ceremony, an event of ritual significance, performed on a special occasion * Convention (meeting), a gathering of individuals engaged in some common interest * Event management, the organization of e ...
and probability as a measure on a class of sets. In Cox's theorem, probability is taken as a primitive (i.e., 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. There are other methods for quantifying uncertainty, such as the Dempster–Shafer theory or possibility theory, but those are essentially different and not compatible with the usually-understood laws of probability.

# Applications

Probability theory is applied in everyday life in
risk In simple terms, risk is the possibility of something bad happening. Risk involves uncertainty about the effects/implications of an activity with respect to something that humans value (such as health, well-being, wealth, property or the environm ...
assessment and modeling. The insurance industry and markets use actuarial science to determine pricing and make trading decisions. Governments apply probabilistic methods in environmental regulation, entitlement analysis, and
financial regulation Financial regulation is a form of regulation or supervision, which subjects financial institutions to certain requirements, restrictions and guidelines, aiming to maintain the stability and integrity of the financial system. This may be handl ...
. An example of the use of probability theory in equity trading 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 can send that commodity's prices up or down, and signals other traders of that opinion. Accordingly, the probabilities are neither assessed independently nor necessarily rationally. The theory of
behavioral finance Behavioral economics studies the effects of psychological, cognitive, emotional, cultural and social factors on the decisions of individuals or institutions, such as how those decisions vary from those implied by classical economic theory. ...
emerged to describe the effect of such
groupthink Groupthink is a psychological phenomenon that occurs within a group of people in which the desire for harmony or conformity in the group results in an irrational or dysfunctional decision-making outcome. Cohesiveness, or the desire for cohesivene ...
on pricing, on policy, and on peace and conflict. In addition to financial assessment, probability can be used to analyze trends in biology (e.g., disease spread) as well as ecology (e.g., biological Punnett squares). As with finance, risk assessment can be used as a statistical tool to calculate the likelihood of undesirable events occurring, and can assist with implementing protocols to avoid encountering such circumstances. Probability is used to design games of chance so that casinos can make a guaranteed profit, yet provide payouts to players that are frequent enough to encourage continued play. Another significant application of probability theory in everyday life is reliability. Many consumer products, such as
automobiles A car or automobile is a motor vehicle with wheels. Most definitions of ''cars'' say that they run primarily on roads, seat one to eight people, have four wheels, and mainly transport people instead of goods. The year 1886 is regarded a ...
and consumer electronics, use reliability theory in product design to reduce the probability of failure. Failure probability may influence a manufacturer's decisions on a product's
warranty In contract law, a warranty is a promise which is not a condition of the contract or an innominate term: (1) it is a term "not going to the root of the contract",Hogg M. (2011). ''Promises and Contract Law: Comparative Perspectives''p. 48 Cambri ...
. The cache language model and other statistical language models that are used in
natural language processing Natural language processing (NLP) is an interdisciplinary subfield of linguistics, computer science, and artificial intelligence concerned with the interactions between computers and human language, in particular how to program computers to pro ...
are also examples of applications of probability theory.

# Mathematical treatment

Consider an experiment that can produce a number of results. The collection of all possible results is called the
sample space In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually de ...
of the experiment, sometimes denoted as $\Omega$. The
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
of the sample space is formed by considering all different collections of possible results. For example, rolling a die can produce six possible results. One collection of possible results gives an odd number on the die. Thus, the subset is an element of the
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
of the sample space of dice rolls. These collections are called "events". In this case, is the event that the die falls on some odd number. If the results that actually occur fall in a given event, the event is said to have occurred. A probability is a way of assigning every event a value between zero and one, with the requirement that the event made up of all possible results (in our example, the event ) is assigned a value of one. To qualify as a probability, the assignment of values must satisfy the requirement that for any collection of mutually exclusive events (events with no common results, such as the events , , and ), the probability that at least one of the events will occur is given by the sum of the probabilities of all the individual events. The probability of an
event Event may refer to: Gatherings of people * Ceremony, an event of ritual significance, performed on a special occasion * Convention (meeting), a gathering of individuals engaged in some common interest * Event management, the organization of eve ...
''A'' is written as $P\left(A\right)$, $p\left(A\right)$, or $\text\left(A\right)$. This mathematical definition of probability can extend to infinite sample spaces, and even uncountable sample spaces, using the concept of a measure. The ''opposite'' or ''complement'' of an event ''A'' is the event ot ''A''(that is, the event of ''A'' not occurring), often denoted as $A\text{'}, A^c$, $\overline, A^\complement, \neg A$, or $A$; its probability is given by . As an example, the chance of not rolling a six on a six-sided die is $= 1 - \tfrac = \tfrac$. For a more comprehensive treatment, see
Complementary event In probability theory, the complement of any event ''A'' is the event ot ''A'' i.e. the event that ''A'' does not occur.Robert R. Johnson, Patricia J. Kuby: ''Elementary Statistics''. Cengage Learning 2007, , p. 229 () The event ''A'' and ...
. If two events ''A'' and ''B'' occur on a single performance of an experiment, this is called the intersection or
joint probability Given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered ...
of ''A'' and ''B'', denoted as $P\left(A \cap B\right)$.

## Independent events

If two events, ''A'' and ''B'' are
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independ ...
then the joint probability is :$P\left(A \mboxB\right) = P\left(A \cap B\right) = P\left(A\right) P\left(B\right).$ For example, if two coins are flipped, then the chance of both being heads is $\tfrac\times\tfrac = \tfrac$.

## Mutually exclusive events

If either event ''A'' or event ''B'' can occur but never both simultaneously, then they are called mutually exclusive events. If two events are
mutually exclusive In logic and probability theory, two events (or propositions) are mutually exclusive or disjoint if they cannot both occur at the same time. A clear example is the set of outcomes of a single coin toss, which can result in either heads or tails ...
, then the probability of ''both'' occurring is denoted as $P\left(A \cap B\right)$ and :$P\left(A \mboxB\right) = P\left(A \cap B\right) = 0$ If two events are
mutually exclusive In logic and probability theory, two events (or propositions) are mutually exclusive or disjoint if they cannot both occur at the same time. A clear example is the set of outcomes of a single coin toss, which can result in either heads or tails ...
, then the probability of ''either'' occurring is denoted as $P\left(A \cup B\right)$ and :$P\left(A\mboxB\right) = P\left(A \cup B\right)= P\left(A\right) + P\left(B\right) - P\left(A \cap B\right) = P\left(A\right) + P\left(B\right) - 0 = P\left(A\right) + P\left(B\right)$ For example, the chance of rolling a 1 or 2 on a six-sided is $P\left(1\mbox2\right) = P\left(1\right) + P\left(2\right) = \tfrac + \tfrac = \tfrac.$

## Not mutually exclusive events

If the events are not mutually exclusive then :$P\left\left(A \hbox B\right\right) = P\left(A \cup B\right) = P\left\left(A\right\right)+P\left\left(B\right\right)-P\left\left(A \mbox B\right\right).$ For example, when drawing a card from a deck of cards, the chance of getting a heart or a face card (J,Q,K) (or both) is $\tfrac + \tfrac - \tfrac = \tfrac$, since among the 52 cards of a deck, 13 are hearts, 12 are face cards, and 3 are both: here the possibilities included in the "3 that are both" are included in each of the "13 hearts" and the "12 face cards", but should only be counted once.

## Conditional probability

''
Conditional probability In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred. This particular method relies on event B occu ...
'' is the probability of some event ''A'', given the occurrence of some other event ''B''. Conditional probability is written $P\left(A \mid B\right)$, and is read "the probability of ''A'', given ''B''". It is defined by :$P\left(A \mid B\right) = \frac.\,$ If $P\left(B\right)=0$ then $P\left(A \mid B\right)$ is formally undefined by this expression. In this case $A$ and $B$ are independent, since $P\left(A \cap B\right) = P\left(A\right)P\left(B\right) = 0$. However, it is possible to define a conditional probability for some zero-probability events using a
σ-algebra 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 includes the empty subset, is closed under complement, and is closed under countable unions and countab ...
of such events (such as those arising from a
continuous random variable In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
). For example, in a bag of 2 red balls and 2 blue balls (4 balls in total), the probability of taking a red ball is $1/2$; however, when taking a second ball, the probability of it being either a red ball or a blue ball depends on the ball previously taken. For example, if a red ball was taken, then the probability of picking a red ball again would be $1/3$, since only 1 red and 2 blue balls would have been remaining. And if a blue ball was taken previously, the probability of taking a red ball will be $2/3$.

## Inverse probability

In
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
and applications, ''Bayes' rule'' relates the
odds Odds provide a measure of the likelihood of a particular outcome. They are calculated as the ratio of the number of events that produce that outcome to the number that do not. Odds are commonly used in gambling and statistics. Odds also have ...
of event $A_1$ to event $A_2$, before (prior to) and after (posterior to)
conditioning Conditioning may refer to: Science, computing, and technology * Air conditioning, the removal of heat from indoor air for thermal comfort ** Automobile air conditioning, air conditioning in a vehicle ** Ice storage air conditioning, air conditi ...
on another event $B$. The odds on $A_1$ to event $A_2$ is simply the ratio of the probabilities of the two events. When arbitrarily many events $A$ are of interest, not just two, the rule can be rephrased as ''posterior is proportional to prior times likelihood'', $P\left(A, B\right)\propto P\left(A\right) P\left(B, A\right)$ where the proportionality symbol means that the left hand side is proportional to (i.e., equals a constant times) the right hand side as $A$ varies, for fixed or given $B$ (Lee, 2012; Bertsch McGrayne, 2012). In this form it goes back to Laplace (1774) and to Cournot (1843); see Fienberg (2005). See Inverse probability and
Bayes' rule In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For exam ...
.

# Relation to randomness and probability in quantum mechanics

In a
deterministic Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and cons ...
universe, based on Newtonian concepts, there would be no probability if all conditions were known (
Laplace's demon In the history of science, Laplace's demon was a notable published articulation of causal determinism on a scientific basis by Pierre-Simon Laplace in 1814. According to determinism, if someone (the demon) knows the precise location and moment ...
), (but there are situations in which sensitivity to initial conditions exceeds our ability to measure them, i.e. know them). In the case of a
roulette Roulette is a casino game named after the French word meaning ''little wheel'' which was likely developed from the Italian game Biribi''.'' In the game, a player may choose to place a bet on a single number, various groupings of numbers, t ...
wheel, if the force of the hand and the period of that force are known, the number on which the ball will stop would be a certainty (though as a practical matter, this would likely be true only of a roulette wheel that had not been exactly levelled – as Thomas A. Bass' Newtonian Casino revealed). 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 analyzing the pattern of outcomes of repeated rolls of a roulette wheel. Physicists face the same situation in the
kinetic theory of gases Kinetic (Ancient Greek: κίνησις “kinesis”, movement or to move) may refer to: * Kinetic theory, describing a gas as particles in random motion * Kinetic energy In physics Physics is the natural science that studies m ...
, where the system, while deterministic ''in principle'', is so complex (with the number of molecules typically the order of magnitude of the
Avogadro constant The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining co ...
) that only a statistical description of its properties is feasible.
Probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
is required to describe quantum phenomena. A revolutionary discovery of early 20th century
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
was the random character of all physical processes that occur at sub-atomic scales and are governed by the laws of
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry ...
. The objective
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
evolves deterministically but, according to the Copenhagen interpretation, it deals with probabilities of observing, the outcome being explained by a
wave function collapse In quantum mechanics, wave function collapse occurs when a wave function—initially in a superposition of several eigenstates—reduces to a single eigenstate due to interaction with the external world. This interaction is called an ''observa ...
when an observation is made. However, the loss of
determinism Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and cons ...
for the sake of
instrumentalism In philosophy of science and in epistemology, instrumentalism is a methodological view that ideas are useful instruments, and that the worth of an idea is based on how effective it is in explaining and predicting phenomena. According to instrument ...
did not meet with universal approval.
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
famously remarked in a letter to
Max Born Max Born (; 11 December 1882 – 5 January 1970) was a German physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a n ...
: "I am convinced that God does not play dice". Like Einstein,
Erwin Schrödinger Erwin Rudolf Josef Alexander Schrödinger (, ; ; 12 August 1887 – 4 January 1961), sometimes written as or , was a Nobel Prize-winning Austrian physicist with Irish citizenship who developed a number of fundamental results in quantum theo ...
, who discovered the wave function, believed quantum mechanics is a
statistical Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
approximation of an underlying deterministic
reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only imaginary. The term is also used to refer to the ontological status of things, indicating their existence. In physical terms, re ...
. In some modern interpretations of the statistical mechanics of measurement,
quantum decoherence Quantum decoherence is the loss of quantum coherence. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the w ...
is invoked to account for the appearance of subjectively probabilistic experimental outcomes.

* Chance (disambiguation) * Class membership probabilities * Contingency * Equiprobability * Heuristics in judgment and decision-making *
Probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
*
Randomness In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual ra ...
*
Statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indus ...
* Estimators *
Estimation theory Estimation theory is a branch of statistics that deals with estimating the values of parameters based on measured empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their valu ...
* Probability density estimation *
Probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ...
* Pairwise independence ;In law *
Balance of probabilities In a legal dispute, one party has the burden of proof to show that they are correct, while the other party had no such burden and is presumed to be correct. The burden of proof requires a party to produce evidence to establish the truth of facts ...

# Bibliography

* Kallenberg, O. (2005) ''Probabilistic Symmetries and Invariance Principles''. Springer-Verlag, New York. 510 pp.  * Kallenberg, O. (2002) ''Foundations of Modern Probability,'' 2nd ed. Springer Series in Statistics. 650 pp.  * Olofsson, Peter (2005) ''Probability, Statistics, and Stochastic Processes'', Wiley-Interscience. 504 pp .

Virtual Laboratories in Probability and Statistics (Univ. of Ala.-Huntsville)
*
Probability and Statistics EBook
* Edwin Thompson Jaynes. ''Probability Theory: The Logic of Science''. Preprint: Washington University, (1996).
HTML index with links to PostScript files
an
PDF
(first three chapters)

* ttp://www.economics.soton.ac.uk/staff/aldrich/Probability%20Earliest%20Uses.htm Probability and Statistics on the Earliest Uses Pages (Univ. of Southampton)
Earliest Uses of Symbols in Probability and Statistics
o

A tutorial on probability and Bayes' theorem devised for first-year Oxford University students

pdf file of An Anthology of Chance Operations (1963) at
UbuWeb UbuWeb is a web-based educational resource for avant-garde material available on the internet, founded in 1996 by poet Kenneth Goldsmith. It offers visual, concrete and sound poetry, expanding to include film and sound art mp3 archives. Phil ...

Introduction to Probability – eBook
, by Charles Grinstead, Laurie Snel
Source
''(
GNU Free Documentation License The GNU Free Documentation License (GNU FDL or simply GFDL) is a copyleft license for free documentation, designed by the Free Software Foundation (FSF) for the GNU Project. It is similar to the GNU General Public License, giving readers the r ...
)'' *
Bruno de Finetti Bruno de Finetti (13 June 1906 – 20 July 1985) was an Italian probabilist statistician and actuary, noted for the "operational subjective" conception of probability. The classic exposition of his distinctive theory is the 1937 "La prévision: ...
,
Probabilità e induzione
', Bologna, CLUEB, 1993. (digital version)

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