Chapter 9 Probability Theory

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Chapter 9Probability Theory

• By Jennifer and John

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Graunt

• (1620-1674)
• Bills of Mortality (1662)
• Studied birth and death records and was able to
  figure the number of people dying from the
  plague.
• Started as an Empirical Science.

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Graunt

• Because men were subject to death from
  occupational accidents, diseases, and war, the
  number of men and women of marriageable age were
  about equal.
• His work was the first to analyze statistics.

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Graunt

• Graunts work led to the development of actuarial
  science, which is used by life insurance
  companies.

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Huygens

• (1629 1695)
• Wrote De Ratiociniis in Ludo Aleae (1657) the
  value(price) of the chance.
• For half a century, it was the only text
  available on probability theory.

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Cardan

• (1501 1576)
• Avid Gambler
• Gambling is a source to which probability theory
  owes its birth and development.
• Wrote treatise Liber Deludo Aleae (Book on
  Games of Chance, 1550).

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Cardan

• First historical record of probability theory.
• Wasnt published until 1663.
• Gave a rough definition of probability as a ratio
  of equally likely events.

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Cardan

• Transition was made from empiricism to the
  theoretical concept.
• In doing so, Cardan probably became the real
  father of modern probability theory.

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The Problem of PointsTartaglia vs. Cardan

• One of the earliest problems concerns the fair
  division of stakes between two players when the
  game is interrupted before its conclusion.
• The problem is so difficult that its solution by
  Pascal in 1654 may well be considered a decisive
  breakthrough in the history of probability theory.

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Blaise PascalThe Greatest Might-Have-Been

• (1623 1662)
• Linked with Fermat as one of the joint founders
  of probability theory.

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Blaise PascalThe Greatest Might-Have-Been

• (1623 1662)
• Linked with Fermat as one of the joint founders
  of probability theory.

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Blaise PascalAn Exceptionally Versatile Man

• Gifted writer author of the first great prose
  classic in the modern French language, Lettres
  Provinciales and Pensées.
• Religious Philosopher
• A rebellious curiosity for math
• Experimental Physicist

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Blaise Pascal

• Essay pour les coniquies (1640) one of the most
  fruitful pages in mathematics.
• Traité du Triangle Arithmétique (1654)
  discoveries relating to binomial coefficients.

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Blaise Pascal

• Pascals work on the cycloid led to his letter
  Traité des sinus du quarte de circle (1658). This
  served as immediate inspiration for Leibniz in
  his invention of the differential and integral
  calculus.

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Creating the Cycloid
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• The year 1654 is a landmark in the history of
  probability theory.

• Bothered by certain problems in gambling, a
  French nobleman sent some questions to Pascal.

• Rolls of the dice Originally considered by
  Cardan who based his study on the premise that
  there are fundamental scientific principles
  governing the likelihood of achieving the double
  six outside of mere luck or chance.

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Pascals Arithmetic Triangle? (1623 1662)

• Chinese Triangle (1304)

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The arithmetic triangle, now generally known as
Pascals triangle, is an infinite numerical table
in triangular form.

Binomial Probability Probability when there are
only two possilble outcomes.
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Uses of Pascals Triangle

• Coefficients in the binomial expansion of (x
  y)n.
• Binomial Probability

• (xy)2 1x2 2xy 1y2
• (xy)3 1x3 3x2y 3xy2 1y3
• (xy)4
• 1x4 4x3y 6x2y2 4xy3 1y4

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(xy)2 1x2 2xy 1y2(xy)3 1x3
3x2y 3xy2 1y3
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Coin Toss
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Pascals Triangle Worksheets

• Pascals Triangle Excel program
• Pascals Triangle quick calculator

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Fermat

• (1601 - 1665 )
• Along with Pascal, Fermat supplied vital links in
  a chain of reasoning that gave us probability
  theory as we know it.

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Buffon

• (1707 1788)
• Author of 36 volume Histoire Naturelle.
• At 20 discovered the binomial theorem.
• Posed the needle problem.

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Buffon-Laplace Needle Problem

• Parallel Lines
• Square grid

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Laplace

• (1749 1827)
• It was largely Laplace who went on to broaden the
  mathematical treatment of probability to areas of
  scientific research well beyond the games of
  chance.

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Laplace

• (1749 1827)
• It was largely Laplace who went on to broaden the
  mathematical treatment of probability to areas of
  scientific research well beyond the games of
  chance.
• Mécanique Céleste

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Mary Fairfax Somerville

• (1780 1870)
• Extraordinary, self educated Scotswoman who
  studied the treatise, Mécanique Céleste, in
  Edinburgh.

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Mary Fairfax Somerville

• Mécanique Céleste was Laplaces
  great achievement.

• It completed the work Newton had begun. It
  showed that all the movements of the planetary
  system were deductible from the law of
  gravitation.

• Laplace is said to have observed that Mary
  Somerville was the single woman who understood
  his great treatise.

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James Bernoulli

• (1654 1705)
• From a family of eight mathematicians.
• Started life in the ministry.
• Ars Conjectandi contains practically all the
  standard results on permutations and combinations
  in the form in which they are still expressed.

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Marquis de LHospital

• (1661 1704)
• French Nobleman
• Paid for tutelage by John Bernoulli.
• Published Bernoullis work as his own.
• Has a famous math concept named after him.
• LHospitals Rule.

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Abraham de Moivre

• (1667 1754)
• French Protestant
• Pioneered the development of analytic geometry
  and the theory of probability.
• The Doctrine of Chances
• Predicted the day of his own death.

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Works Consulted

• Burton, David M., The History of Mathematics An
  Introduction, 5th ed., New York, McGraw-Hill,
  2003.
• See Web Site List.

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Probability Quiz
Hit me!!!
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Thank you