Probability is a way of expressing knowledge or belief that an event In probability theory, an event is a set of outcomes to which a probability is assigned. Typically, when the sample space is finite, any subset of the sample space is an event (i.e. all elements of the power set of the sample space are defined as events). However, this approach does not work well in cases where the sample space is infinite, most will occur or has occurred. In mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions the concept has been given an exact meaning in probability theory Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or evolve over time in an apparently random, that is used extensively in such areas of study An academic discipline, or field of study, is a branch of knowledge which is taught and researched at the college or university level. Disciplines are defined , and recognized by the academic journals in which research is published, and the learned societies and academic departments or faculties to which their practitioners belong as mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions, statistics Statistics is the formal science of making effective use of numerical data relating to groups of individuals or experiments. It deals with all aspects of this, including not only the collection, analysis and interpretation of such data, but also the planning of the collection of data, in terms of the design of surveys and experiments, finance Finance is the science of funds management. The general areas of finance are business finance, personal finance, and public finance. Finance includes saving money and often includes lending money. The field of finance deals with the concepts of time, money, and risk and how they are interrelated. It also deals with how money is spent and budgeted, gambling Gambling is the wagering of money or something of material value on an event with an uncertain outcome with the primary intent of winning additional money and/or material goods. Typically, the outcome of the wager is evident within a short period, science Science is a systematic enterprise of gathering knowledge about nature and organizing and condensing that knowledge into testable laws and theories. As knowledge has increased, some methods have proved more reliable than others, and today the scientific method is the standard for science. It includes the use of careful observation, experimentation,, and philosophy Philosophy is the study of general and fundamental problems concerning matters such as existence, knowledge, values, reason, mind, and language. It is distinguished from other ways of addressing fundamental questions by its critical, generally systematic approach and its reliance on rational argument. The word "philosophy" comes from the to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems Complex systems is the subject of a diverse variety of sciences and professional practice methods. It is often overshadowed by the representation of natural physical organization with systems of equations, the main subject below. In the study of complex systems that are less usefully represented with equations various other kinds of narratives and.

Interpretations

The word probability does not have a consistent direct definition A definition is a passage that explains the meaning of a term , or a type of thing. The term to be defined is the definiendum (plural definienda). A term may have many different senses or meanings. For each such specific sense, a definiens (plural definientia) is a cluster of words that defines it. In fact, there are two broad categories of probability interpretations, whose adherents possess different (and sometimes conflicting) views about the fundamental nature of probability:

1. Frequentists Frequency probability is the interpretation of probability that defines an event's probability as the limit of its relative frequency in a large number of trials. The development of the frequentist account was motivated by the problems and paradoxes of the previously dominant viewpoint, the classical interpretation. The shift from the classical talk about probabilities only when dealing with experiments Experiment is the step in the scientific method that arbitrates between competing models or hypotheses. Experimentation is also used to test existing theories or new hypotheses in order to support them or disprove them. An experiment or test can be carried out using the scientific method to answer a question or investigate a problem. First an that are random Randomness is a concept of non-order or non-coherence in a sequence of symbols or steps, such that there is no intelligible pattern or combination. Randomness has somewhat disparate meanings as used in several different fields. It also has common meanings which may have loose connections with some of those more definite meanings. The Oxford and well-defined In mathematics, the term well-defined is used to specify that a certain concept or object is defined in a mathematical or logical way using a set of base axioms in an entirely unambiguous way and satisfies the properties it is required to satisfy. Usually definitions are stated unambiguously, and it is clear they satisfy the required properties. The probability of a random event denotes the relative frequency of occurrence of an experiment's outcome, when repeating the experiment. Frequentists consider probability to be the relative frequency "in the long run" of outcomes.^[1]
2. Bayesians Bayesian probability is one of the most popular interpretations of the concept of probability. The Bayesian interpretation of probability can be seen as an extension of logic that enables reasoning with uncertain statements. To evaluate the probability of a hypothesis, the Bayesian probabilist specifies some prior probability, which is then, however, assign probabilities to any statement In logic a statement is a declarative sentence that is either true or false. A statement is distinct from a sentence in that a sentence is only one formulation of a statement, whereas there may be many other formulations expressing the same statement. The term "statement" may to refer to a sentence or the idea expressed by a sentence whatsoever, even when no random process is involved. Probability, for a Bayesian, is a way to represent an individual's As commonly used, an individual is a person or any specific object in a collection. In the 15th century and earlier, and also today within the fields of statistics and metaphysics, individual means "indivisible", typically describing any numerically singular thing, but sometimes meaning "a person." . From the seventeenth degree of belief in a statement, or an objective degree of rational belief, given the evidence Evidence in its broadest sense includes everything that is used to determine or demonstrate the truth of an assertion. Giving or procuring evidence is the process of using those things that are either presumed to be true, or (b) were themselves proven via evidence, to demonstrate an assertion's truth. Evidence is the currency by which one fulfills.

Etymology

The word Probability derives In linguistics, derivation is "Used to form new words, as with happi-ness and un-happy from happy, or determination from determine. A contrast is intended with the process of inflection, which uses another kind of affix in order to form variants of the same word, as with determine/determine-s/determin-ing/determin-ed from probity, a measure of the authority Authority, from the Latin word auctoritas, means invention, advice, opinion, influence, or command. Essentially authority is imposed by superiors upon inferiors either by force of arms or by force of argument (sapiential authority). Usually authority has components of both compulsion and persuasion. For this reason, as used in Roman law, authority of a witness A witness is someone who has firsthand knowledge about a crime or significant event through their senses , and can help certify important considerations to the crime or event. A witness who has seen the event firsthand is known as an "eye-witness". Witnesses are often called before a court of law to testify in trials in a legal case A legal case is a dispute between opposing parties resolved by a court, or by some equivalent legal process. A legal case may be either civil or criminal. There is a defendant and an accuser in Europe Europe is, by convention, one of the world's seven continents. Comprising the westernmost peninsula of Eurasia, Europe is generally divided from Asia to its east by the water divide of the Ural Mountains, the Ural River, the Caspian Sea, the Caucasus region (Specification of borders) and the Black Sea to the southeast. Europe is bordered by the, and often correlated with the witness's nobility Nobility is an aristocratic social class with privileges, titles, and status acquired through heredity, by purchase, or by grant. The privileges associated with nobility may constitute substantial advantages over, or relative to, non-nobles, or may be largely honorary , but are maintained, or at least officially acknowledged, by law or government. In a sense, this differs much from the modern meaning of probability, which, in contrast, is used as a measure of the weight of empirical evidence Empirical research is research that derives its data by means of direct observation or experiment, such research is used to answer a question or test a hypothesis . The results are based upon actual evidence as opposed to theory or conjecture, as such they can be replicated in follow-up studies. Empirical research articles are published in peer-, and is arrived at from inductive reasoning Inductive reasoning, also known as induction or inductive logic, is a type of reasoning that involves moving from a set of specific facts to a general conclusion. It uses premises from objects that have been examined to establish a conclusion about an object that has not been examined. It can also be seen as a form of theory-building, in which and statistical inference Statistical inference or statistical induction comprises the use of statistics and random sampling to make inferences concerning some unknown aspect of a population. It is distinguished from descriptive statistics.^[2][3]

History

The scientific study of probability is a modern development. Gambling Gambling is the wagering of money or something of material value on an event with an uncertain outcome with the primary intent of winning additional money and/or material goods. Typically, the outcome of the wager is evident within a short period shows that there has been an interest in quantifying the ideas of probability for millennia, but exact mathematical descriptions of use in those problems only arose much later.

According to Richard Jeffrey, "Before the middle of the seventeenth century, the term 'probable' (Latin probabilis) meant approvable, and was applied in that sense, univocally, to opinion and to action. A probable action or opinion was one such as sensible people would undertake or hold, in the circumstances."^[4] However, in legal contexts especially, 'probable' could also apply to propositions for which there was good evidence.^[5]

Aside from some elementary considerations made by Girolamo Cardano Gerolamo Cardano or Girolamo Cardano was an Italian Renaissance mathematician, physician, astrologer and gambler in the 16th century, the doctrine of probabilities dates to the correspondence of Pierre de Fermat Pierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to modern calculus. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of the and Blaise Pascal Blaise Pascal was a French mathematician, physicist, and Catholic philosopher. He was a child prodigy who was educated by his father, a Tax Collector in Rouen. Pascal's earliest work was in the natural and applied sciences where he made important contributions to the construction of mechanical calculators, the study of fluids, and clarified the (1654). Christiaan Huygens Christiaan Huygens, FRS was a prominent Dutch mathematician, astronomer, physicist, horologist, and writer of early science fiction. His work included early telescopic studies elucidating the nature of the rings of Saturn and the discovery of its moon Titan, the invention of the pendulum clock and other investigations in timekeeping, and studies (1657) gave the earliest known scientific treatment of the subject. Jakob Bernoulli's Jacob Bernoulli (Basel, 27 December 1654 – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family Ars Conjectandi (posthumous, 1713) and Abraham de Moivre's Abraham de Moivre was a French mathematician famous for de Moivre's formula, which links complex numbers and trigonometry, and for his work on the normal distribution and probability theory. He was elected a Fellow of the Royal Society in 1697, and was a friend of Isaac Newton, Edmund Halley, and James Stirling. Among his fellow Huguenot exiles in Doctrine of Chances The Doctrine of Chances was the first textbook on probability theory, written by 18th-century French mathematician Abraham de Moivre and first published in 1718. De Moivre wrote in English because he resided in England at the time, having fled France to escape the persecution of Huguenots. The book's title came to be synonymous with probability (1718) treated the subject as a branch of mathematics. See Ian Hacking's Ian Hacking, CC, FRSC, FBA is a Canadian philosopher, specializing in the philosophy of science The Emergence of Probability and James Franklin's James Franklin, Australian historian of ideas and philosopher, was born in 1953 in Sydney, Australia, and educated at St. Joseph's College, Hunters Hill, NSW. His undergraduate work was at the University of Sydney , where he attended St John's College and he was influenced by philosophers David Stove and David Armstrong. He completed his PhD in 198 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's Roger Cotes FRS 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 formulas known as Newton–Cotes formulas and first introduced what is known today as Euler's formula. He was the first Plumian Professor Opera Miscellanea (posthumous, 1722), but a memoir prepared by Thomas Simpson Thomas Simpson was a British mathematician, inventor and eponym of Simpson's rule to approximate definite integrals. However, this rule was also found 200 years earlier from Johannes Kepler, in the so-called Keplersche Fassregel in 1755 (printed 1756) first applied the theory to the discussion of errors of observation. The reprint (1757) of this memoir lays down the axioms that positive and negative errors are equally probable, and that there are certain assignable limits within which all errors may be supposed to fall; continuous errors are discussed and a probability curve is given.

Pierre-Simon Laplace Pierre-Simon, marquis de Laplace was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics. He summarized and extended the work of his predecessors in his five volume Mécanique Céleste (Celestial Mechanics) (1799–1825). This work translated the geometric study of classical (1774) made the first attempt to deduce a rule for the combination of observations from the principles of the theory of probabilities. He represented the law of probability of errors by a curve y = φ(x), x being any error and y its probability, and laid down three properties of this curve:

1. it is symmetric as to the y-axis;
2. the x-axis is an asymptote In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors. In some contexts, such as algebraic geometry, an asymptote is, the probability of the error being 0;
3. the area enclosed is 1, it being certain that an error exists.

He also gave (1781) a formula for the law of facility of error (a term due to Lagrange, 1774), but one which led to unmanageable equations. Daniel Bernoulli Daniel Bernoulli was a Dutch-Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics. Bernoulli's work is still studied at length by many schools (1778) introduced the principle of the maximum product of the probabilities of a system of concurrent errors.

The method of least squares The method of least squares is used to approximately solve overdetermined systems, i.e. systems of equations in which there are more equations than unknowns. Least squares is often applied in statistical contexts, particularly regression analysis is due to Adrien-Marie Legendre Adrien-Marie Legendre was a French mathematician. He made important contributions to statistics, number theory, abstract algebra and mathematical analysis (1805), who introduced it in his Nouvelles méthodes pour la détermination des orbites des comètes (New Methods for Determining the Orbits of Comets). In ignorance of Legendre's contribution, an Irish-American writer, Robert Adrain He was born in Carrickfergus, Ireland, but left Ireland after the failure of the uprising of the United Irishmen in 1798 and moved to Princeton, New Jersey. He taught mathematics at various schools in the United States, editor of "The Analyst" (1808), first deduced the law of facility of error,

h being a constant depending on precision of observation, and c a scale factor ensuring that the area under the curve equals 1. He gave two proofs, the second being essentially the same as John Herschel's Sir John Frederick William Herschel, 1st Baronet KH, FRS was an English mathematician, astronomer, chemist, and experimental photographer/inventor, who in some years also did valuable botanical work. He was the son of astronomer Sir Friedrich Wilhelm Herschel and the father of 12 children (1850). Gauss gave the first proof which seems to have been known in Europe (the third after Adrain's) in 1809. Further proofs were given by Laplace (1810, 1812), Gauss (1823), James Ivory (1825, 1826), Hagen (1837), Friedrich Bessel (1838), W. F. Donkin (1844, 1856), and Morgan Crofton (1870). Other contributors were Ellis (1844), De Morgan (1864), Glaisher (1872), and Giovanni Schiaparelli (1875). Peters's (1856) formula for r, the probable error of a single observation, is well known.

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.

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

The modern theory of probability based on the measure theory was developed by Andrey Kolmogorov (1931).

On the geometric side (see integral geometry) contributors to The Educational Times were influential (Miller, Crofton, McColl, Wolstenholme, Watson, and Artemas Martin).

Mathematical treatment

In mathematics, a probability of an event A is represented by a real number in the range from 0 to 1 and written as P(A), p(A) or Pr(A).^[6] An impossible event has a probability of 0, and a certain event has a probability of 1. However, the converses are not always true: probability 0 events are not always impossible, nor probability 1 events certain. The rather subtle distinction between "certain" and "probability 1" is treated at greater length in the article on "almost surely".

The opposite or complement of an event A is the event [not A] (that is, the event of A not occurring); its probability is given by P(not A) = 1 - P(A).^[7] As an example, the chance of not rolling a six on a six-sided die is 1 – (chance of rolling a six) . See Complementary event for a more complete treatment.

If both the events A and B occur on a single performance of an experiment this is called the intersection or joint probability of A and B, denoted as . If two events, A and B are independent then the joint probability is

for example, if two coins are flipped the chance of both being heads is ^[8]

If either event A or event B or both events occur on a single performance of an experiment this is called the union of the events A and B denoted as . If two events are mutually exclusive then the probability of either occurring is

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

If the events are not mutually exclusive then

For example, when drawing a single card at random from a regular deck of cards, the chance of getting a heart or a face card (J,Q,K) (or one that is both) is , because of 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 is the probability of some event A, given the occurrence of some other event B. Conditional probability is written P(A|B), and is read "the probability of A, given B". It is defined by

^[9]

If P(B) = 0 then is undefined.

Theory

Like other theories, the theory of probability is a representation of probabilistic concepts in formal terms—that is, in terms that can be considered separately from their meaning. These formal terms are manipulated by the rules of mathematics and logic, and any results are then interpreted or translated back into the problem domain.

There have been at least two successful attempts to formalize probability, namely the Kolmogorov formulation and the Cox formulation. In Kolmogorov's formulation (see probability space), sets are interpreted as events and probability itself as a measure on a class of sets. In Cox's theorem, probability is taken as a primitive (that is, not further analyzed) and the emphasis is on constructing a consistent assignment of probability values to propositions. In both cases, the laws of probability are the same, except for technical details.

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 laws of probability as they are usually understood.

Applications

Two major applications of probability theory in everyday life are in risk assessment and in trade on commodity markets. Governments typically apply probabilistic methods in environmental regulation where it is called "pathway analysis", often measuring well-being using methods that are stochastic in nature, and choosing projects to undertake based on statistical analyses of their probable effect on the population as a whole.

A good example is the effect of the perceived probability of any widespread Middle East conflict on oil prices - which have ripple effects in the economy as a whole. An assessment by a commodity trader that a war is more likely vs. less likely sends prices up or down, and signals other traders of that opinion. Accordingly, the probabilities are not assessed independently nor necessarily very rationally. The theory of behavioral finance emerged to describe the effect of such groupthink on pricing, on policy, and on peace and conflict.

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

Another significant application of probability theory in everyday life is reliability. Many consumer products, such as automobiles and consumer electronics, utilize reliability theory in the design of the product in order to reduce the probability of failure. The probability of failure may be closely associated with the product's warranty.

Relation to randomness

In a deterministic universe, based on Newtonian concepts, there is no probability if all conditions are known. In the case of a roulette wheel, if the force of the hand and the period of that force are known, then the number on which the ball will stop would be a certainty. Of course, this also assumes knowledge of inertia and friction of the wheel, weight, smoothness and roundness of the ball, variations in hand speed during the turning and so forth. A probabilistic description can thus be more useful than Newtonian mechanics for analyzing the pattern of outcomes of repeated rolls of roulette wheel. Physicists face the same situation in kinetic theory of gases, where the system, while deterministic in principle, is so complex (with the number of molecules typically the order of magnitude of Avogadro constant 6.02·10²³) that only statistical description of its properties is feasible.

A revolutionary discovery of 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 wave function itself evolves deterministically as long as no observation is made, but, according to the prevailing Copenhagen interpretation, the randomness caused by the wave function collapsing when an observation is made, is fundamental. This means that probability theory is required to describe nature. Others never came to terms with the loss of determinism. Albert Einstein famously remarked in a letter to Max Born: Jedenfalls bin ich überzeugt, daß der Alte nicht würfelt. (I am convinced that God does not play dice). Although alternative viewpoints exist, such as that of quantum decoherence being the cause of an apparent random collapse, at present there is a firm consensus among physicists that probability theory is necessary to describe quantum phenomena.^{[citation needed]}

See also

• Black Swan theory
• Calculus of predispositions
• Class membership probabilities
• Decision theory
• Equiprobable
• Fuzzy measure theory
• Game theory
• Gaming mathematics
• Information theory
• Important publications in probability
• Measure theory
• Negative probability
• Probabilistic argumentation
• Probabilistic logic
• Random fields
• Random variable
• List of scientific journals in probability
• List of statistical topics
• Stochastic process
• Wiener process

Notes

1. ^ The Logic of Statistical Inference, Ian Hacking, 1965
2. ^ The Emergence of Probability: A Philosophical Study of Early Ideas about Probability, Induction and Statistical Inference, Ian Hacking, Cambridge University Press, 2006, ISBN 0521685575, 9780521685573
3. ^ The Cambridge History of Seventeenth-century Philosophy, Daniel Garber, 2003
4. ^ Jeffrey, R.C., Probability and the Art of Judgment, Cambridge University Press. (1992). pp. 54-55 . ISBN 0-521-39459-7
5. ^ Franklin, J., The Science of Conjecture: Evidence and Probability Before Pascal, Johns Hopkins University Press. (2001). pp. 22, 113, 127
6. ^ Olofsson, Peter. (2005) Page 8.
7. ^ Olofsson, page 9
8. ^ Olofsson, page 35.
9. ^ Olofsson, page 29.

References

• Kallenberg, O. (2005) Probabilistic Symmetries and Invariance Principles. Springer -Verlag, New York. 510 pp. ISBN 0-387-25115-4
• Kallenberg, O. (2002) Foundations of Modern Probability, 2nd ed. Springer Series in Statistics. 650 pp. ISBN 0-387-95313-2
• Olofsson, Peter (2005) Probability, Statistics, and Stochastic Processes, Wiley-Interscience. 504 pp ISBN 0-471-67969-0.

Quotations

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

External links

• Probability and Statistics EBook
• Edwin Thompson Jaynes. Probability Theory: The Logic of Science. Preprint: Washington University, (1996). — HTML index with links to PostScript files and PDF (first three chapters)
• People from the History of Probability and Statistics (Univ. of Southampton)
• Probability and Statistics on the Earliest Uses Pages (Univ. of Southampton)
• Earliest Uses of Symbols in Probability and Statistics on Earliest Uses of Various Mathematical Symbols
• 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
• Probability Theory Guide for Non-Mathematicians
• Understanding Risk and Probability with BBC raw

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