Probability is a Way of Expressing Knowledge or Belief That an Event Will Occur or Has Occurred

7/27/2019 Probability is a Way of Expressing Knowledge or Belief That an Event Will Occur or Has Occurred

1/26

Probability is a way of expressing knowledge or belief that aneventwill occur or has occurred. In

mathematicsthe concept has been given an exact meaning inprobability theory, that is used

extensively in suchareas of studyasmathematics,statistics,finance,gambling,science, and

philosophyto draw conclusions about the likelihood of potential events and the underlying

mechanics ofcomplex systems.

History of probability

From Wikipedia, the free encyclopedia

Jump to:navigation,search.

------------------An introduction to probability theory and mathematical statistics that emphasizes the

probabilistic foundations required to understand probability models and statistical methods. Topicscovered will include the probability axioms, basic combinatorics, discrete and continuous random

variables, probability distributions, mathematical expectation, common families of probability

distributions, and the central limit theorem.

History of science

Background

Theories/sociologyHistoriographyPseudoscience

By era

In early culturesin Classical Antiquity

In the Middle AgesIn the Renaissance

Scientific Revolution

By topic

Natural sciences

Astronomy

BiologyBotanyChemistryEcologyGeography

Geology

Paleontology
http://en.wikipedia.org/wiki/Event_(probability_theory)http://en.wikipedia.org/wiki/Event_(probability_theory)http://en.wikipedia.org/wiki/Event_(probability_theory)http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Areas_of_studyhttp://en.wikipedia.org/wiki/Areas_of_studyhttp://en.wikipedia.org/wiki/Areas_of_studyhttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Statisticshttp://en.wikipedia.org/wiki/Statisticshttp://en.wikipedia.org/wiki/Statisticshttp://en.wikipedia.org/wiki/Financehttp://en.wikipedia.org/wiki/Financehttp://en.wikipedia.org/wiki/Financehttp://en.wikipedia.org/wiki/Gamblinghttp://en.wikipedia.org/wiki/Gamblinghttp://en.wikipedia.org/wiki/Gamblinghttp://en.wikipedia.org/wiki/Sciencehttp://en.wikipedia.org/wiki/Sciencehttp://en.wikipedia.org/wiki/Sciencehttp://en.wikipedia.org/wiki/Philosophyhttp://en.wikipedia.org/wiki/Philosophyhttp://en.wikipedia.org/wiki/Complex_systemshttp://en.wikipedia.org/wiki/Complex_systemshttp://en.wikipedia.org/wiki/Complex_systemshttp://en.wikipedia.org/wiki/History_of_probability#column-onehttp://en.wikipedia.org/wiki/History_of_probability#column-onehttp://en.wikipedia.org/wiki/History_of_probability#column-onehttp://en.wikipedia.org/wiki/History_of_probability#searchInputhttp://en.wikipedia.org/wiki/History_of_probability#searchInputhttp://en.wikipedia.org/wiki/History_of_probability#searchInputhttp://en.wikipedia.org/wiki/History_of_sciencehttp://en.wikipedia.org/wiki/History_of_sciencehttp://en.wikipedia.org/wiki/Theories_and_sociology_of_the_history_of_sciencehttp://en.wikipedia.org/wiki/Theories_and_sociology_of_the_history_of_sciencehttp://en.wikipedia.org/wiki/Historiography_of_sciencehttp://en.wikipedia.org/wiki/History_of_pseudosciencehttp://en.wikipedia.org/wiki/History_of_science_in_early_cultureshttp://en.wikipedia.org/wiki/History_of_science_in_early_cultureshttp://en.wikipedia.org/wiki/History_of_science_in_Classical_Antiquityhttp://en.wikipedia.org/wiki/Science_in_the_Middle_Ageshttp://en.wikipedia.org/wiki/History_of_science_in_the_Renaissancehttp://en.wikipedia.org/wiki/Scientific_Revolutionhttp://en.wikipedia.org/wiki/History_of_natural_sciencehttp://en.wikipedia.org/wiki/History_of_astronomyhttp://en.wikipedia.org/wiki/History_of_astronomyhttp://en.wikipedia.org/wiki/History_of_biologyhttp://en.wikipedia.org/wiki/History_of_botanyhttp://en.wikipedia.org/wiki/History_of_chemistryhttp://en.wikipedia.org/wiki/History_of_ecologyhttp://en.wikipedia.org/wiki/History_of_ecologyhttp://en.wikipedia.org/wiki/History_of_geographyhttp://en.wikipedia.org/wiki/History_of_geologyhttp://en.wikipedia.org/wiki/History_of_geologyhttp://en.wikipedia.org/wiki/History_of_paleontologyhttp://en.wikipedia.org/wiki/History_of_paleontologyhttp://en.wikipedia.org/wiki/File:Libr0310.jpghttp://en.wikipedia.org/wiki/History_of_paleontologyhttp://en.wikipedia.org/wiki/History_of_geologyhttp://en.wikipedia.org/wiki/History_of_geographyhttp://en.wikipedia.org/wiki/History_of_ecologyhttp://en.wikipedia.org/wiki/History_of_chemistryhttp://en.wikipedia.org/wiki/History_of_botanyhttp://en.wikipedia.org/wiki/History_of_biologyhttp://en.wikipedia.org/wiki/History_of_astronomyhttp://en.wikipedia.org/wiki/History_of_natural_sciencehttp://en.wikipedia.org/wiki/Scientific_Revolutionhttp://en.wikipedia.org/wiki/History_of_science_in_the_Renaissancehttp://en.wikipedia.org/wiki/Science_in_the_Middle_Ageshttp://en.wikipedia.org/wiki/History_of_science_in_Classical_Antiquityhttp://en.wikipedia.org/wiki/History_of_science_in_early_cultureshttp://en.wikipedia.org/wiki/History_of_pseudosciencehttp://en.wikipedia.org/wiki/Historiography_of_sciencehttp://en.wikipedia.org/wiki/Theories_and_sociology_of_the_history_of_sciencehttp://en.wikipedia.org/wiki/History_of_sciencehttp://en.wikipedia.org/wiki/History_of_probability#searchInputhttp://en.wikipedia.org/wiki/History_of_probability#column-onehttp://en.wikipedia.org/wiki/Complex_systemshttp://en.wikipedia.org/wiki/Philosophyhttp://en.wikipedia.org/wiki/Sciencehttp://en.wikipedia.org/wiki/Gamblinghttp://en.wikipedia.org/wiki/Financehttp://en.wikipedia.org/wiki/Statisticshttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Areas_of_studyhttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Event_(probability_theory)


2/26

Probabilityhas a dual aspect: on the one hand the probability or

likelihood of hypotheses given the evidence for them, and on the

other hand the behavior ofstochastic processessuch as the throwing

of dice or coins. The study of the former is historically older in, for

example, the law of evidence, while the mathematical treatment of

dice began with the work ofPascalandFermatin the 1650s.

Probabilityis distinguished fromstatistics. (SeeHistory of

Statistics). While statistics deals with data and inferences from it,

(stochastic) probability deals with the stochastic (random) processes

which lie behind data or outcomes.

Contents

[hide]

1 Etymology

2 Origins

3 18th Century

4 19th Century

5 20th Century

6 Bibliography

7 References

8 External links

[edit] Etymology

Probable and likely and their cognates in other modern languages derive from medieval

learnedLatinprobabilis and verisimilis, deriving fromCiceroand generally applied to an

opinion to meanplausible orgenerally approved.[1]

[edit] Origins

See also:Timeline of probability and statistics

Ancient and medievallaw of evidencedeveloped a grading of degrees of proof, probabilities,

presumptionsandhalf-proofto deal with the uncertainties of evidence in court.[2]In

Renaissancetimes, betting was discussed in terms of odds such as "ten to one" and maritime

insurancepremiums were estimated based on intuitive risks, but there was no theory on how

to calculate such odds or premiums.[3]

The mathematical methods of probability arose in the correspondence ofPierre de Fermat

andBlaise Pascal(1654) on such questions as the fair division of the stake in an interrupted

game of chance.Christiaan Huygens(1657) gave a comprehensive treatment of the subject.[4]

[edit] 18th Century

Physics

Mathematics

AlgebraCalculus

CombinatoricsGeometry Logic

ProbabilityStatisticsTrigonometrySocial sciences

AnthropologyEconomicsLinguisticsPolitical sciencePsychology

Sociology

TechnologyAgricultural scienceComputer scienceMaterials science

Medicine

Navigational pages

TimelinesPortalCategories
http://en.wikipedia.org/wiki/Probabilityhttp://en.wikipedia.org/wiki/Probabilityhttp://en.wikipedia.org/wiki/Stochastic_processeshttp://en.wikipedia.org/wiki/Stochastic_processeshttp://en.wikipedia.org/wiki/Stochastic_processeshttp://en.wikipedia.org/wiki/Blaise_Pascalhttp://en.wikipedia.org/wiki/Blaise_Pascalhttp://en.wikipedia.org/wiki/Blaise_Pascalhttp://en.wikipedia.org/wiki/Pierre_de_Fermathttp://en.wikipedia.org/wiki/Pierre_de_Fermathttp://en.wikipedia.org/wiki/Pierre_de_Fermathttp://en.wikipedia.org/wiki/Probabilityhttp://en.wikipedia.org/wiki/Probabilityhttp://en.wikipedia.org/wiki/Statisticshttp://en.wikipedia.org/wiki/Statisticshttp://en.wikipedia.org/wiki/Statisticshttp://en.wikipedia.org/wiki/History_of_Statisticshttp://en.wikipedia.org/wiki/History_of_Statisticshttp://en.wikipedia.org/wiki/History_of_Statisticshttp://en.wikipedia.org/wiki/History_of_Statisticshttp://toggletoc%28%29/http://toggletoc%28%29/http://toggletoc%28%29/http://en.wikipedia.org/wiki/History_of_probability#Etymologyhttp://en.wikipedia.org/wiki/History_of_probability#Etymologyhttp://en.wikipedia.org/wiki/History_of_probability#Originshttp://en.wikipedia.org/wiki/History_of_probability#Originshttp://en.wikipedia.org/wiki/History_of_probability#18th_Centuryhttp://en.wikipedia.org/wiki/History_of_probability#18th_Centuryhttp://en.wikipedia.org/wiki/History_of_probability#19th_Centuryhttp://en.wikipedia.org/wiki/History_of_probability#19th_Centuryhttp://en.wikipedia.org/wiki/History_of_probability#20th_Centuryhttp://en.wikipedia.org/wiki/History_of_probability#20th_Centuryhttp://en.wikipedia.org/wiki/History_of_probability#Bibliographyhttp://en.wikipedia.org/wiki/History_of_probability#Bibliographyhttp://en.wikipedia.org/wiki/History_of_probability#Referenceshttp://en.wikipedia.org/wiki/History_of_probability#Referenceshttp://en.wikipedia.org/wiki/History_of_probability#External_linkshttp://en.wikipedia.org/wiki/History_of_probability#External_linkshttp://en.wikipedia.org/w/index.php?title=History_of_probability&action=edit&section=1http://en.wikipedia.org/w/index.php?title=History_of_probability&action=edit&section=1http://en.wikipedia.org/wiki/Medieval_Latinhttp://en.wikipedia.org/wiki/Medieval_Latinhttp://en.wikipedia.org/wiki/Medieval_Latinhttp://en.wikipedia.org/wiki/Cicerohttp://en.wikipedia.org/wiki/Cicerohttp://en.wikipedia.org/wiki/Cicerohttp://en.wikipedia.org/wiki/History_of_probability#cite_note-0http://en.wikipedia.org/wiki/History_of_probability#cite_note-0http://en.wikipedia.org/wiki/History_of_probability#cite_note-0http://en.wikipedia.org/w/index.php?title=History_of_probability&action=edit&section=2http://en.wikipedia.org/w/index.php?title=History_of_probability&action=edit&section=2http://en.wikipedia.org/wiki/Timeline_of_probability_and_statisticshttp://en.wikipedia.org/wiki/Timeline_of_probability_and_statisticshttp://en.wikipedia.org/wiki/Timeline_of_probability_and_statisticshttp://en.wikipedia.org/wiki/Law_of_evidencehttp://en.wikipedia.org/wiki/Law_of_evidencehttp://en.wikipedia.org/wiki/Law_of_evidencehttp://en.wikipedia.org/wiki/Presumptionhttp://en.wikipedia.org/wiki/Presumptionhttp://en.wikipedia.org/wiki/Half-proofhttp://en.wikipedia.org/wiki/Half-proofhttp://en.wikipedia.org/wiki/Half-proofhttp://en.wikipedia.org/wiki/History_of_probability#cite_note-1http://en.wikipedia.org/wiki/History_of_probability#cite_note-1http://en.wikipedia.org/wiki/History_of_probability#cite_note-1http://en.wikipedia.org/wiki/Renaissancehttp://en.wikipedia.org/wiki/Renaissancehttp://en.wikipedia.org/wiki/Insurancehttp://en.wikipedia.org/wiki/Insurancehttp://en.wikipedia.org/wiki/History_of_probability#cite_note-2http://en.wikipedia.org/wiki/History_of_probability#cite_note-2http://en.wikipedia.org/wiki/History_of_probability#cite_note-2http://en.wikipedia.org/wiki/Pierre_de_Fermathttp://en.wikipedia.org/wiki/Pierre_de_Fermathttp://en.wikipedia.org/wiki/Pierre_de_Fermathttp://en.wikipedia.org/wiki/Blaise_Pascalhttp://en.wikipedia.org/wiki/Blaise_Pascalhttp://en.wikipedia.org/wiki/Blaise_Pascalhttp://en.wikipedia.org/wiki/Christiaan_Huygenshttp://en.wikipedia.org/wiki/Christiaan_Huygenshttp://en.wikipedia.org/wiki/Christiaan_Huygenshttp://en.wikipedia.org/wiki/History_of_probability#cite_note-3http://en.wikipedia.org/wiki/History_of_probability#cite_note-3http://en.wikipedia.org/wiki/History_of_probability#cite_note-3http://en.wikipedia.org/w/index.php?title=History_of_probability&action=edit&section=3http://en.wikipedia.org/w/index.php?title=History_of_probability&action=edit&section=3http://en.wikipedia.org/wiki/History_of_physicshttp://en.wikipedia.org/wiki/History_of_mathematicshttp://en.wikipedia.org/wiki/History_of_algebrahttp://en.wikipedia.org/wiki/History_of_calculushttp://en.wikipedia.org/wiki/History_of_combinatoricshttp://en.wikipedia.org/wiki/History_of_geometryhttp://en.wikipedia.org/wiki/History_of_geometryhttp://en.wikipedia.org/wiki/History_of_logichttp://en.wikipedia.org/wiki/History_of_logichttp://en.wikipedia.org/wiki/History_of_statisticshttp://en.wikipedia.org/wiki/History_of_trigonometryhttp://en.wikipedia.org/wiki/History_of_the_social_scienceshttp://en.wikipedia.org/wiki/History_of_anthropologyhttp://en.wikipedia.org/wiki/History_of_economic_thoughthttp://en.wikipedia.org/wiki/History_of_linguisticshttp://en.wikipedia.org/wiki/History_of_political_sciencehttp://en.wikipedia.org/wiki/History_of_psychologyhttp://en.wikipedia.org/wiki/History_of_psychologyhttp://en.wikipedia.org/wiki/History_of_sociologyhttp://en.wikipedia.org/wiki/History_of_sociologyhttp://en.wikipedia.org/wiki/History_of_technologyhttp://en.wikipedia.org/wiki/History_of_agricultural_sciencehttp://en.wikipedia.org/wiki/History_of_agricultural_sciencehttp://en.wikipedia.org/wiki/History_of_computer_sciencehttp://en.wikipedia.org/wiki/History_of_materials_sciencehttp://en.wikipedia.org/wiki/History_of_medicinehttp://en.wikipedia.org/wiki/List_of_timelines#Sciencehttp://en.wikipedia.org/wiki/Portal:History_of_sciencehttp://en.wikipedia.org/wiki/Category:History_of_sciencehttp://en.wikipedia.org/wiki/Category:History_of_sciencehttp://en.wikipedia.org/wiki/Portal:History_of_sciencehttp://en.wikipedia.org/wiki/List_of_timelines#Sciencehttp://en.wikipedia.org/wiki/History_of_medicinehttp://en.wikipedia.org/wiki/History_of_materials_sciencehttp://en.wikipedia.org/wiki/History_of_computer_sciencehttp://en.wikipedia.org/wiki/History_of_agricultural_sciencehttp://en.wikipedia.org/wiki/History_of_technologyhttp://en.wikipedia.org/wiki/History_of_sociologyhttp://en.wikipedia.org/wiki/History_of_psychologyhttp://en.wikipedia.org/wiki/History_of_political_sciencehttp://en.wikipedia.org/wiki/History_of_linguisticshttp://en.wikipedia.org/wiki/History_of_economic_thoughthttp://en.wikipedia.org/wiki/History_of_anthropologyhttp://en.wikipedia.org/wiki/History_of_the_social_scienceshttp://en.wikipedia.org/wiki/History_of_trigonometryhttp://en.wikipedia.org/wiki/History_of_statisticshttp://en.wikipedia.org/wiki/History_of_logichttp://en.wikipedia.org/wiki/History_of_geometryhttp://en.wikipedia.org/wiki/History_of_combinatoricshttp://en.wikipedia.org/wiki/History_of_calculushttp://en.wikipedia.org/wiki/History_of_algebrahttp://en.wikipedia.org/wiki/History_of_mathematicshttp://en.wikipedia.org/wiki/History_of_physicshttp://en.wikipedia.org/w/index.php?title=History_of_probability&action=edit&section=3http://en.wikipedia.org/wiki/History_of_probability#cite_note-3http://en.wikipedia.org/wiki/Christiaan_Huygenshttp://en.wikipedia.org/wiki/Blaise_Pascalhttp://en.wikipedia.org/wiki/Pierre_de_Fermathttp://en.wikipedia.org/wiki/History_of_probability#cite_note-2http://en.wikipedia.org/wiki/Insurancehttp://en.wikipedia.org/wiki/Renaissancehttp://en.wikipedia.org/wiki/History_of_probability#cite_note-1http://en.wikipedia.org/wiki/Half-proofhttp://en.wikipedia.org/wiki/Presumptionhttp://en.wikipedia.org/wiki/Law_of_evidencehttp://en.wikipedia.org/wiki/Timeline_of_probability_and_statisticshttp://en.wikipedia.org/w/index.php?title=History_of_probability&action=edit&section=2http://en.wikipedia.org/wiki/History_of_probability#cite_note-0http://en.wikipedia.org/wiki/Cicerohttp://en.wikipedia.org/wiki/Medieval_Latinhttp://en.wikipedia.org/w/index.php?title=History_of_probability&action=edit&section=1http://en.wikipedia.org/wiki/History_of_probability#External_linkshttp://en.wikipedia.org/wiki/History_of_probability#Referenceshttp://en.wikipedia.org/wiki/History_of_probability#Bibliographyhttp://en.wikipedia.org/wiki/History_of_probability#20th_Centuryhttp://en.wikipedia.org/wiki/History_of_probability#19th_Centuryhttp://en.wikipedia.org/wiki/History_of_probability#18th_Centuryhttp://en.wikipedia.org/wiki/History_of_probability#Originshttp://en.wikipedia.org/wiki/History_of_probability#Etymologyhttp://toggletoc%28%29/http://en.wikipedia.org/wiki/History_of_Statisticshttp://en.wikipedia.org/wiki/History_of_Statisticshttp://en.wikipedia.org/wiki/Statisticshttp://en.wikipedia.org/wiki/Probabilityhttp://en.wikipedia.org/wiki/Pierre_de_Fermathttp://en.wikipedia.org/wiki/Blaise_Pascalhttp://en.wikipedia.org/wiki/Stochastic_processeshttp://en.wikipedia.org/wiki/Probability


3/26

Jacob Bernoulli'sArs Conjectandi(posthumous, 1713) andAbraham de Moivre'sThe

Doctrine of Chances(1718) put probability on a sound mathematical footing, showing how to

calculate a wide range of complex probabilities. Bernoulli proved a version of the

fundamentallaw of large numbers, which states that in a large number of trials, the average

of the outcomes is likely to be very close to the expected value - for example, in 1000 throws

of a fair coin, it is likely that there are close to 500 heads (and the larger the number ofthrows, the closer to half-and-half the proportion is likely to be).

[edit] 19th Century

The power of probabilistic methods in dealing with uncertainty was shown byGauss's

determination of the orbit ofCeresfrom a few observations. Thetheory of errorsused the

method of least squaresto correct error-prone observations, especially in astronomy, based on

the assumption of anormal distributionof errors to determine the most likely true value.

Towards the end of the nineteenth century, a major success of explanation in terms ofprobabilities was theStatistical mechanicsofLudwig BoltzmannandJ. Willard Gibbswhich

explained properties of gases such as temperature in terms of the random motions of large

numbers of particles.

The field of the history of probability itself was established byIsaac Todhunter's monumental

History of the Mathematical Theory of Probability from the Time of Pascal to that of

Lagrange (1865).

[edit] 20th Century

Probability and statistics became closely connected through the work onhypothesis testingof

R. A. FisherandJerzy Neyman, which is now widely applied in biological and psychological

experiments and inclinical trialsof drugs. A hypothesis, for example that a drug is usually

effective, gives rise to aprobability distributionthat would be observed if the hypothesis is

true. If observations approximately agree with the hypothesis, it is confirmed, if not, the

hypothesis is rejected.[5]

The theory of stochastic processes broadened into such areas asMarkov processesand

Brownian motion, the random movement of tiny particles suspended in a fluid. That provided

a model for the study of random fluctuations in stock markets, leading to the use of

sophisticated probability models inmathematical finance, including such successes as thewidely-usedBlack-Scholesformula for thevaluation of options.[6]

The twentieth century also saw long-running disputes on theinterpretations of probability. In

the mid-centuryfrequentismwas dominant, holding that probability means long-run relative

frequency in a large number of trials. At the end of the century there was some revival of the

Bayesianview, according to which the fundamental notion of probability is how well a

proposition is supported by the evidence for it.

The mathematical treatment of probabilities, especially when there are infinitely many

possible outcomes, was facilitated byKolmogorov's axioms(1931).
http://en.wikipedia.org/wiki/Jacob_Bernoullihttp://en.wikipedia.org/wiki/Jacob_Bernoullihttp://en.wikipedia.org/wiki/Ars_Conjectandihttp://en.wikipedia.org/wiki/Ars_Conjectandihttp://en.wikipedia.org/wiki/Ars_Conjectandihttp://en.wikipedia.org/wiki/Abraham_de_Moivrehttp://en.wikipedia.org/wiki/Abraham_de_Moivrehttp://en.wikipedia.org/wiki/Abraham_de_Moivrehttp://en.wikipedia.org/wiki/The_Doctrine_of_Chanceshttp://en.wikipedia.org/wiki/The_Doctrine_of_Chanceshttp://en.wikipedia.org/wiki/The_Doctrine_of_Chanceshttp://en.wikipedia.org/wiki/The_Doctrine_of_Chanceshttp://en.wikipedia.org/wiki/Law_of_large_numbershttp://en.wikipedia.org/wiki/Law_of_large_numbershttp://en.wikipedia.org/wiki/Law_of_large_numbershttp://en.wikipedia.org/w/index.php?title=History_of_probability&action=edit&section=4http://en.wikipedia.org/w/index.php?title=History_of_probability&action=edit&section=4http://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttp://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttp://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttp://en.wikipedia.org/wiki/Cereshttp://en.wikipedia.org/wiki/Cereshttp://en.wikipedia.org/wiki/Cereshttp://en.wikipedia.org/wiki/Theory_of_errorshttp://en.wikipedia.org/wiki/Theory_of_errorshttp://en.wikipedia.org/wiki/Theory_of_errorshttp://en.wikipedia.org/wiki/Method_of_least_squareshttp://en.wikipedia.org/wiki/Method_of_least_squareshttp://en.wikipedia.org/wiki/Normal_distributionhttp://en.wikipedia.org/wiki/Normal_distributionhttp://en.wikipedia.org/wiki/Normal_distributionhttp://en.wikipedia.org/wiki/Statistical_mechanicshttp://en.wikipedia.org/wiki/Statistical_mechanicshttp://en.wikipedia.org/wiki/Statistical_mechanicshttp://en.wikipedia.org/wiki/Ludwig_Boltzmannhttp://en.wikipedia.org/wiki/Ludwig_Boltzmannhttp://en.wikipedia.org/wiki/Ludwig_Boltzmannhttp://en.wikipedia.org/wiki/J._Willard_Gibbshttp://en.wikipedia.org/wiki/J._Willard_Gibbshttp://en.wikipedia.org/wiki/J._Willard_Gibbshttp://en.wikipedia.org/wiki/Isaac_Todhunterhttp://en.wikipedia.org/wiki/Isaac_Todhunterhttp://en.wikipedia.org/wiki/Isaac_Todhunterhttp://en.wikipedia.org/w/index.php?title=History_of_probability&action=edit&section=5http://en.wikipedia.org/w/index.php?title=History_of_probability&action=edit&section=5http://en.wikipedia.org/wiki/Statistical_hypothesis_testinghttp://en.wikipedia.org/wiki/Statistical_hypothesis_testinghttp://en.wikipedia.org/wiki/Statistical_hypothesis_testinghttp://en.wikipedia.org/wiki/Ronald_Fisherhttp://en.wikipedia.org/wiki/Ronald_Fisherhttp://en.wikipedia.org/wiki/Jerzy_Neymanhttp://en.wikipedia.org/wiki/Jerzy_Neymanhttp://en.wikipedia.org/wiki/Jerzy_Neymanhttp://en.wikipedia.org/wiki/Clinical_trialshttp://en.wikipedia.org/wiki/Clinical_trialshttp://en.wikipedia.org/wiki/Clinical_trialshttp://en.wikipedia.org/wiki/Probability_distributionhttp://en.wikipedia.org/wiki/Probability_distributionhttp://en.wikipedia.org/wiki/Probability_distributionhttp://en.wikipedia.org/wiki/History_of_probability#cite_note-4http://en.wikipedia.org/wiki/History_of_probability#cite_note-4http://en.wikipedia.org/wiki/History_of_probability#cite_note-4http://en.wikipedia.org/wiki/Markov_processhttp://en.wikipedia.org/wiki/Markov_processhttp://en.wikipedia.org/wiki/Markov_processhttp://en.wikipedia.org/wiki/Brownian_motionhttp://en.wikipedia.org/wiki/Brownian_motionhttp://en.wikipedia.org/wiki/Mathematical_financehttp://en.wikipedia.org/wiki/Mathematical_financehttp://en.wikipedia.org/wiki/Mathematical_financehttp://en.wikipedia.org/wiki/Black-Scholeshttp://en.wikipedia.org/wiki/Black-Scholeshttp://en.wikipedia.org/wiki/Black-Scholeshttp://en.wikipedia.org/wiki/Valuation_of_optionshttp://en.wikipedia.org/wiki/Valuation_of_optionshttp://en.wikipedia.org/wiki/History_of_probability#cite_note-5http://en.wikipedia.org/wiki/History_of_probability#cite_note-5http://en.wikipedia.org/wiki/History_of_probability#cite_note-5http://en.wikipedia.org/wiki/Probability_interpretationshttp://en.wikipedia.org/wiki/Probability_interpretationshttp://en.wikipedia.org/wiki/Probability_interpretationshttp://en.wikipedia.org/wiki/Frequency_probabilityhttp://en.wikipedia.org/wiki/Frequency_probabilityhttp://en.wikipedia.org/wiki/Frequency_probabilityhttp://en.wikipedia.org/wiki/Bayesian_probabilityhttp://en.wikipedia.org/wiki/Bayesian_probabilityhttp://en.wikipedia.org/wiki/Probability_axiomshttp://en.wikipedia.org/wiki/Probability_axiomshttp://en.wikipedia.org/wiki/Probability_axiomshttp://en.wikipedia.org/wiki/Probability_axiomshttp://en.wikipedia.org/wiki/Bayesian_probabilityhttp://en.wikipedia.org/wiki/Frequency_probabilityhttp://en.wikipedia.org/wiki/Probability_interpretationshttp://en.wikipedia.org/wiki/History_of_probability#cite_note-5http://en.wikipedia.org/wiki/Valuation_of_optionshttp://en.wikipedia.org/wiki/Black-Scholeshttp://en.wikipedia.org/wiki/Mathematical_financehttp://en.wikipedia.org/wiki/Brownian_motionhttp://en.wikipedia.org/wiki/Markov_processhttp://en.wikipedia.org/wiki/History_of_probability#cite_note-4http://en.wikipedia.org/wiki/Probability_distributionhttp://en.wikipedia.org/wiki/Clinical_trialshttp://en.wikipedia.org/wiki/Jerzy_Neymanhttp://en.wikipedia.org/wiki/Ronald_Fisherhttp://en.wikipedia.org/wiki/Statistical_hypothesis_testinghttp://en.wikipedia.org/w/index.php?title=History_of_probability&action=edit&section=5http://en.wikipedia.org/wiki/Isaac_Todhunterhttp://en.wikipedia.org/wiki/J._Willard_Gibbshttp://en.wikipedia.org/wiki/Ludwig_Boltzmannhttp://en.wikipedia.org/wiki/Statistical_mechanicshttp://en.wikipedia.org/wiki/Normal_distributionhttp://en.wikipedia.org/wiki/Method_of_least_squareshttp://en.wikipedia.org/wiki/Theory_of_errorshttp://en.wikipedia.org/wiki/Cereshttp://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttp://en.wikipedia.org/w/index.php?title=History_of_probability&action=edit&section=4http://en.wikipedia.org/wiki/Law_of_large_numbershttp://en.wikipedia.org/wiki/The_Doctrine_of_Chanceshttp://en.wikipedia.org/wiki/The_Doctrine_of_Chanceshttp://en.wikipedia.org/wiki/Abraham_de_Moivrehttp://en.wikipedia.org/wiki/Ars_Conjectandihttp://en.wikipedia.org/wiki/Jacob_Bernoulli


4/26

[edit] Bibliography

Bernstein, Peter L.(1996).Against the Gods: The Remarkable Story of Risk. New

York: Wiley.ISBN0471121045.

Daston, Lorraine(1988). Classical Probability in the Enlightenment. Princeton:

Princeton University Press.ISBN0691084971.

Franklin, James(2001). The Science of Conjecture: Evidence and Probability Before

Pascal. Baltimore, MD: Johns Hopkins University Press.ISBN0801865697.

Hacking, Ian(2006). The Emergence of Probability (2nd ed). New York: Cambridge

University Press.ISBN9780521866552.

Hald, Anders(2003).A History of Probability and Statistics and Their Applications

before 1750. Hoboken, NJ: Wiley.ISBN0471471291.

Hald, Anders(1998).A History of Mathematical Statistics from 1750 to 1930. New

York: Wiley.ISBN0471179124.

Heyde, C. C.; Seneta, E. (eds) (2001). Statisticians of the Centuries. New York:

Springer.ISBN0387953299. von Plato, Jan (1994). Creating Modern Probability: Its Mathematics, Physics and

Philosophy in Historical Perspective. New York: Cambridge University Press.ISBN

9780521597357.

Salsburg, David (2001). The Lady Tasting Tea: How Statistics Revolutionized Science

in the Twentieth Century.ISBN 0-7167-4106-7

Stigler, Stephen M.(1990). The History of Statistics: The Measurement of

Uncertainty before 1900. Belknap Press/Harvard University Press.ISBN 0-674-

40341-X.

[edit] References

1. ^J. Franklin, The Science of Conjecture: Evidence and Probability Before Pascal,113, 126.

2. ^Franklin, The Science of Conjecture, ch. 2.3. ^Franklin, Science of Conjecture, ch. 11.4. ^Hacking,Emergence of Probability; Franklin, Science of Conjecture, ch. 12.5. ^Salsburg, The Lady Tasting Tea.6. ^Bernstein,Against the Gods, ch. 18.

statistics

The scientific study of probability is a modern development.Gamblingshows 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' (Latinprobabilis) 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]
http://en.wikipedia.org/w/index.php?title=History_of_probability&action=edit&section=6http://en.wikipedia.org/w/index.php?title=History_of_probability&action=edit&section=6http://en.wikipedia.org/wiki/Peter_L._Bernsteinhttp://en.wikipedia.org/wiki/Peter_L._Bernsteinhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Special:BookSources/0471121045http://en.wikipedia.org/wiki/Special:BookSources/0471121045http://en.wikipedia.org/wiki/Special:BookSources/0471121045http://en.wikipedia.org/wiki/Lorraine_Dastonhttp://en.wikipedia.org/wiki/Lorraine_Dastonhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Special:BookSources/0691084971http://en.wikipedia.org/wiki/Special:BookSources/0691084971http://en.wikipedia.org/wiki/Special:BookSources/0691084971http://en.wikipedia.org/wiki/James_Franklin_(philosopher)http://en.wikipedia.org/wiki/James_Franklin_(philosopher)http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Special:BookSources/0801865697http://en.wikipedia.org/wiki/Special:BookSources/0801865697http://en.wikipedia.org/wiki/Special:BookSources/0801865697http://en.wikipedia.org/wiki/Ian_Hackinghttp://en.wikipedia.org/wiki/Ian_Hackinghttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Special:BookSources/9780521866552http://en.wikipedia.org/wiki/Special:BookSources/9780521866552http://en.wikipedia.org/wiki/Special:BookSources/9780521866552http://en.wikipedia.org/wiki/Anders_Haldhttp://en.wikipedia.org/wiki/Anders_Haldhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Special:BookSources/0471471291http://en.wikipedia.org/wiki/Special:BookSources/0471471291http://en.wikipedia.org/wiki/Special:BookSources/0471471291http://en.wikipedia.org/wiki/Anders_Haldhttp://en.wikipedia.org/wiki/Anders_Haldhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Special:BookSources/0471179124http://en.wikipedia.org/wiki/Special:BookSources/0471179124http://en.wikipedia.org/wiki/Special:BookSources/0471179124http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Special:BookSources/0387953299http://en.wikipedia.org/wiki/Special:BookSources/0387953299http://en.wikipedia.org/wiki/Special:BookSources/0387953299http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Special:BookSources/9780521597357http://en.wikipedia.org/wiki/Special:BookSources/9780521597357http://en.wikipedia.org/wiki/Special:BookSources/0716741067http://en.wikipedia.org/wiki/Special:BookSources/0716741067http://en.wikipedia.org/wiki/Special:BookSources/0716741067http://en.wikipedia.org/wiki/Stephen_Stiglerhttp://en.wikipedia.org/wiki/Stephen_Stiglerhttp://en.wikipedia.org/wiki/Special:BookSources/067440341Xhttp://en.wikipedia.org/wiki/Special:BookSources/067440341Xhttp://en.wikipedia.org/wiki/Special:BookSources/067440341Xhttp://en.wikipedia.org/wiki/Special:BookSources/067440341Xhttp://en.wikipedia.org/w/index.php?title=History_of_probability&action=edit&section=7http://en.wikipedia.org/w/index.php?title=History_of_probability&action=edit&section=7http://en.wikipedia.org/wiki/History_of_probability#cite_ref-0http://en.wikipedia.org/wiki/History_of_probability#cite_ref-0http://en.wikipedia.org/wiki/History_of_probability#cite_ref-1http://en.wikipedia.org/wiki/History_of_probability#cite_ref-1http://en.wikipedia.org/wiki/History_of_probability#cite_ref-2http://en.wikipedia.org/wiki/History_of_probability#cite_ref-2http://en.wikipedia.org/wiki/History_of_probability#cite_ref-3http://en.wikipedia.org/wiki/History_of_probability#cite_ref-3http://en.wikipedia.org/wiki/History_of_probability#cite_ref-4http://en.wikipedia.org/wiki/History_of_probability#cite_ref-4http://en.wikipedia.org/wiki/History_of_probability#cite_ref-5http://en.wikipedia.org/wiki/History_of_probability#cite_ref-5http://en.wikipedia.org/wiki/History_of_statisticshttp://en.wikipedia.org/wiki/History_of_statisticshttp://en.wikipedia.org/wiki/Gamblinghttp://en.wikipedia.org/wiki/Gamblinghttp://en.wikipedia.org/wiki/Gamblinghttp://en.wikipedia.org/wiki/Probability#cite_note-Jeffrey-3http://en.wikipedia.org/wiki/Probability#cite_note-Jeffrey-3http://en.wikipedia.org/wiki/Probability#cite_note-Franklin-4http://en.wikipedia.org/wiki/Probability#cite_note-Franklin-4http://en.wikipedia.org/wiki/Probability#cite_note-Franklin-4http://en.wikipedia.org/wiki/Probability#cite_note-Franklin-4http://en.wikipedia.org/wiki/Probability#cite_note-Jeffrey-3http://en.wikipedia.org/wiki/Gamblinghttp://en.wikipedia.org/wiki/History_of_statisticshttp://en.wikipedia.org/wiki/History_of_probability#cite_ref-5http://en.wikipedia.org/wiki/History_of_probability#cite_ref-4http://en.wikipedia.org/wiki/History_of_probability#cite_ref-3http://en.wikipedia.org/wiki/History_of_probability#cite_ref-2http://en.wikipedia.org/wiki/History_of_probability#cite_ref-1http://en.wikipedia.org/wiki/History_of_probability#cite_ref-0http://en.wikipedia.org/w/index.php?title=History_of_probability&action=edit&section=7http://en.wikipedia.org/wiki/Special:BookSources/067440341Xhttp://en.wikipedia.org/wiki/Special:BookSources/067440341Xhttp://en.wikipedia.org/wiki/Stephen_Stiglerhttp://en.wikipedia.org/wiki/Special:BookSources/0716741067http://en.wikipedia.org/wiki/Special:BookSources/9780521597357http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Special:BookSources/0387953299http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Special:BookSources/0471179124http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Anders_Haldhttp://en.wikipedia.org/wiki/Special:BookSources/0471471291http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Anders_Haldhttp://en.wikipedia.org/wiki/Special:BookSources/9780521866552http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Ian_Hackinghttp://en.wikipedia.org/wiki/Special:BookSources/0801865697http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/James_Franklin_(philosopher)http://en.wikipedia.org/wiki/Special:BookSources/0691084971http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Lorraine_Dastonhttp://en.wikipedia.org/wiki/Special:BookSources/0471121045http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Peter_L._Bernsteinhttp://en.wikipedia.org/w/index.php?title=History_of_probability&action=edit&section=6


5/26

Aside from some elementary considerations made byGirolamo Cardanoin the 16th century,

the doctrine of probabilities dates to the correspondence ofPierre de FermatandBlaise

Pascal(1654).Christiaan Huygens(1657) gave the earliest known scientific treatment of the

subject.Jakob Bernoulli'sArs Conjectandi(posthumous, 1713) andAbraham de Moivre's

Doctrine of Chances(1718) treated the subject as a branch of mathematics. SeeIan Hacking's

The Emergence of Probability andJames Franklin'sThe Science of Conjecture for histories ofthe early development of the very concept of mathematical probability.

The theory of errors may be traced back toRoger Cotes'sOpera Miscellanea (posthumous,

1722), but a memoir prepared byThomas Simpsonin 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(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 curvey= (x),x being any error andy its probability, and laiddown three properties of this curve:

1. it is symmetric as to they-axis;

2. thex-axis is anasymptote, 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(1778) introduced the

principle of the maximum product of the probabilities of a system of concurrent errors.

Themethod of least squaresis due toAdrien-Marie Legendre(1805), who introduced it in his

Nouvelles mthodes pour la dtermination des orbites des comtes (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,

h being a constant depending on precision of observation, and c a scale factor ensuring thatthe area under the curve equals 1. He gave two proofs, the second being essentially the same

asJohn Herschel's(1850).Gaussgave 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), andMorgan Crofton(1870). Other contributors were Ellis (1844),De

Morgan(1864),Glaisher(1872), andGiovanni Schiaparelli(1875). Peters's (1856) formula

forr, the probable error of a single observation, is well known.

In the nineteenth century authors on the general theory includedLaplace,Sylvestre Lacroix

(1816), Littrow (1833),Adolphe Quetelet(1853),Richard Dedekind(1860), Helmert (1872),
http://en.wikipedia.org/wiki/Girolamo_Cardanohttp://en.wikipedia.org/wiki/Girolamo_Cardanohttp://en.wikipedia.org/wiki/Girolamo_Cardanohttp://en.wikipedia.org/wiki/Pierre_de_Fermathttp://en.wikipedia.org/wiki/Pierre_de_Fermathttp://en.wikipedia.org/wiki/Pierre_de_Fermathttp://en.wikipedia.org/wiki/Blaise_Pascalhttp://en.wikipedia.org/wiki/Blaise_Pascalhttp://en.wikipedia.org/wiki/Blaise_Pascalhttp://en.wikipedia.org/wiki/Blaise_Pascalhttp://en.wikipedia.org/wiki/Christiaan_Huygenshttp://en.wikipedia.org/wiki/Christiaan_Huygenshttp://en.wikipedia.org/wiki/Christiaan_Huygenshttp://en.wikipedia.org/wiki/Jakob_Bernoullihttp://en.wikipedia.org/wiki/Jakob_Bernoullihttp://en.wikipedia.org/wiki/Ars_Conjectandihttp://en.wikipedia.org/wiki/Ars_Conjectandihttp://en.wikipedia.org/wiki/Ars_Conjectandihttp://en.wikipedia.org/wiki/Abraham_de_Moivrehttp://en.wikipedia.org/wiki/Abraham_de_Moivrehttp://en.wikipedia.org/wiki/Abraham_de_Moivrehttp://en.wikipedia.org/wiki/Doctrine_of_Chanceshttp://en.wikipedia.org/wiki/Doctrine_of_Chanceshttp://en.wikipedia.org/wiki/Ian_Hackinghttp://en.wikipedia.org/wiki/Ian_Hackinghttp://en.wikipedia.org/wiki/Ian_Hackinghttp://en.wikipedia.org/wiki/James_Franklin_(philosopher)http://en.wikipedia.org/wiki/James_Franklin_(philosopher)http://en.wikipedia.org/wiki/James_Franklin_(philosopher)http://en.wikipedia.org/wiki/Roger_Coteshttp://en.wikipedia.org/wiki/Roger_Coteshttp://en.wikipedia.org/wiki/Roger_Coteshttp://en.wikipedia.org/wiki/Thomas_Simpsonhttp://en.wikipedia.org/wiki/Thomas_Simpsonhttp://en.wikipedia.org/wiki/Thomas_Simpsonhttp://en.wikipedia.org/wiki/Pierre-Simon_Laplacehttp://en.wikipedia.org/wiki/Pierre-Simon_Laplacehttp://en.wikipedia.org/wiki/Asymptotehttp://en.wikipedia.org/wiki/Asymptotehttp://en.wikipedia.org/wiki/Asymptotehttp://en.wikipedia.org/wiki/Daniel_Bernoullihttp://en.wikipedia.org/wiki/Daniel_Bernoullihttp://en.wikipedia.org/wiki/Daniel_Bernoullihttp://en.wikipedia.org/wiki/Method_of_least_squareshttp://en.wikipedia.org/wiki/Method_of_least_squareshttp://en.wikipedia.org/wiki/Method_of_least_squareshttp://en.wikipedia.org/wiki/Adrien-Marie_Legendrehttp://en.wikipedia.org/wiki/Adrien-Marie_Legendrehttp://en.wikipedia.org/wiki/Adrien-Marie_Legendrehttp://en.wikipedia.org/wiki/Robert_Adrainhttp://en.wikipedia.org/wiki/Robert_Adrainhttp://en.wikipedia.org/wiki/Robert_Adrainhttp://en.wikipedia.org/wiki/John_Herschelhttp://en.wikipedia.org/wiki/John_Herschelhttp://en.wikipedia.org/wiki/John_Herschelhttp://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttp://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttp://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttp://en.wikipedia.org/wiki/James_Ivory_(mathematician)http://en.wikipedia.org/wiki/James_Ivory_(mathematician)http://en.wikipedia.org/wiki/James_Ivory_(mathematician)http://en.wikipedia.org/wiki/Friedrich_Besselhttp://en.wikipedia.org/wiki/Friedrich_Besselhttp://en.wikipedia.org/wiki/Friedrich_Besselhttp://en.wikipedia.org/w/index.php?title=W._F._Donkin&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=W._F._Donkin&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=W._F._Donkin&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=W._F._Donkin&action=edit&redlink=1http://en.wikipedia.org/wiki/Morgan_Croftonhttp://en.wikipedia.org/wiki/Morgan_Croftonhttp://en.wikipedia.org/wiki/Morgan_Croftonhttp://en.wikipedia.org/wiki/Augustus_De_Morganhttp://en.wikipedia.org/wiki/Augustus_De_Morganhttp://en.wikipedia.org/wiki/Augustus_De_Morganhttp://en.wikipedia.org/wiki/Augustus_De_Morganhttp://en.wikipedia.org/wiki/James_Whitbread_Lee_Glaisherhttp://en.wikipedia.org/wiki/James_Whitbread_Lee_Glaisherhttp://en.wikipedia.org/wiki/James_Whitbread_Lee_Glaisherhttp://en.wikipedia.org/wiki/Giovanni_Schiaparellihttp://en.wikipedia.org/wiki/Giovanni_Schiaparellihttp://en.wikipedia.org/wiki/Giovanni_Schiaparellihttp://en.wikipedia.org/wiki/Laplacehttp://en.wikipedia.org/wiki/Laplacehttp://en.wikipedia.org/wiki/Laplacehttp://en.wikipedia.org/wiki/Sylvestre_Lacroixhttp://en.wikipedia.org/wiki/Sylvestre_Lacroixhttp://en.wikipedia.org/wiki/Sylvestre_Lacroixhttp://en.wikipedia.org/wiki/Adolphe_Quetelethttp://en.wikipedia.org/wiki/Adolphe_Quetelethttp://en.wikipedia.org/wiki/Adolphe_Quetelethttp://en.wikipedia.org/wiki/Richard_Dedekindhttp://en.wikipedia.org/wiki/Richard_Dedekindhttp://en.wikipedia.org/wiki/Richard_Dedekindhttp://en.wikipedia.org/wiki/Richard_Dedekindhttp://en.wikipedia.org/wiki/Adolphe_Quetelethttp://en.wikipedia.org/wiki/Sylvestre_Lacroixhttp://en.wikipedia.org/wiki/Laplacehttp://en.wikipedia.org/wiki/Giovanni_Schiaparellihttp://en.wikipedia.org/wiki/James_Whitbread_Lee_Glaisherhttp://en.wikipedia.org/wiki/Augustus_De_Morganhttp://en.wikipedia.org/wiki/Augustus_De_Morganhttp://en.wikipedia.org/wiki/Morgan_Croftonhttp://en.wikipedia.org/w/index.php?title=W._F._Donkin&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=W._F._Donkin&action=edit&redlink=1http://en.wikipedia.org/wiki/Friedrich_Besselhttp://en.wikipedia.org/wiki/James_Ivory_(mathematician)http://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttp://en.wikipedia.org/wiki/John_Herschelhttp://en.wikipedia.org/wiki/Robert_Adrainhttp://en.wikipedia.org/wiki/Adrien-Marie_Legendrehttp://en.wikipedia.org/wiki/Method_of_least_squareshttp://en.wikipedia.org/wiki/Daniel_Bernoullihttp://en.wikipedia.org/wiki/Asymptotehttp://en.wikipedia.org/wiki/Pierre-Simon_Laplacehttp://en.wikipedia.org/wiki/Thomas_Simpsonhttp://en.wikipedia.org/wiki/Roger_Coteshttp://en.wikipedia.org/wiki/James_Franklin_(philosopher)http://en.wikipedia.org/wiki/Ian_Hackinghttp://en.wikipedia.org/wiki/Doctrine_of_Chanceshttp://en.wikipedia.org/wiki/Abraham_de_Moivrehttp://en.wikipedia.org/wiki/Ars_Conjectandihttp://en.wikipedia.org/wiki/Jakob_Bernoullihttp://en.wikipedia.org/wiki/Christiaan_Huygenshttp://en.wikipedia.org/wiki/Blaise_Pascalhttp://en.wikipedia.org/wiki/Blaise_Pascalhttp://en.wikipedia.org/wiki/Pierre_de_Fermathttp://en.wikipedia.org/wiki/Girolamo_Cardano


6/26

Hermann Laurent(1873), Liagre, Didion, andKarl Pearson.Augustus De Morganand

George Booleimproved the exposition of the theory.

On the geometric side (seeintegral geometry) contributors toThe Educational Timeswere

influential (Miller, Crofton, McColl, Wolstenholme, Watson, andArtemas Martin).

[edit] Mathematical treatment

Further information:Probability theory

In mathematics, a probability of aneventA 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 orcomplementof an eventA is the event [notA] (that is, the event ofA not

occurring); its probability is given by P(notA) = 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 eventfor a more complete treatment.

If both the eventsA andB occur on a single performance of an experiment this is called the

intersection orjoint probabilityofA andB, denoted as . If two events,A andB

areindependentthen the joint probability is

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

If either eventA or eventB or both events occur on a single performance of an experiment

this is called the union of the eventsA andB denoted as . If two events are

mutually exclusivethen 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
http://en.wikipedia.org/wiki/Hermann_Laurenthttp://en.wikipedia.org/wiki/Hermann_Laurenthttp://en.wikipedia.org/wiki/Karl_Pearsonhttp://en.wikipedia.org/wiki/Karl_Pearsonhttp://en.wikipedia.org/wiki/Karl_Pearsonhttp://en.wikipedia.org/wiki/Augustus_De_Morganhttp://en.wikipedia.org/wiki/Augustus_De_Morganhttp://en.wikipedia.org/wiki/Augustus_De_Morganhttp://en.wikipedia.org/wiki/George_Boolehttp://en.wikipedia.org/wiki/George_Boolehttp://en.wikipedia.org/wiki/Integral_geometryhttp://en.wikipedia.org/wiki/Integral_geometryhttp://en.wikipedia.org/wiki/Integral_geometryhttp://en.wikipedia.org/w/index.php?title=The_Educational_Times&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=The_Educational_Times&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=The_Educational_Times&action=edit&redlink=1http://en.wikipedia.org/wiki/Artemas_Martinhttp://en.wikipedia.org/wiki/Artemas_Martinhttp://en.wikipedia.org/wiki/Artemas_Martinhttp://en.wikipedia.org/w/index.php?title=Probability&action=edit&section=4http://en.wikipedia.org/w/index.php?title=Probability&action=edit&section=4http://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Event_(probability_theory)http://en.wikipedia.org/wiki/Event_(probability_theory)http://en.wikipedia.org/wiki/Event_(probability_theory)http://en.wikipedia.org/wiki/Probability#cite_note-5http://en.wikipedia.org/wiki/Probability#cite_note-5http://en.wikipedia.org/wiki/Probability#cite_note-5http://en.wikipedia.org/wiki/Almost_surelyhttp://en.wikipedia.org/wiki/Almost_surelyhttp://en.wikipedia.org/wiki/Almost_surelyhttp://en.wikipedia.org/wiki/Probability#cite_note-6http://en.wikipedia.org/wiki/Probability#cite_note-6http://en.wikipedia.org/wiki/Probability#cite_note-6http://en.wikipedia.org/wiki/Complementary_eventhttp://en.wikipedia.org/wiki/Complementary_eventhttp://en.wikipedia.org/wiki/Joint_distributionhttp://en.wikipedia.org/wiki/Joint_distributionhttp://en.wikipedia.org/wiki/Joint_distributionhttp://en.wikipedia.org/wiki/Statistical_independencehttp://en.wikipedia.org/wiki/Statistical_independencehttp://en.wikipedia.org/wiki/Statistical_independencehttp://en.wikipedia.org/wiki/Probability#cite_note-7http://en.wikipedia.org/wiki/Probability#cite_note-7http://en.wikipedia.org/wiki/Probability#cite_note-7http://en.wikipedia.org/wiki/Mutually_exclusive_eventshttp://en.wikipedia.org/wiki/Mutually_exclusive_eventshttp://en.wikipedia.org/wiki/Mutually_exclusive_eventshttp://en.wikipedia.org/wiki/Probability#cite_note-7http://en.wikipedia.org/wiki/Statistical_independencehttp://en.wikipedia.org/wiki/Joint_distributionhttp://en.wikipedia.org/wiki/Complementary_eventhttp://en.wikipedia.org/wiki/Probability#cite_note-6http://en.wikipedia.org/wiki/Almost_surelyhttp://en.wikipedia.org/wiki/Probability#cite_note-5http://en.wikipedia.org/wiki/Event_(probability_theory)http://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/w/index.php?title=Probability&action=edit&section=4http://en.wikipedia.org/wiki/Artemas_Martinhttp://en.wikipedia.org/w/index.php?title=The_Educational_Times&action=edit&redlink=1http://en.wikipedia.org/wiki/Integral_geometryhttp://en.wikipedia.org/wiki/George_Boolehttp://en.wikipedia.org/wiki/Augustus_De_Morganhttp://en.wikipedia.org/wiki/Karl_Pearsonhttp://en.wikipedia.org/wiki/Hermann_Laurent


7/26

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 probabilityis the probability of some eventA, given the occurrence of some

other eventB. Conditional probability is writtenP(A|B), and is read "the probability ofA,

givenB". It is defined by

[9]

IfP(B) = 0 then isundefined.

Summary of probabilities

Event Probability

A

not A

A or

B

A and

B

A

given

B

[edit] Theory

Main article:Probability theory

Like othertheories, thetheory of probabilityis a representation of probabilistic concepts in

formal termsthat 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.
http://en.wikipedia.org/wiki/Conditional_probabilityhttp://en.wikipedia.org/wiki/Conditional_probabilityhttp://en.wikipedia.org/wiki/Probability#cite_note-8http://en.wikipedia.org/wiki/Probability#cite_note-8http://en.wikipedia.org/wiki/Defined_and_undefinedhttp://en.wikipedia.org/wiki/Defined_and_undefinedhttp://en.wikipedia.org/wiki/Defined_and_undefinedhttp://en.wikipedia.org/w/index.php?title=Probability&action=edit&section=5http://en.wikipedia.org/w/index.php?title=Probability&action=edit&section=5http://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Theoryhttp://en.wikipedia.org/wiki/Theoryhttp://en.wikipedia.org/wiki/Theoryhttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Theoryhttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/w/index.php?title=Probability&action=edit&section=5http://en.wikipedia.org/wiki/Defined_and_undefinedhttp://en.wikipedia.org/wiki/Probability#cite_note-8http://en.wikipedia.org/wiki/Conditional_probability


8/26

There have been at least two successful attempts to formalize probability, namely the

Kolmogorovformulation and theCoxformulation. In Kolmogorov's formulation (see

probability space),setsare interpreted aseventsand probability itself as ameasureon a class

of sets. InCox'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 probabilityare the same, except for technical details.

There are other methods for quantifying uncertainty, such as theDempster-Shafer theoryor

possibility theory, but those are essentially different and not compatible with the laws of

probability as they are usually understood.

[edit] Applications

Two major applications of probability theory in everyday life are inriskassessment and in

trade oncommodity markets. Governments typically apply probabilistic methods in

environmental regulationwhere it is called "pathway analysis", oftenmeasuring well-beingusing methods that arestochasticin 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 ofbehavioral financeemerged to

describe the effect of suchgroupthinkon pricing, on policy, and on peace and conflict.

It can reasonably be said that the discovery of rigorous methods to assess and combineprobability 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 ademocracy.

Another significant application of probability theory in everyday life isreliability. Many

consumer products, such asautomobilesand consumer electronics, utilizereliability theoryin

the design of the product in order to reduce the probability of failure. The probability of

failure may be closely associated with the product'swarranty.

[edit] Relation to randomnessMain article:Randomness

In adeterministicuniverse, based onNewtonianconcepts, 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 theoryof gases, where the system, while deterministic in principle, is so complex
http://en.wikipedia.org/wiki/Kolmogorovhttp://en.wikipedia.org/wiki/Kolmogorovhttp://en.wikipedia.org/wiki/Richard_Threlkeld_Coxhttp://en.wikipedia.org/wiki/Richard_Threlkeld_Coxhttp://en.wikipedia.org/wiki/Richard_Threlkeld_Coxhttp://en.wikipedia.org/wiki/Probability_spacehttp://en.wikipedia.org/wiki/Probability_spacehttp://en.wikipedia.org/wiki/Set_(mathematics)http://en.wikipedia.org/wiki/Set_(mathematics)http://en.wikipedia.org/wiki/Set_(mathematics)http://en.wikipedia.org/wiki/Event_(probability_theory)http://en.wikipedia.org/wiki/Event_(probability_theory)http://en.wikipedia.org/wiki/Event_(probability_theory)http://en.wikipedia.org/wiki/Measure_(mathematics)http://en.wikipedia.org/wiki/Measure_(mathematics)http://en.wikipedia.org/wiki/Measure_(mathematics)http://en.wikipedia.org/wiki/Cox%27s_theoremhttp://en.wikipedia.org/wiki/Cox%27s_theoremhttp://en.wikipedia.org/wiki/Cox%27s_theoremhttp://en.wikipedia.org/wiki/Probability_axiomshttp://en.wikipedia.org/wiki/Probability_axiomshttp://en.wikipedia.org/wiki/Dempster-Shafer_theoryhttp://en.wikipedia.org/wiki/Dempster-Shafer_theoryhttp://en.wikipedia.org/wiki/Dempster-Shafer_theoryhttp://en.wikipedia.org/wiki/Possibility_theoryhttp://en.wikipedia.org/wiki/Possibility_theoryhttp://en.wikipedia.org/w/index.php?title=Probability&action=edit&section=6http://en.wikipedia.org/w/index.php?title=Probability&action=edit&section=6http://en.wikipedia.org/wiki/Riskhttp://en.wikipedia.org/wiki/Riskhttp://en.wikipedia.org/wiki/Riskhttp://en.wikipedia.org/wiki/Commodity_marketshttp://en.wikipedia.org/wiki/Commodity_marketshttp://en.wikipedia.org/wiki/Commodity_marketshttp://en.wikipedia.org/wiki/Environmental_regulationhttp://en.wikipedia.org/wiki/Environmental_regulationhttp://en.wikipedia.org/w/index.php?title=Pathway_analysis&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Pathway_analysis&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Pathway_analysis&action=edit&redlink=1http://en.wikipedia.org/wiki/Measuring_well-beinghttp://en.wikipedia.org/wiki/Measuring_well-beinghttp://en.wikipedia.org/wiki/Measuring_well-beinghttp://en.wikipedia.org/wiki/Stochastichttp://en.wikipedia.org/wiki/Stochastichttp://en.wikipedia.org/wiki/Stochastichttp://en.wikipedia.org/wiki/Behavioral_financehttp://en.wikipedia.org/wiki/Behavioral_financehttp://en.wikipedia.org/wiki/Behavioral_financehttp://en.wikipedia.org/wiki/Groupthinkhttp://en.wikipedia.org/wiki/Groupthinkhttp://en.wikipedia.org/wiki/Groupthinkhttp://en.wikipedia.org/wiki/Democracyhttp://en.wikipedia.org/wiki/Democracyhttp://en.wikipedia.org/wiki/Democracyhttp://en.wikipedia.org/wiki/Reliability_theory_of_aging_and_longevityhttp://en.wikipedia.org/wiki/Reliability_theory_of_aging_and_longevityhttp://en.wikipedia.org/wiki/Reliability_theory_of_aging_and_longevityhttp://en.wikipedia.org/wiki/Automobileshttp://en.wikipedia.org/wiki/Automobileshttp://en.wikipedia.org/wiki/Automobileshttp://en.wikipedia.org/wiki/Reliability_theoryhttp://en.wikipedia.org/wiki/Reliability_theoryhttp://en.wikipedia.org/wiki/Reliability_theoryhttp://en.wikipedia.org/wiki/Warrantyhttp://en.wikipedia.org/wiki/Warrantyhttp://en.wikipedia.org/wiki/Warrantyhttp://en.wikipedia.org/w/index.php?title=Probability&action=edit&section=7http://en.wikipedia.org/w/index.php?title=Probability&action=edit&section=7http://en.wikipedia.org/wiki/Randomnesshttp://en.wikipedia.org/wiki/Randomnesshttp://en.wikipedia.org/wiki/Randomnesshttp://en.wikipedia.org/wiki/Determinismhttp://en.wikipedia.org/wiki/Determinismhttp://en.wikipedia.org/wiki/Determinismhttp://en.wikipedia.org/wiki/Newtonian_mechanicshttp://en.wikipedia.org/wiki/Newtonian_mechanicshttp://en.wikipedia.org/wiki/Newtonian_mechanicshttp://en.wikipedia.org/wiki/Kinetic_theoryhttp://en.wikipedia.org/wiki/Kinetic_theoryhttp://en.wikipedia.org/wiki/Kinetic_theoryhttp://en.wikipedia.org/wiki/Newtonian_mechanicshttp://en.wikipedia.org/wiki/Determinismhttp://en.wikipedia.org/wiki/Randomnesshttp://en.wikipedia.org/w/index.php?title=Probability&action=edit&section=7http://en.wikipedia.org/wiki/Warrantyhttp://en.wikipedia.org/wiki/Reliability_theoryhttp://en.wikipedia.org/wiki/Automobileshttp://en.wikipedia.org/wiki/Reliability_theory_of_aging_and_longevityhttp://en.wikipedia.org/wiki/Democracyhttp://en.wikipedia.org/wiki/Groupthinkhttp://en.wikipedia.org/wiki/Behavioral_financehttp://en.wikipedia.org/wiki/Stochastichttp://en.wikipedia.org/wiki/Measuring_well-beinghttp://en.wikipedia.org/w/index.php?title=Pathway_analysis&action=edit&redlink=1http://en.wikipedia.org/wiki/Environmental_regulationhttp://en.wikipedia.org/wiki/Commodity_marketshttp://en.wikipedia.org/wiki/Riskhttp://en.wikipedia.org/w/index.php?title=Probability&action=edit&section=6http://en.wikipedia.org/wiki/Possibility_theoryhttp://en.wikipedia.org/wiki/Dempster-Shafer_theoryhttp://en.wikipedia.org/wiki/Probability_axiomshttp://en.wikipedia.org/wiki/Cox%27s_theoremhttp://en.wikipedia.org/wiki/Measure_(mathematics)http://en.wikipedia.org/wiki/Event_(probability_theory)http://en.wikipedia.org/wiki/Set_(mathematics)http://en.wikipedia.org/wiki/Probability_spacehttp://en.wikipedia.org/wiki/Richard_Threlkeld_Coxhttp://en.wikipedia.org/wiki/Kolmogorov


9/26

(with the number of molecules typically the order of magnitude ofAvogadro constant

6.021023) that only statistical description of its properties is feasible.

A revolutionary discovery of 20th centuryphysicswas the random character of all physical

processes that occur at sub-atomic scales and are governed by the laws ofquantum

mechanics. Thewave functionitself evolves deterministically as long as no observation ismade, but, according to the prevailingCopenhagen interpretation, the randomness caused by

thewave function collapsingwhen an observation is made, is fundamental. This means that

probability theoryis required to describe nature. Others never came to terms with the loss of

determinism.Albert Einsteinfamouslyremarkedin a letter toMax Born:Jedenfalls bin ich

berzeugt, da der Alte nicht wrfelt. (I am convinced that God does not play dice). Although

alternative viewpoints exist, such as that ofquantum decoherencebeing the cause of an

apparentrandom collapse, at present there is a firm consensus amongphysiciststhat

probability theory is necessary to describe quantum phenomena.[citation needed]

[edit] See alsoThe probability of an event (seesample space) is a number lying in the interval 0p1, with 0

corresponding to an event that never occurs and 1 to an event that is certain to occur. For an

experiment withNequally likely outcomes the probability of an eventA is n/N, where n is the

number of outcomes in which the eventA occurs. For some experiments, such as throwing a

drawing pin and seeing whether it lands point up, there is no possible set of equally likely

outcomes. In the 'frequentist' view of probability, the probability of getting 'point up' is the

limit, in some sense, of the relativefrequencyas the number of experiments tends to infinity.

In the context ofBayesian inference, each observer has his or her own a prioridistribution

for the probability, which is then modified a posteriori in the light of whatever results have

been obtained.

Probabilities can also be used, more generally, to describe degrees of belief in propositions that do

not involve random variables -- for example `the probability that 2050 will be the warmest year on

record, assuming people don't change their lifestyle', or `the probability that the Hubble constant

lies between 41 and 43, given measurements of the Sunyaev-Zel'dovich effect'. Degrees of belief can

be mapped onto probabilities if they satisfy some simple consistency rules known as the Cox

axioms . Thus probabilities can be used to describe assumptions, and to describe inferences given

those assumptions. The rules of probability ensure that if two people make the same assumptions

and receive the same data then they will draw identical conclusions. This more general use of

probability is known as the Bayesian viewpoint. It is also known as the subjective interpretation of

probability, since the probabilities depend on assumptions. Advocates of a Bayesian approach to

data modelling and pattern recognition do not view this subjectivity as a defect, since in their view,

you can't do inference without making assumptions. In this book it will be taken for granted that a

Bayesian approach makes sense, but the reader is warned that this is not yet a globally held view --

the field of statistics has been dominated for most of the 20th century by non-Bayesian methods in

which probabilities are onlyINTRODUCTION

The history of probability theory dates back to the 17th century and at that time was related to

games of chance. In the 18th century the probability theory was known to have applications

beyond the scope of games of chance. Some of the applications in which probability theory is
http://en.wikipedia.org/wiki/Avogadro_constanthttp://en.wikipedia.org/wiki/Avogadro_constanthttp://en.wikipedia.org/wiki/Avogadro_constanthttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Wave_functionhttp://en.wikipedia.org/wiki/Wave_functionhttp://en.wikipedia.org/wiki/Wave_functionhttp://en.wikipedia.org/wiki/Copenhagen_interpretationhttp://en.wikipedia.org/wiki/Copenhagen_interpretationhttp://en.wikipedia.org/wiki/Copenhagen_interpretationhttp://en.wikipedia.org/wiki/Wave_function_collapsehttp://en.wikipedia.org/wiki/Wave_function_collapsehttp://en.wikipedia.org/wiki/Wave_function_collapsehttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Albert_Einsteinhttp://en.wikipedia.org/wiki/Albert_Einsteinhttp://en.wikipedia.org/wiki/Albert_Einsteinhttp://de.wikipedia.org/wiki/Albert_Einstein#Quellenangaben_und_Anmerkungenhttp://de.wikipedia.org/wiki/Albert_Einstein#Quellenangaben_und_Anmerkungenhttp://de.wikipedia.org/wiki/Albert_Einstein#Quellenangaben_und_Anmerkungenhttp://en.wikipedia.org/wiki/Max_Bornhttp://en.wikipedia.org/wiki/Max_Bornhttp://en.wikipedia.org/wiki/Max_Bornhttp://en.wikipedia.org/wiki/Quantum_decoherencehttp://en.wikipedia.org/wiki/Quantum_decoherencehttp://en.wikipedia.org/wiki/Quantum_decoherencehttp://en.wikipedia.org/wiki/Physicisthttp://en.wikipedia.org/wiki/Physicisthttp://en.wikipedia.org/wiki/Physicisthttp://en.wikipedia.org/wiki/Wikipedia:Citation_neededhttp://en.wikipedia.org/wiki/Wikipedia:Citation_neededhttp://en.wikipedia.org/wiki/Wikipedia:Citation_neededhttp://en.wikipedia.org/w/index.php?title=Probability&action=edit&section=8http://en.wikipedia.org/w/index.php?title=Probability&action=edit&section=8http://www.answers.com/topic/sample-spacehttp://www.answers.com/topic/sample-spacehttp://www.answers.com/topic/sample-spacehttp://www.answers.com/topic/frequency-421http://www.answers.com/topic/frequency-421http://www.answers.com/topic/frequency-421http://www.answers.com/topic/subjective-probabilityhttp://www.answers.com/topic/subjective-probabilityhttp://www.answers.com/topic/subjective-probabilityhttp://www.answers.com/topic/distributionhttp://www.answers.com/topic/distributionhttp://www.answers.com/topic/distributionhttp://www.answers.com/topic/distributionhttp://www.answers.com/topic/subjective-probabilityhttp://www.answers.com/topic/frequency-421http://www.answers.com/topic/sample-spacehttp://en.wikipedia.org/w/index.php?title=Probability&action=edit&section=8http://en.wikipedia.org/wiki/Wikipedia:Citation_neededhttp://en.wikipedia.org/wiki/Physicisthttp://en.wikipedia.org/wiki/Quantum_decoherencehttp://en.wikipedia.org/wiki/Max_Bornhttp://de.wikipedia.org/wiki/Albert_Einstein#Quellenangaben_und_Anmerkungenhttp://en.wikipedia.org/wiki/Albert_Einsteinhttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Wave_function_collapsehttp://en.wikipedia.org/wiki/Copenhagen_interpretationhttp://en.wikipedia.org/wiki/Wave_functionhttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Avogadro_constant


10/26

applied are situations with outcomes such as life or death and boy or girl. Statistics and

probability are currently applied to insurance, annuities, biology, and social investigations.

The treatment of probability in this chapter is limited to simple applications. These

applications will be, to a large extent, based on games of chance, which lend themselves to an

understanding of basic ideas of probability.

BASIC CONCEPTS

If a coin were tossed, the chance it would land heads up is just as likely as the chance it

would land tails up; that is, the coin has no more reason to land heads up than it has to

land tails up. Every toss of the coin is called a trial.

We define probability as the ratio of the different number of ways a trial can succeed

(or fail) to the total number of ways in which it may result. We will let p represent the

probability of success and q represent the probability of failure.

One commonly misunderstood concept of probability is the effect prior trials have on a

single trial. That is, after a coin has been tossed many times and every trial resulted in

the coin falling heads up, will the next toss of the coin result in tails up? The answer is

"not necessarily" and will be explained later in this chapter

allowed to describe random variables.

ProbabilityMany things in everyday life, from stock price to lottery, are random phenomena for which

the outcome is uncertain. The concept of probability provides us with the idea on how to

measure the chances of possible outcomes. Probability enables us to quantify uncertainty,

which is described in terms of mathematics. Here we introduce basic notions that help us to

find probabilities of interest.

Chapter 2

Probability SpaceIn this chapter we describe the probability model of \choosing an object at random." Ex-amples will help us come up with a good denition. We explain that the key idea is toassociate a likelihood, which we callprobability, to sets of outcomes, not to individualoutcomes. These sets are events. The description of the events and of their probabilityconstitute aprobability space that characterizes completely a random experiment.

2.1 Choosing At RandomFirst consider picking a card out of a 52-card deck. We could say that the odds of pickingany particular card are the same as that of picking any other card, assuming that the deck


11/26

has been well shued. We then decide to assign a \probability" of 1/52 to each card. Thatprobability represents the odds that a given card is picked. One interpretation is that if werepeat the experiment \choosing a card from the deck" a large number Nof times (replacingthe card previously picked every time and re-shuing the deck before the next selection),then a given card, say the ace of diamonds, is selected approximated N=52 times. Note thatthis is only an interpretation. There is nothing that tells us that this is indeed the case;moreover, if it is the case, then there is certainly nothing yet in our theory that allows us toexpect that result. Indeed, so far, we have simply assigned the number 1/52 to each card1314 CHAPTER 2. PROBABILITY SPACEin the deck. Our interpretation comes from what we expect from the physical experiment.This remarkable \statistical regularity" of the physical experiment is a consequence of somedeeper properties of the sequences of successive cards picked from a deck. We will comebackto these deeper properties when we study independence. You may object that the denitionof probability involves implicitly that of \equally likely events." That is correct as far asthe interpretation goes. The mathematical denition does not require such a notion.

Second, consider the experiment of throwing a dart on a dartboard. The likelihood ofhitting a specic point on the board, measured with pinpoint accuracy, is essentially zero.Accordingly, in contrast with the previous example, we cannot assign numbers to individualoutcomes of the experiment. The way to proceed is to assign numbers to sets of possibleoutcomes. Thus, one can look at a subset of the dartboard and assign some probabilitythat represents the odds that the dart will land in that set. It is not simple to assign thenumbers to all the sets in a way that these numbers really correspond to the odds of a givendart player. Even if we forget about trying to model an actual player, it is not that simpleto assign numbers to all the subsets of the dartboard. At the very least, to be meaningful,the numbers assigned to the dierent subsets must obey some basic consistency rules. Forinstance, ifA and B are two subsets of the dartboard such thatA B, then the numberP(B) assigned to B must be at least as large as the numberP(A) assigned toA. Also, ifA

and B are disjoint, then P(A [ B) = P(A) + P(B). Finally, P() = 1, if designates theset of all possible outcomes (the dartboard, possibly extended to cover all bases). This is thebasic story: probability is dened on sets of possible outcomes and it is additive. [However,it turns out that one more property is required: countable additivity (see below).]Note that we can lump our two examples into one. Indeed, the rst case can be viewedas a particular case of the second where we would dene P(A) =jAj=52, whereA is anysubset of the deck of cards andjAjis the number of cards in the deck. This denition iscertainly additive and it assigns the probability 1=52 to any one card.2.2. EVENTS 15Some care is required when dening what we mean by a random choice. See Bertrand'sparadox in Appendix E for an illustration of a possible confusion. Another example of thepossible confusion with statistics is Simpson's paradox in Appendix F.

2.2 EventsThe sets of outcomes to which one assigns a probability are called events. It is notnecessary(and often not possible, as we may explain later) for every set of outcomes to be an event.For instance, assume that we are only interested in whether the card that we pick isblack or red. In that case, it suces to dene P(A) = 0:5 = P(Ac) whereA is the set of allthe black cards andAcis the complement of that set, i.e., the set of all the red cards. Ofcourse, we know that P() = 1 where is the set of all the cards and P(;) = 0, where ;is the empty set. In this case, there are four events: ;;; A;Ac.More generally, ifA and B are events, then we wantAc;A \ B, andA [ B to beevents also. Indeed, if we want to dene the probability that the outcome is inA and the

probability that it is in B, it is reasonable to ask that we can also dene the probability thatthe outcome is not inA, that it is inA and B, and that it is inA or in B (or in both). By


12/26

extension, set operations that are performed on a nite collection of events should alwaysproduce an event. For instance, ifA;B;C;D are events, then [(AnB) \ C] [D should alsobe an event. We say that the set of events is closed under nite set operations. [We explainbelow that we need to extend this property to countable operations.] With these properties,it makes sense to write for disjoint eventsA and B that P(A[B) = P(A)+P(B). Indeed,A [ B is an event, so that P(A [ B) is dened.You will notice that if we wantA (withA 6= andA 6= ;) to be an event, thenthe smallest collection of events is necessarily f;;; A;Acg.If you want to see why, generally for uncountable sample spaces, all sets of outcomes16 CHAPTER 2. PROBABILITY SPACEmay not be events, check Appendix C.

2.3 Countable AdditivityThis topic is the rst serious hurdle that you face when studying probability theory. Ifyou understand this section, you increase considerably your appreciation of the theory.Otherwise, many issues will remain obscure and fuzzy.We want to be able to say that if the eventsAn forn = 1; 2; : : :, are such thatAn An+1for all n and ifA := [nAn, then P(An) " P(A) as n ! 1. Why is this useful? This

property, called -additivityis the key to being able to approximate events. The propertyspecies that the probability is continuous: if we approximate the events, then we alsoapproximate their probability.This strategy of \lling the gaps" by taking limits is central in mathematics. Youremember that real numbers are dened as limits of rational numbers. Similarly, integralsare dened as limits of sums. The key idea is that dierent approximations should give thesame result. For this to work, we need the continuity property above.To be able to write the continuity property, we need to assume thatA := [nAn is anevent whenever the eventsAn forn = 1; 2; : : :, are such thatAn An+1. More generally,we need the set of events to be closed under countable set operations.For instance, if we dene P([0; x]) =xforx 2[0; 1], then we can dene P([0; a)) = abecause ifis small enough, thenAn := [0; a =n] is such thatAn An+1 and [0; a) :=

[nAn. We will discuss many more interesting examples.You may wish to review the meaning of countability (see Appendix ??).2.4. PROBABILITY SPACE17

2.4 Probability SpacePutting together the observations of the sections above, we have dened a probability spaceas follows.Denition 2.4.1. Probability SpaceA probability space is a triplet f;F; Pgwhere is a nonempty set, called the sample space; Fis a collection of subsets of closed under countable set operations - such a collectionis called a -eld. The elements ofFare called events;

Pis a countably additive function from Finto [0; 1] such that P() = 1, called aprobability measure.Examples will clarify this denition. The main point is that one denes the probabilityof sets of outcomes (the events). The probability should be countably additive (to becontinuous). Accordingly (to be able to write down this property), and also quite intuitively,the collection of events should be closed under countable set operations.

2.5 ExamplesThroughout the course, we will make use of simple examples of probability space. Wereviewsome of those here.2.5.1 Choosing uniformly in f1; 2; : : : ;NgWe say that we pick a value !uniformly in f1; 2; : : : ;Ngwhen the Nvalues are equallylikely to be selected. In this case, the sample space is = f1; 2; : : : ;Ng. For any subsetA , one denes P(A) =jAj=NwherejAjis the number of elements inA. For instance,


13/26

P(f2; 5g) = 2=N.18 CHAPTER 2. PROBABILITY SPACE2.5.2 Choosing uniformly in [0; 1]Here, = [0; 1] and one has, for example, P([0; 0:3]) = 0:3 and P([0:2; 0:7]) = 0:5. Thatis, P(A) is the \length" of the setA. Thus, if!is picked uniformly in [0; 1], then one can

write P([0:2; 0:7]) = 0:5.It turns out that one cannot dene the length of every subset of [0; 1], as we explainin Appendix C. The collection of sets whose length is dened is the smallest -eld thatcontains the intervals. This collection is called the Borel -eld of [0; 1]. More generally, thesmallest -eld of< that contains the intervals is the Borel -eld of


14/26

experiment \picking ve cards without replacement from a perfectly shued 52-card deck."1. One can choose to be all the permutations ofA := f1; 2; : : : ; 52g. The interpretationof! 2 is then the shued deck. Each permutation is equally likely, so thatp!= 1=(52!)for! 2. When we pick the ve cards, these cards are (!1; !2; : : : ; !5), the top 5 cards ofthe deck.

probability

Probability is a branch of mathematics that deals with calculating the likelihood of a given

event's occurrence, which is expressed as a number between 1 and 0. An event with a

probability of 1 can be considered a certainty: for example, the probability of a coin toss

resulting in either "heads" or "tails" is 1, because there are no other options, assuming the

coin lands flat. An event with a probability of .5 can be considered to have equal odds of

occurring or not occurring: for example, the probability of a coin toss resulting in "heads" is

.5, because the toss is equally as likely to result in "tails." An event with a probability of 0

can be considered an impossibility: for example, the probability that the coin will land (flat)without either side facing up is 0, because either "heads" or "tails" must be facing up. A little

paradoxical, probability theory applies precise calculations to quantify uncertain measures of

random events.

In its simplest form, probability can be expressed mathematically as: the number of

occurrences of a targeted event divided by the number of occurrencesplus the number of

failures of occurrences (this adds up to the total of possible outcomes):

p(a) = p(a)/[p(a) + p(b)]

Calculating probabilities in a situation like a coin toss is straightforward, because theoutcomes are mutually exclusive: either one event or the other must occur. Each coin toss is

an independentevent; the outcome of one trial has no effect on subsequent ones. No matter

how many consecutive times one side lands facing up, the probability that it will do so at the

next toss is always .5 (50-50). The mistaken idea that a number of consecutive results (six

"heads" for example) makes it more likely that the next toss will result in a "tails" is known

as thegambler's fallacy , one that has led to the downfall of many a bettor.

Probability theory had its start in the 17th century, when two French mathematicians, Blaise

Pascal and Pierre de Fermat carried on a correspondence discussing mathematical problems

dealing with games of chance. Contemporary applications of probability theory run the gamut

of human inquiry, and include aspects of computer programming, astrophysics, music,

weather prediction, and medicine.


15/26

Fascinating facts about Blaise Pascalinventor of a mechanical adding machine in 1642. Blaise Pascal

Inventor:Blaise Pascal

Criteria:First to invent. First practical.

Birth:June 19, 1623 in Clermont-Ferrand, France

Death:August 19, 1662 in Paris, France

Nationality:French

Blaise Pascal, French philosopher, mathematician, and physicist, considered one of the great mindsin Western intellectual history. Inventor of the first mechanical adding machine.

Blaise Pascal was born in Clermont-Ferrand on June 19, 1623, and his family settled in Paris in1629. Under the tutelage of his father, Pascal soon proved himself a mathematical prodigy, and atthe age of 16 he formulated one of the basic theorems of projective geometry, known as Pascal'stheorem and described in his Essai pour les coniques (Essay on Conics, 1639).

In 1642 he invented the first mechanical adding machine. Pascal proved by experimentation in 1648that the level of the mercury column in a barometer is determined by an increase or decrease in thesurrounding atmospheric pressure rather than by a vacuum, as previously believed. This discoveryverified the hypothesis of the Italian physicist Evangelista Torricelli concerning the effect ofatmospheric pressure on the equilibrium of liquids. Six years later, in conjunction with the Frenchmathematician Pierre de Fermat, Pascal formulated the mathematical theory of probability, whichhas become important in such fields as actuarial, mathematical, and social statistics and as afundamental element in the calculations of modern theoretical physics.

Pascal's other important scientific contributions include the derivation of Pascal's law or principle,which states that fluids transmit pressures equally in all directions, and his investigations in thegeometry of infinitesimal. His methodology reflected his emphasis on empirical experimentation asopposed to analytical, a priori methods, and he believed that human progress is perpetuated by theaccumulation of scientific discoveries resulting from such experimentation.

Pascal espoused Jansenism and in 1654 entered the Jansenist community at Port Royal, where heled a rigorously ascetic life until his death eight years later. In 1656 he wrote the famous 18 Lettresprovinciales (Provincial Letters), in which he attacked the Jesuits for their attempts to reconcile 16th-century naturalism with orthodox Roman Catholicism.

His most positive religious statement appeared posthumously (he died August 19, 1662); it waspublished in fragmentary form in 1670 as Apologie de la religion Chrtienne (Apology of theChristian Religion). In these fragments, which later were incorporated into his major work, he posedthe alternatives of potential salvation and eternal damnation, with the implication that only byconversion to Jansenism could salvation be achieved. Pascal asserted that whether or not salvationwas achieved, humanity's ultimate destiny is an afterlife belonging to a supernatural realm that canonly be known intuitively. Pascal's final important work was Penses sur la religion et sur quelquesautres sujets (Thoughts on Religion and on Other Subjects), also published in 1670. In the PensesPascal attempted to explain and justify the difficulties of human life by the doctrine of original sin,and he contended that revelation can be comprehended only by faith, which in turn is justified byrevelation.

Pascal's writings urging acceptance of the Christian life contain frequent applications of the

calculations of probability; he reasoned that the value of eternal happiness is infinite and thatalthough the probability of gaining such happiness by religion may be small it is infinitely greater than
http://www.ideafinder.com/features/classact/index.htmhttp://www.ideafinder.com/features/classact/index.htm


16/26

by any other course of human conduct or belief. A reclassification of the Penses, a careful workbegun in 1935 and conti

Documents

Probability is a Way of Expressing Knowledge or Belief That an Event Will Occur or Has Occurred