The Development of Probability Theory: Pascal, Bernoulli, and Laplace

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The Development of Probability Theory Pascal,
Bernoulli, and Laplace

• By Richie and Jason

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What is Probability Theory?

• A branch of mathematics concerned with the
  analysis of random events. The outcome of a
  random event cannot be determined before it
  occurs, but it may be any one of several possible
  outcomes. The actual outcome is considered to be
  determined by chance.

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Origin of Probability

• Began as an empirical science became
  mathematical science later
• Games of chance, insurance rates, and mortality
  tables led to the idea of probability
• Exact origin is hard to determine started with
  the gathering of data

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Dice Play

• Began most likely as a religious ceremony found
  in almost every primitive culture earliest dice
  usage found in Northern Iraq (3000 B.C.)
• Main type of gambling in Europe for nearly 1000
  years
• http//nces.ed.gov/nceskids/Probability/
• NOTE before dice, tarsal bone was used in a
  manner of gambling

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Card Play

• Most likely started in China between the 7th and
  10th centuries, Egyptians and Indians were also
  credited
• Did not find way into Europe until 1300s first
  cards in Europe were for upper class only
• French developed present-day cards

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Insurance and Mortality Rates

• Insurance
• Used in protecting merchant ships, which dates
  back to the Roman time period.
• 14th century marine insurance companies began in
  Italy and Holland, idea spread by the 16th
  century.

• Mortality Rates
• Companies needed something to base their life
  insurance policies on.
• John Graunt collected data on the weekly and
  yearly returns on the number of burials in
  several London parishes.

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John Graunt

• First to draw an extensive set of statistical
  inferences from mass data.
• Natural and Political Observations Made upon the
  Bills of Mortality
• Tallies were started to be kept as a way to keep
  up with the number of deaths due to the Black
  Plague.
• Graunt collected many bills from previous years
  and reduced them to tables in his book. (p 414)

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Christiaan and Ludwig Huygens

• Ludwig wrote to Christiaan saying that he would
  live to 56 and ½ years, and that he himself would
  live to be 55.
• Christian constructed a graph of his own
  mortality, which was the earliest known graph
  from statistical data.
• Lead to the notions of the probability of death
  in a given time, and the probability of surviving
  to a given age.

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Cardans Liber de Ludo Aleae

• First theoretical concept of probability
• More than likely, Cardan is the true father of
  probability

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Fra Luca Paciolis Contribution

• Summa de Arithatica, Geometrica, Proportioni, et
  Proportionalita
• Not original work, maybe Arabic
• Cardan contradicts work, and Tartaglia then
  contradicts Fra Lucas and Cardans work

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Blaise Pascal
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• Exceptionally versatile man 1)gifted writer,
  2)religious philosopher, 3)creative
  mathematician, 4)experimental physicist
• Greatest might-have-been in math
• Never went to a proper school or university
• Spent most of his life devoted to religion
  (Pascals Wager)

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Pascals Triangle

• Published in Triangle Arithmetique
• Infinite numerical table in triangular form nth
  row of triangle lists successive coefficients in
  binomial expansion of (xy)n
• Can be used in determining the probability of the
  likelihood of an event occurring

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Pascals Triangle (cont.)

• Pascal published in Traite du Triange
  Arithmetique (1653)
• Similar work dates back to China in 1050 and
  Persia in the 13th century
• Printed in Europe 1st in 1527, Peter Apians
  Rechung
• Michael Stifel (1544) worked to the 17th line,
  but noted that it would go on

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Pascals Work on the Cycloid

• P 424, area under a curve which is traced out by
  a point on the circumference of a wheel, as the
  wheel rolls along a straight line.
• Pascal found the solution after starting to work
  on the problem because he had a tooth ache. He
  worked for 8 days.

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Gilles Personne de Roberval

• Already found the solution of the quadrature of
  the cycloid curve.
• He didnt publish it when he first figured it out
  because he was fearful of being challenged and he
  wanted to have that as a weapon to use against
  any challengers

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Antoine Gombaud, Chevalier de Mere

• Soldier with a brilliant record, linguist, and a
  classical scholar
• Devoted life to teaching of good manners
• Made precarious living at cards and dice
• Initiated correspondence between Pascal and
  Fermat who are commonly credited as being the
  founders of the Probability Theory

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Dice Problem

• -Chevalier de Mere knew bet he could throw a 6 in
  four throws of a die next move was to bet that
  he could throw two dice and get a double six in
  24 attempts
• -Was unsatisfied with this assumption and asked
  for Pascals help to work true probabilities
• -Led to correspondence between Pascal and Laplace
  that is thought of as beginning of probability
  theory

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Francesco Maurolico

• Photismi de Lumine et Umbra- Maurolicos book on
  his optical studies.
• Analysis of lenses, spherical mirrors, and the
  human eye.
• Maurolico was one of the first to prove by
  induction. P 436

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Christian Huygens

• Devoted himself to a life of scientific enquiry
  by 1666, his knowledge in physics, astronomy, and
  math gained him a senior position at Academie des
  Sciences
• Contribution was the expectation of the value
  (price) of the chance to win in a game
• His pamphlet about this expectation was the only
  text available on probability for 50 years

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Bernoulli Family

• During a century, this gifted family produced no
  fewer than 8 mathematicians.
• The family fled Antwerp in 1583 because their
  protestant faith wasnt allowed by the Spaniards
  who had just settled in and took over.
• James and John were the first two brothers. They
  provided a link between 17th and 18th century
  mathematicians.

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James Bernoulli

• With very religious parents, James Bernoulli
  originally obtained a degree in theology in order
  to please them.
• He then rebelled and sought additional training
  in Math Astronomy
• His first book was entitled Invito Patre Sidera
  Verso I Study the Stars Against my Fathers
  Will.
• James was one of the first to achieve full
  understanding of Leibnizs abbreviated
  presentation of integral calculus and his book
  Acta Eruditorum.

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James continued

• James then taught his brother John the ideas hed
  derived from Leibniz, John then instructed
  LHospital in calculus, who then passed it on the
  Huygens.
• The Bernoullis expanded Leibnizs work and
  created almost all of what we call elementary
  calculus.
• During the last 20 years of his life, James wrote
  a book, which he never finished. He left it to
  his nephew Nicholas who felt unworthy of
  finishing it, so he published it as it is. (Ars
  Conjectandi).

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Ars Conjectandi

• Four Parts
• Reproduction of Huygenss De Ratiociniis in Ludo
  Aleae with alternate proofs. Thought to be of
  more value than the original.
• All the standard results on permutations and
  combinations in the form in which they are
  expressed. P 444
• 24 problems relating to the various games of
  chance that were popular in Bernoullis day.
• Bernoullis theorem p 444.

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Abraham De Moivre

• Doctrine of Chances or, a Method of Calculating
  the Probability of Events in Play (1718)
• Spoke of throwing dice, dropping balls of color
  from a bag, and life annuities
• Predicted his own death, like Cardan
• Work of De Moivre, along w/ James Bernoulli,
  opened the door to a broad range of new
  applications for probability theory

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Pierre Simon Laplace

• Traite de Mecanique Celeste, was written and
  published in five large volumes over 26 years.
• It completed the work Newton had begun showing
  that all the movements of the known members of
  the planetary system were deducible from the law
  of gravitation.
• Contained the work of Newton, Clairaunt,
  dAlembert, Euler, and Lagrange.
• Irregularities were found, and most scientist of
  the time thought that a divine intervention would
  be necessary to keep all the planets in their
  order, but Laplace proved them otherwise.

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Laplaces place in probability

• Published the Theorie Analytique des
  Probabilites. Presented the solution of almost
  every classical problem of probability theory.
• It was thought to be too hard for people who
  werent knowledgeable in mathematics, so he
  rewrote it to acquaint a broader circle of
  readers with the fundamentals of probability
  theory and its applications without resorting to
  higher mathematics. P 455

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Mary Fairfax Somerville

• Had no formal training in education, and yet
  Laplace said she was the only female who
  understood his great treatise
• Asked by Laplace to work in secret on full
  explanations and diagrams to make Laplaces work
  comprehensible
• Hailed as the Queen of Nineteenth-Century
  Science