Probability Fallacies
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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 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 mathematical formalization in probability theory, which is used widely in
areas of study Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open s ...
such as
statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation 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 ...
, science,
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 of fina ...
, gambling, artificial intelligence, machine learning, computer science,
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. Some ...
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 s ...
.


Interpretations

When dealing with experiments that are random and well-defined 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 assign numbers to describe some objective or physical state of affairs. The most popular version of objective probability is frequentist probability, 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, 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, which includes expert knowledge as well as experimental data to produce probabilities. The expert knowledge is represented by some (subjective) prior probability distribution. These data are incorporated in a likelihood function. The product of the prior and the likelihood, when normalized, results in a posterior probability distribution that incorporates all the information known to date. By Aumann's agreement theorem, 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.''The N ...
of a witness in a legal case in Europe, and often correlated with the witness's nobility. In a sense, this differs much from the modern meaning of ''probability'', which in contrast is a measure of the weight of empirical evidence, 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'' re ...
and
statistical inference Statistical inference is the process of using data analysis to infer properties of an underlying probability distribution, distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical ...
. 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, "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 polymath Gerolamo Cardano demonstrated the efficacy of defining odds 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 and
Blaise Pascal Blaise Pascal ( , , ; ; 19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, philosopher, and Catholic Church, Catholic writer. He was a child prodigy who was educated by his father, a tax collector in Rouen. Pa ...
(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 of ...
(1657) gave the earliest known scientific treatment of the subject. Jakob Bernoulli's '' Ars Conjectandi'' (posthumous, 1713) and Abraham de Moivre's ''
Doctrine of Chances 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 First ...
'' (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'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. 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, 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 mechan ...
(1778) introduced the principle of the maximum product of the probabilities of a system of concurrent errors. Adrien-Marie Legendre (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(x) = 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 wor ...
's (1850). Gauss 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 (1838), W.F. Donkin (1844, 1856), and
Morgan Crofton Morgan Crofton (1826, Dublin, Ireland – 1915, Brighton, England) was an Irish mathematician who contributed to the field of geometric probability theory. He also worked with James Joseph Sylvester and contributed an article on probability to ...
(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 '' ...
(1844), De Morgan (1864),
Glaisher Glaisher is a surname, and may refer to: *Cecilia Glaisher (1828–1892), photographer and illustrator *James Glaisher (1809–1903), English meteorologist and astronomer *James Whitbread Lee Glaisher (1848–1928), English mathematician and astron ...
(1872), and Giovanni Schiaparelli (1875). Peters's (1856) formula for ''r'', the
probable error In statistics, probable error defines the half-range of an interval about a central point for the distribution, such that half of the values from the distribution will lie within the interval and half outside.Dodge, Y. (2006) ''The Oxford Dictiona ...
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, a ...
(1816), Littrow (1833), Adolphe Quetelet (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 Paul Matthieu Hermann Laurent (2 September 1841, in Luxembourg City – 19 February 1908, in Paris, France) was a French mathematician. Despite his large body of works, Laurent series expansions for complex functions were ''not'' named after him, ...
(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 st ...
. Augustus De Morgan and George Boole improved the exposition of the theory. In 1906,
Andrey Markov Andrey Andreyevich Markov, first name also spelled "Andrei", in older works also spelled Markoff) (14 June 1856 – 20 July 1922) was a Russian mathematician best known for his work on stochastic processes. A primary subject of his research lat ...
introduced the notion of Markov chains, 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 simil ...
was developed by Andrey Kolmogorov 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, 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 formulation and the Cox formulation. In Kolmogorov's formulation (see also probability space), sets are interpreted as events and probability as a
measure Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Mea ...
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 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 o ...
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 assessment and modeling. The insurance industry and
markets Market is a term used to describe concepts such as: * Market (economics), system in which parties engage in transactions according to supply and demand * Market economy *Marketplace, a physical marketplace or public market Geography *Märket, a ...
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 handled ...
. 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 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 A game of chance is in contrast with a game of skill. It is a game whose outcome is strongly influenced by some randomizing device. Common devices used include dice, spinning tops, playing cards, roulette wheels, or numbered balls drawn from ...
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 as ...
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 A cache language model is a type of statistical language model. These occur in the natural language processing subfield of computer science and assign probabilities to given sequences of words by means of a probability distribution. Statistical lan ...
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 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 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 ''A'' is written as P(A), p(A), or \text(A). 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', 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 probability of ''A'' and ''B'', denoted as P(A \cap B).


Independent events

If two events, ''A'' and ''B'' are independent then the joint probability is :P(A \mboxB) = P(A \cap B) = P(A) P(B). 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, then the probability of ''both'' occurring is denoted as P(A \cap B) and :P(A \mboxB) = P(A \cap B) = 0 If two events are mutually exclusive, then the probability of ''either'' occurring is denoted as P(A \cup B) and :P(A\mboxB) = P(A \cup B)= P(A) + P(B) - P(A \cap B) = P(A) + P(B) - 0 = P(A) + P(B) For example, the chance of rolling a 1 or 2 on a six-sided is P(1\mbox2) = P(1) + P(2) = \tfrac + \tfrac = \tfrac.


Not mutually exclusive events

If the events are not mutually exclusive then :P\left(A \hbox B\right) = P(A \cup B) = P\left(A\right)+P\left(B\right)-P\left(A \mbox B\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'' is the probability of some event ''A'', given the occurrence of some other event ''B''. Conditional probability is written P(A \mid B), and is read "the probability of ''A'', given ''B''". It is defined by :P(A \mid B) = \frac.\, If P(B)=0 then P(A \mid B) is formally undefined by this expression. In this case A and B are independent, since P(A \cap B) = P(A)P(B) = 0. However, it is possible to define a conditional probability for some zero-probability events using a σ-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. And if a blue ball was taken previously, the probability of taking a red ball will be 2/3.


Inverse probability

In probability theory and applications, ''Bayes' rule'' relates the odds 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 condition ...
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(A, B)\propto P(A) P(B, A) 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 exampl ...
.


Summary of probabilities


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 consi ...
universe, based on Newtonian concepts, there would be no probability if all conditions were known ( Laplace's demon), (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, the ...
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 ''The Eudaemonic Pie'' is a non-fiction book about gambling by American author Thomas A. Bass. The book was initially published in April 1985 by Houghton Mifflin. Overview The book focuses on a group of University of California, Santa Cruz, ph ...
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, the energy of an object that it possesses due to its motion Art and enter ...
, 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 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 consi ...
for the sake of instrumentalism did not meet with universal approval. Albert Einstein 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, 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, industria ...
approximation of an underlying deterministic reality. 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 wave ...
is invoked to account for the appearance of subjectively probabilistic experimental outcomes.


See also

*
Chance (disambiguation) Chance may refer to: Mathematics and Science * In mathematics, likelihood of something (by way of the Likelihood function and/or Probability density function). * Chance (statistics magazine), ''Chance'' (statistics magazine) Places * Chance, ...
* Class membership probabilities * Contingency * Equiprobability * Heuristics in judgment and decision-making * Probability theory * Randomness *
Statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
* 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 In statistics, probability density estimation or simply density estimation is the construction of an estimate, based on observed data, of an unobservable underlying probability density function. The unobservable density function is thought of ...
* Probability density function * Pairwise independence ;In law * Balance of probabilities


Notes


References


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 .


External links


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
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,
Probabilità e induzione
', Bologna, CLUEB, 1993. (digital version)

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