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Presentation 4

Presentation 4. Chapter 9 The Development of Probability Theory: Pascal, Bernoulli, and Laplace. Section 9.1 The Origins of Probability Theory. Grunt’s Bills of Mortality. Probability Theory Started before that the correspondence between Blaise Pascal and Pierre de Fermat (1654)

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Presentation 4

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  1. Presentation 4 Chapter 9 The Development of Probability Theory: Pascal, Bernoulli, and Laplace

  2. Section 9.1The Origins of Probability Theory

  3. Grunt’s Bills of Mortality • Probability Theory • Started before that the correspondence between Blaise Pascal and Pierre de Fermat (1654) • Empirical Science • 2 lines of investigation • Wagering problems connected with the Game of Chance • Statistical data for Insurance Rates and Mortality Tables

  4. Oldest type of insurance • Protecting merchant vessels • Lloyd’s of London (before 1688) • John Graunt (1620-1674) • First to draw an extensive set of statistical inferences from mass data • Natural and Political Observations Made upon the Bills of Mortality • Discipline we now call mathematical statistics

  5. Bills of Mortality • Number of Burials in London • Information: sex of a person, not their age at death, cause: accident disease • Natural and Political Observations • Series of Tables • Male births > female births • Women live longer then Men • Death from disease constant • Noted number of children’s deaths and the elderlly

  6. London Life Table Age • 0 • 6 • 16 • 26 • 36 • 46 • 56 • 66 • 76 Survivors • 100 • 64 • 40 • 25 • 16 • 10 • 6 • 3 • 1 • Constant death rate after age 6 • A new and graphical method of representing the age pattern of mortality • The deaths in each decade really amount to about 3/8 of the survivors at the beginning of the decade. • Info: • Royal Society of London founded 1660 • Graunt one of charter fellows • 119 fellows consist of doctors (medicine and divinity), noblemen, lawyers , civil servants, literary men and a shopkeeper.

  7. Ludwig and Christiaan Huygens • Holland • Question they had was “length of life” • Made a table • Mortality curve (first graph produced from statistical data) • Denote the probability of death in a given time • Denote the probability of surviving to a given age • De Ratiociniis in LudoAleae • “On Reasoning in Games of Chance” • First text on Probability Theory to be published

  8. The other source probability theory owes its birth/development is GAMBLING • Dice Play • Tarsal Bone (hind foot of a hooved animal) see which side fell up • Astragali • GirolamoCardan • Liber de LudoAleae • “Book on Games of Chance” • Defined probability • Emperor Claudius (Rome) wrote “How to Win at Dice”

  9. Card Playing • Egyptians, Chinese, Indians, Europe through the Crusades • Western society in 1300 • Johann Gutenberg’s presses – 78 card tarot deck • French developed present day cards (Spades, Harts, Diamonds and Clubs) 1500

  10. De Mere’s Problem of Points • 1654 landmark in the history of probability theory (some say date which this science began) • Member of French Nobility • Chevalier de Mere (1607-1684) • Made a living at cards and dice. • Problem with gambling – wrote to Pascal who wrote to Fermat

  11. YOU TRY • Coin • Dice • Deck of Cards

  12. Section 9.2Pascal’s Arithmetic Triangle

  13. Blaise Pascal (1623-1662) • At 3 Mother died raised by Father (judge) • Father was first member of “Father Marin Mersenne’s Academy” (math & science topics) • Blaise attended • Father appointed to high administrative post in the government – required calculations • “Pascaline”

  14. Pascal’s Triangle • Successive coefficients in the binomial expansion of (x + y)n • Example: (x + y)2 = x2 + 2xy + y2 1 1 1 1 2 1 1 x2 + 2xy + 1 y2 1 3 3 1 1 4 6 4 1

  15. Chinense / Arabs • Chu Shih-Chien (1303) • Treatise “The Precious Mirror of the Four Elements” • ChiaHsien 1050 • Omar Khayyam • Rubaiyat in On Demonstrations of Problems of Algebra & Almucabola (arithmetic triangle) • Al-Tusi • Collection on Arithmetic by means of Board & Dust • Approximated the value of the square root  an + r by a+r/[(a+1)n – r] the denominator was calculated by the binomial expansion

  16. Peter Apian • Rechnung(first triangular arrangement of the binomial coefficients to be printed) • Michael Stifel also discovered it • Arithmetica Integra (calculated through 17th line) • different form 1 2 3 3 4 6 5 10 10 6 1520 7 21 35 35 • Where each column after the first starts two places lower than the preceding one.

  17. Cardan • Opus Novum de Proportionibus(to the 17th line citing Stifel as original discoverer) • Tartaglia • GeneraleTarattato(gave a number of triangles throught 8th line) • Pascal – not originator but he was first to make any sort of systematic study of the relations it exhibited.

  18. Triangle Arithmetique

  19. n = n! rr!(n-r)!

  20. Christian Kramp n! • Introduced first • Elements d’ArithmetiqueUniverselle(1808)

  21. Pascal’s TriangleYOU TRY • How many patterns can you find? • Look at diagonals - what do you see? • Numbers/ patterns • GOOGLE it

  22. Mathematical Induction • Pascal gave explanation of Mathematical Induction • “Reasoning by Recurrence” • First to recognize value of this logical process in Triangle Arithmetique • Francesco Maurolico (1494-1575) • Born in Sicily of Greek parents, was a Priest • Most of his books published after death

  23. OpusculaMathematica • ArithmeticorumLibri Duo • The sum of the first n odd numbers is equal to the nth square number 1+3+5+7 + (2n-1) = n2 MI – This is the statement P(n) First observe P(1) is true – that is, 1 = 12 Now, obtain P(k+1) from P(k)

  24. Assume that Adding the next odd integer (2k +1) to both sides 1+3+5+7+…+(2k-1)+(2k+1) = k2 k2 + (2k+1) k2 + 2k + 1 (k+1) 2 = (k+1) 2 the next term squared 1+3+5+7 + (2k-1) = k2

  25. YOU TRY

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