A Short History of Probability

8/9/2019 A Short History of Probability

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A Short History of Probability

Dr. Alan M. Polansky

Division of StatisticsNorthern Illinois University


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History of Probability 2

French Society in the 1650’s

! Gambling was popularand fashionable

! Not restricted by law! As the games became

more complicated andthe stakes becamelarger there was aneed for mathematicalmethods for computingchances.


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Enter the Mathematicians

! A well-known gambler,the chevalier De Mereconsulted Blaise Pascalin Paris about a somequestions about somegames of chance.

! Pascal began tocorrespond with hisfriend Pierre Fermatabout these problems.


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Classical Probability

! The correspondence between Pascal and Fermatis the origin of the mathematical study ofprobability.

! The method they developed is now called theclassical approach to computing probabilities.

! The method: Suppose a game has n equallylikely outcomes, of which m outcomescorrespond to winning. Then the probability ofwinning is m/n.


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Problems with the Classical Method

! The classical method requires a gameto be broken down into equally likely

outcomes.! It is not always possible to do this.! It is not always clear when possibilities are

equally likely.


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Experience

! Another method, known as the frequencymethod had also been used for some time.

! This method consists of repeating a game alarge number of times under the sameconditions. The probability of winning is thenapproximately equal to the proportion of wins

in the repeats.! This method was used by Pascal and Fermat

to verify results obtained by the classicalmethod .


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Early Generalizations

! James Bernoulliproved that thefrequency methodand the classicalmethod areconsistent with oneanother in his book

Ars Conjectandi in1713.


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Early Generalizations

! Abraham De Moivreprovided many tools tomake the classical methodmore useful, including themultiplication rule , in hisbook The Doctrine of

Chances in 1718.! The book was popular,

eventually going throughthree editions.


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From Games to Science

! Throughout the 18 th century, theapplication of probability moved from

games of chance to scientific problems:! Mathematical theory of life insurance - life

tables.! Biological problems - what is the

probability of being born female or male?


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Applied Probability

! Pierre-Simon Laplace presenteda mathematical theory ofprobability with an emphasis onscientific applications in his1812 book Theorie Analytiquedes Probabilities .

! Unfortunately, Laplace onlyconsidered the classicalmethod , leaving no indicationon how the method was to beapplied to general problems.


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Stagnation the Frustration

! After the publication of Laplace’s book, themathematical development of probabilitystagnated for many years.

! By 1850, many mathematicians found theclassical method to be unrealistic for generaluse and were attempting to redefineprobability in terms of the frequency method .

! These attempts were never fully acceptedand the stagnation continued.


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Axiomatic Development! Andrey Kolmogorov

developed the first rigorousapproach to probability in his

1933 monographGrundbegriffe derWahrscheinlichkeitsrechnun .

! He built up probabilitytheory from fundamentalaxioms in a way comparablewith Euclid's treatment of

geometry.


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Probability Today

! Modern research in probability theory isclosely related to the mathematical field ofmeasure theory .

! Modern innovators in the field include PatrickBillingsley (University of Chicago), Yuan ShihChow (Columbia), Kai Lai Chung (Stanford),Samuel Karlin (Stanford), Rolf-Dieter Reiss ,Sheldon Ross (Berkeley), Henry Teicher(Rutgers) and many many more…

A Short History of Probability

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