TheInfoList Probability is the branch of
mathematics Mathematics (from Greek: ) includes the study of such topics as quantity (number theory), structure (algebra), space (geometry), and change (analysis). It has no generally accepted definition. Mathematicians seek and use patterns to formulate ...
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 event ...
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 Greek ''ax├Ł┼Źma'' () 'that which is thought worthy or fit' or 'that ...
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 of ...
, which is used widely in
areas of study An academic discipline or academic field is a subdivision of knowledge that is taught and researched at the college or university level. Disciplines are defined (in part) and recognized by the academic journals in which research is published, an ...
such as
statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical ...
,
mathematics Mathematics (from Greek: ) includes the study of such topics as quantity (number theory), structure (algebra), space (geometry), and change (analysis). It has no generally accepted definition. Mathematicians seek and use patterns to formulate ...
,
science Science (from the Latin word ''scientia'', meaning "knowledge") is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe."... modern science is a discovery as w ...
,
finance Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investments. Savers and investors have money available which could ...
,
gambling A gambling stand in Paris Gambling (also known as betting) is the wagering of money or something of value (referred to as "the stakes") on an event with an uncertain outcome, with the primary intent of winning money or material goods. Gambling ...
,
artificial intelligence Artificial intelligence (AI) is intelligence demonstrated by machines, unlike the natural intelligence displayed by humans and animals, which involves consciousness and emotionality. The distinction between the former and the latter categorie ... ,
machine learning#REDIRECT machine learning#REDIRECT machine learning {{Redirect category shell, 1= {{R from other capitalisation ...
{{Redirect category shell, 1= {{R from other capitalisation ...
,
computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of algorithmic processes, comp ...
,
game theory Game theory is the study of mathematical models of strategic interaction among rational decision-makers.Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppviiŌĆōxi It has ...
, and
philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, existence, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. The term was proba ...
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 sys ...
.

# Interpretations

When dealing with
experiment An experiment is a procedure carried out to support, refute, or validate a hypothesis. Experiments provide insight into cause-and-effect by demonstrating what outcome occurs when a particular factor is manipulated. Experiments vary greatly in ...
s that are
random In common parlance, 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 ran ...
and
well-defined In mathematics, an expression is called well-defined or ''unambiguous'' if its definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well-defined'', ''ill-defined'' or ''ambiguous''. A function is wel ...
in a purely theoretical setting (like tossing a fair coin), probabilities can be numerically described by the number of desired outcomes, divided by the total number of all outcomes. For example, tossing a fair 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 Ayn Rand. Rand first expressed Objectivism in her fiction, most notably ''The Fountainhead'' (1943) and ''Atlas Shrugged'' (1957), and later in non-fiction essays and book ...
assign numbers to describe some objective or physical state of affairs. The most popular version of objective probability is
frequentist probabilityFrequentist inference is a type of statistical inference that draws conclusions from sample data by emphasizing the frequency or proportion of the data. An alternative name is frequentist statistics. This is the inference framework in which the well ...
, 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 one interpretation of the concept of probability. Theorists who adopt this interpretation think of probability as a physical propensity, or disposition, or tendency of a given type of physical situation to yie ...
, 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." 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 of ...
, 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 into ac ...
. These data are incorporated in a
likelihood function In statistics, the likelihood function (often simply called the likelihood) measures the goodness of fit of a statistical model to a sample of data for given values of the unknown parameters. It is formed from the joint probability distribution of t ...
. The product of the prior and the likelihood, when normalized, results in a
posterior probability distribution In Bayesian statistics, the posterior probability of a random event or an uncertain proposition is the conditional probability that is assigned after the relevant evidence or background is taken into account. "Posterior", in this context, means ...
that incorporates all the information known to date. By
Aumann's agreement theorem In game theory, Aumann's agreement theorem is a theorem which demonstrates that rational agents with common knowledge of each other's beliefs cannot agree to disagree. It was first formulated in the 1976 paper titled "Agreeing to Disagree" by Robert ...
, 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 ''probabilitas'', which can also mean " probity", a measure of the
authority In the fields of sociology and political science, authority is the legitimate power that a person or a group of persons possess and practice over other people. In a civil state, ''authority'' is made formal by way of a judicial branch and an execu ...
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, ei ...
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 more ...
in
Europe Europe is a continent located entirely in the Northern Hemisphere and mostly in the Eastern Hemisphere. It comprises the westernmost peninsulas of the continental landmass of Eurasia, and is bordered by the Arctic Ocean to the north, the Atlant ...
, and often correlated with the witness's
nobility Nobility is a social class normally ranked immediately below royalty and found in some societies that have a formal aristocracy. Nobility has often been an estate of the realm that possessed more acknowledged privilege and higher social st ...
. 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 is the information received by means of the senses, particularly by observation and documentation of patterns and behavior through experimentation. The term comes from the Greek word for experience, ß╝É╬╝ŽĆ╬Ą╬╣Žü╬»╬▒ (''empeir ...
, and is arrived at from
inductive reasoning Inductive reasoning is a method of reasoning in which the premises are viewed as supplying ''some'' evidence, but not full assurance, of the truth of the conclusion. It is also described as a method where one's experiences and observations, incl ...
and
statistical inference#REDIRECT Statistical inference#REDIRECT Statistical inference {{R from other capitalisation ...
{{R from other capitalisation ...
.
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 (from Greek: ) includes the study of such topics as quantity (number theory), structure (algebra), space (geometry), and change (analysis). It has no generally accepted definition. Mathematicians seek and use patterns to formulate ...
.
Gambling A gambling stand in Paris Gambling (also known as betting) is the wagering of money or something of value (referred to as "the stakes") on an event with an uncertain outcome, with the primary intent of winning money or material goods. 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 ŌĆō 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 prob ...
, "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. 54ŌĆō55 . 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) 140px, Al-Kindi's ''Book of Cryptographic Messages'' contains the earliest known use of
statistical inference#REDIRECT Statistical inference#REDIRECT Statistical inference {{R from other capitalisation ...
{{R from other capitalisation ...
(9th century) The early form of statistical inference were developed by Mathematics in medieval Islam, Middle Eastern mathematicians studying cryptography between the 8th and 13th centuries. Al-Khalil ibn Ahmad al-Farahidi, Al-Khalil (717ŌĆō786) wrote the ''Book of Cryptographic Messages'' which contains the first use of
permutations and combinations In combinatorics, the twelvefold way is a systematic classification of 12 related enumerative problems concerning two finite sets, which include the classical problems of counting permutations, combinations, multisets, and partitions either of a se ...
to list all possible
Arabic Arabic (, ' or , ' or ) is a Semitic language that first emerged in the 1st to 4th centuries CE.Semitic languages: an international handbook / edited by Stefan Weninger; in collaboration with Geoffrey Khan, Michael P. Streck, Janet C. E.Wats ... words with and without vowels.
Al-Kindi Abu Y┼½suf Ya╩╗q┼½b ibn ╩╝IsßĖź─üq aß╣Ż-ß╣óabb─üßĖź al-Kind─½ (; ar, žŻž©┘ł ┘Ŗ┘łž│┘ü ┘Ŗž╣┘é┘łž© ž©┘å žźž│žŁž¦┘é ž¦┘äžĄž©┘æž¦žŁ ž¦┘ä┘ā┘åž»┘Ŗ; la, Alkindus; c. 801ŌĆō873 AD) was an Arab Muslim philosopher, polymath, mathematician, physician and ...
(801ŌĆō873) made the earliest known use of
statistical inference#REDIRECT Statistical inference#REDIRECT Statistical inference {{R from other capitalisation ...
{{R from other capitalisation ...
in his work on
cryptanalysis cipher machine Cryptanalysis (from the Greek language, Greek ''krypt├│s'', "hidden", and ''anal├Įein'', "to analyze") is the study of analyzing information systems in order to study the hidden aspects of the systems. Cryptanalysis is used to brea ...
and
frequency analysis In cryptanalysis, frequency analysis (also known as counting letters) is the study of the frequency of letters or groups of letters in a ciphertext. The method is used as an aid to breaking classical ciphers. Frequency analysis is based on the ...
. An important contribution of
Ibn Adlan ╩╗Af─½f al-D─½n ╩╗Al─½ ibn ╩╗Adl─ün al-Mawsil─½ ( ar, ž╣┘ü┘Ŗ┘ü ┘äž»┘Ŗ┘å ž╣┘ä┘Ŗ ž©┘å ž╣ž»┘äž¦┘å ž¦┘ä┘ģ┘łžĄ┘ä┘Ŗ ; 1187ŌĆō1268 CE), born in Mosul, was an Arab cryptologist, linguist and poet who is known for his early contributions to cryptanalysis, ...
(1187ŌĆō1268) was on
sample sizeSample size determination is the act of choosing the number of observations or replicates to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a population fr ...
for use of frequency analysis.  The sixteenth-century
Italian Italian may refer to: * Anything of, from, or related to the country and nation of Italy ** Italians, an ethnic group or simply a citizen of the Italian Republic ** Italian language, a Romance language *** Regional Italian, regional variants of the ...
polymath
Gerolamo Cardano Gerolamo (also Girolamo or Geronimo) Cardano (; french: link=no, J├®r├┤me Cardan; la, Hieronymus Cardanus; 24 September 1501 (O. S.)ŌĆō 21 September 1576 (O. S.)) was an Italian polymath, whose interests and proficiencies ranged through those of m ...
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 can be demon ...
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 ŌĆō 12 January 1665) was a French lawyer at the ''Parlement'' of Toulouse, France, and a mathematician who is given credit for early developments that led to infinitesimal calculus, incl ... and
Blaise Pascal Blaise Pascal ( , , ; ; 19 June 1623 ŌĆō 19 August 1662) was a French mathematician, physicist, inventor, philosopher, writer and Catholic theologian. He was a child prodigy who was educated by his father, a tax collector in Rouen. Pascal's ... (1654).
Christiaan Huygens Christiaan Huygens ( , also , ; la, Hugenius; 14 April 1629 ŌĆō 8 July 1695), also spelled Huyghens, was a Dutch mathematician, physicist, astronomer and inventor, who is widely regarded as one of the greatest scientists of all time and a majo ... (1657) gave the earliest known scientific treatment of the subject.
Jakob Bernoulli Jacob Bernoulli (also known as James or Jacques; ŌĆō 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 Leibn ... 's ''
Ars Conjectandi ''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 co ...
'' (posthumous, 1713) and
Abraham de Moivre Abraham de Moivre (; 26 May 166727 November 1754) was a French mathematician known for de Moivre's formula, a formula that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory. He moved to En ... 's ''
Doctrine of Chances In law, the doctrine of chances is a rule of evidence that allows evidence to show that it is unlikely a defendant would be repeatedly, innocently involved in similar, suspicious circumstances. Normally, under Federal Rule of Evidence 404, evidenc ...
'' (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 a m ... '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 In statistics, propagation of uncertainty (or propagation of error) is the effect of variables' uncertainties (or errors, more specifically random errors) on the uncertainty of a function based on them. When the variables are the values of experime ...
may be traced back to
Roger Cotes Roger Cotes FRS (10 July 1682 ŌĆō 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 quadrature ...
's ''Opera Miscellanea'' (posthumous, 1722), but a memoir prepared by
Thomas Simpson Thomas Simpson FRS (20 August 1710 ŌĆō 14 May 1761) was a British mathematician and inventor known for the eponymous Simpson's rule to approximate definite integrals. The attribution, as often in mathematics, can be debated: this rule had been fo ... 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 ŌĆō 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 a ... . 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 errorŌĆödisregarding 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 A journal, from the Old French ''journal'' (meaning "daily"), may refer to: *Bullet journal, a method of personal organizations *Diary, a record of what happened over the course of a day or other period *Daybook, also known as a general journal, a ...
, 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 (; ŌĆō 17 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 mechanic ...
(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 ŌĆō 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 named a ...
(1805) developed the
method of least squares The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the resid ...
, and introduced it in his ''Nouvelles m├®thodes pour la d├®termination des orbites des com├©tes'' (''New Methods for Determining the Orbits of Comets''). In ignorance of Legendre's contribution, an Irish-American writer, , 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 ŌĆō 11 May 1871) was an English polymath, mathematician, astronomer, chemist, inventor, experimental photographer who invented the blueprint, and did botanical work. Hersche ...
's (1850).
Gauss Johann Carl Friedrich Gauss (; german: Gau├¤ ; 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 referred ... 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 James Francis Ivory (born June 7, 1928) is an American film director, producer, and screenwriter. For many years, he worked extensively with Indian-born film producer Ismail Merchant, his domestic as well as professional partner, and with screenw ...
(1825, 1826), Hagen (1837),
Friedrich Bessel Friedrich Wilhelm Bessel (; 22 July 1784 ŌĆō 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 of ...
(1838), W.F. Donkin (1844, 1856), and Morgan Crofton (1870). Other contributors were Ellis (1844),
De MorganDe Morgan or de Morgan is a surname, and may refer to: *Augustus De Morgan (1806–1871), British mathematician and logician. ** De Morgan's laws (or De Morgan's theorem), a set of rules from propositional logic. ** The De Morgan Medal, a trienni ... (1864),
Glaisher (1872), and
Giovanni Schiaparelli Giovanni Virginio Schiaparelli ( , also , ; 14 March 1835 ŌĆō 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 Observatory, un ...
(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, 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. In 1906, Andrey Markov introduced the notion of Markov chains, which played an important role in stochastic processes theory and its applications. The modern theory of probability based on the Measure (mathematics), measure theory was developed by Andrey Kolmogorov in 1931. On the geometric side, contributors to ''The Educational Times'' were influential (Miller, Crofton, McColl, Wolstenholme, Watson, and Artemas Martin). See integral geometry for more info.

# Theory

Like other theory, theories, the probability theory, theory of probability is a representation of its concepts in formal termsŌĆöthat 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 formulation and the Richard Threlkeld Cox, Cox formulation. In Kolmogorov's formulation (see also probability space), Set (mathematics), sets are interpreted as Event (probability theory), events and probability as a Measure (mathematics), 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 probability axioms, laws of probability are the same, except for technical details. There are other methods for quantifying uncertainty, such as the DempsterŌĆōShafer 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 assessment and Statistical model, modeling. The insurance industry and Market (economics), markets use actuarial science to determine pricing and make trading decisions. Governments apply probabilistic methods in environmental regulation, entitlement analysis (reliability theory of aging and longevity), and financial regulation. A good 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 emerged to describe the effect of such groupthink 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. The discovery of rigorous methods to assess and combine probability assessments has changed society. Another significant application of probability theory in everyday life is reliability (statistics), reliability. Many consumer products, such as automobiles 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. The cache language model and other Statistical Language Model, statistical language models that are used in natural language processing 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 of the experiment, sometimes denoted as $\Omega$. The power set 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 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 Function (mathematics), 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 event ...
''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 [not ''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. If two events ''A'' and ''B'' occur on a single performance of an experiment, this is called the intersection or Joint distribution, joint probability of ''A'' and ''B'', denoted as $P\left(A \cap B\right)$.

## Independent events

If two events, ''A'' and ''B'' are Independence (probability theory), independent 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 events, mutually exclusive, 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 events, mutually exclusive, 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 single card at random from a regular deck of cards, the chance of getting a heart or a face card (J,Q,K) (or one that is 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'' 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 (mathematics), undefined by this expression. However, it is possible to define a conditional probability for some zero-probability events using a Žā-algebra of such events (such as those arising from a continuous random variable). 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.

## 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 of ...
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 can be demon ...
of event $A_1$ to event $A_2$, before (prior to) and after (posterior to) Conditional probability, conditioning 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.

# Relation to randomness and probability in quantum mechanics

In a determinism, deterministic universe, based on Newtonian mechanics, Newtonian concepts, there would be no probability if all conditions were known (Laplace's demon), (but there are situations in which chaos theory, sensitivity to initial conditions exceeds our ability to measure them, i.e. know them). In the case of a roulette 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 ŌĆō as Thomas A. Bass' eudaemons, 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 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 the Avogadro constant ) that only a statistical description of its properties is feasible. Probability theory is required to describe quantum phenomena. A revolutionary discovery of early 20th century physics was the random character of all physical processes that occur at sub-atomic scales and are governed by the laws of quantum mechanics. The objective wave function evolves deterministically but, according to the Copenhagen interpretation, it deals with probabilities of observing, the outcome being explained by a wave function collapse when an observation is made. However, the loss of determinism for the sake of instrumentalism did not meet with universal approval. Albert Einstein famously :de:Albert Einstein#Quellenangaben und Anmerkungen, remarked in a letter to Max Born: "I am convinced that God does not play dice". Like Einstein, Erwin Schr├Čdinger, who Schr├Čdinger equation#Historical background and development, discovered the wave function, believed quantum mechanics is a statistical approximation of an underlying deterministic reality. In some modern interpretations of the statistical mechanics of measurement, quantum decoherence is invoked to account for the appearance of subjectively probabilistic experimental outcomes.

* Chance (disambiguation) * Class membership probabilities * Contingency (philosophy), Contingency * Equiprobability * Heuristics in judgment and decision-making * Probability theory * Randomness * Statistics * Estimators * Estimation theory * Probability density function *Pairwise independence ;In law * Balance of probabilities

# Bibliography

* Olav Kallenberg, 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). ŌĆ
HTML index with links to PostScript files
an
PDF
(first three chapters)

* [http://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
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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
Introduction to Probability ŌĆō eBook