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Chapter IX The Development of Probability Theory Pascal_ Bernoulli

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Chapter IX The Development of Probability Theory Pascal_  Bernoulli Powered By Docstoc
					   Chapter IX


The Development of
 Probability Theory:
  Pascal, Bernoulli,
Bernoulli, and Laplace
Origins of Probability Theory
• Came about
  – The solution of wagering problems connected
    with games of chance
  – The process of statistical data for such matters
    as insurance rates and mortality tables
• 14th Century
  – Needed an estimate on mortality rates
    because towns were losing money to people
    who died early
   John Graunt (1620-1674)
• Wrote Natural and Political Observations
 Made upon the Bills of Mortality
• Bills of Mortality
  – weekly and yearly returns on the number of
    burials in several London perishes
  – Counted deaths because of the plague
  – Did not distinguish age, only sex and cause
    (disease or accident)
• Used statistics for the 57 years from 1604-
  1661
• Came up with table on pg 404
        Christian Huygens
          (1629-1695)
• Used Graunt’s figures to determine when
  people would die
• Estimated his death at 55 and his
  brother’s at 56½
• Huygens wrote De Ratiociniis in Ludo
  Aleae (On Reasoning in Games of
  Chance)1st text on probablity theory
  published
                 Gaming
• Many believed nothing was random, a
  divine being (God or gods) influenced
  earthly events
• They could interfere with the throwing of
  dice.
• Gambling was know in Egypt 3500BC with
  a game called “Hounds and Jackals.”
• Games were used during the famine
  – One day of pure games to keep their minds
    off food
  – One day for food
            Restrictions
• Greeks and Romans issued laws
  forbidding gaming except certain seasons
  because of popularity
• Forbidden by Jews under penalty of death
  because the gamblers always expects to
  win, and therefore get something for
  nothing. Rabbis considered this immoral
  and akin to robbery
             Restrictions
• Christian Churches banned gaming
  because of the accompanying vices of
  dringking and swearing
• 3rd Crusade (AD1190) – no persons below
  the rank of knight could gamble and
  those who could, weren’t allowed to lose
  more that 20 shillings in a 24 hour period.
                 Cards
• Exact origin is murky
• Credited to Egyptians, Chinese, and
  Indians (~1300’s)
• 1st were hand painted
• Gutenberg made the 78-card tarot deck
• About 1500, present day cards were made
• Original portraits of actual people David
  (spades), Charlemagne (hearts),
  Alexander the Great (clubs), Julius Cæsar
  (diamonds)
                 Cardan
• Wrote Liber de Ludo Aleae – 1st reasoned
  considerations relating to games of
  chance
• Stated that the probability of a particular
  outcome was the sum of the possible
  ways of achieving that outcome divided
  by the totality of possible outcomes of an
  event
                Cardan
• Found a transition from empiricism to
  theoretical concept of a fair die
• Probably became the real father of
  modern probability theory
   Blaise Pascal (1623-1662)
• Closely linked with Fermat as one of the
  joint founders of probability theory
• A gifted writer, religious philosopher,
  creative mathematician, and experimental
  physicist
• Theologian
          History of Pascal
• Raised without ever going to school or
  studying at a university
• Taught exclusively by his father
• Kept from studying math until age 15
  – All math books were locked up and the
    subject was forbidden
          History of Pascal
• At 12, wanted to know what geometry
  was and worked on it alone drawing
  charcoal figures on floor tiles
• His father caught him trying to prove his
  guess that the sum of the angles of a
  triangle is two right angles (Euclid’s Prop
  32)
• He than was given a copy of Euclid’s
  Elements and mastered them without
  assistance
    Essay pour les coniques
• Pascal’s mystic hexagon
• In this essay, Pascal deduced no fewer
  that 400 propositions on conic sections
  from the mystic hexagon theorem
              Calculator
• Saw his father with lots of addition work
  and thought of a contraption to add and
  multiply eight figures
• Named the Pascaline (like calculators of
  1400’s)
• Too expensive to buy so it was more a
  curiosity than a useful device
                 Cycloid -1658
• Cycloid – the curve traced out by a point
  on the circumference of a wheel, as the
  wheel rolls along a straight line.
 http://archives.math.utk.edu/visual.calculus/0/parametric.5/

• Pascal found out how to get area under
  one arch and the volume of the solid
  obtained revolving the curve about the
  base line
             Cycloid -1658
• The work of Pascal on the cycloid had one
  by product of surprising importance
  – Inspired Leibniz in his invention of he
    differential and integral calculus
• One of Pascal’s letters involved certain
  calculations that resembled the evaluation
  of the definition of the sine function
               De Méré
• De Méré made a precarious living at cards
  and dive
• One of De Méré’s gaming problems gave
  Pascal new insight
  – Page 417
             Pascal’s Triangle
• The arithmetic triangle, now generally known as
  Pascal’s triangle, is an infinite numerical table in
  “triangular form,” where the nth row of the
  triangle lists the successive coefficients in the
  binomial expansion of x  y    n

• Was thought of many times before Pascal
  –   Chu Shih-Chen in 1303
  –   Chia Hsien in 1050
  –   Omar Khayyam
  –   Al-Tusi
• Printed by Peter Apian in 1527
           Pascal’s Triangle
• Linked to Pascal because he was the first to
  make any sort of systematic study of the relation
  it exhibited.
• Wrote Traité du Triangle Arithmétique – an
  exposition of the properties and relations
  between the binomial coefficients
             Recursive formula:
x  y      x   x y   x y   
                          n-1       n- 2 2
                                                     xy   y
                                                       n-1
       n     n   n   n          n              n             n    n
             0       1          2              n1           n
   – Stated in this symbolism, the property of the
     numbers  n  on which the triangle is based
                           
                r

                         n      n        n1
                         r 1   r        r
   – This rule does not give a formula for the
     binomial coefficients, but tells us only how to
     build them from 2 numbers on the previous
     base.
Explicit formula for      
                          n
                          r


     
      n
      r   
                n!
            r!n  r !
            Consequences
• Pascal listed 19 consequences of the
  binomial coefficients that could be
  discovered for the triangle.
• Tells us that each number in the triangle is
  the sum of entries in the preceding
  diagonal, beginning with the entry above
  the given number
           Consequence XII

     nr
    n
    r 1
      r 1
      n
      r
• Proof on pg 427
• Important because it’s proof involves the
  1st explicit formulation of the
  demonstrative procedure known as
 induction
                  Induction
• The proof of a proposition P(n) by
  mathematical induction consists of
  showing that:
  1. The statement P(n0) is true for some
     particular integer n0.
  2. The assumed truth of P(k) implies the truth
     of P(k+1)
Activity:
   Prove sum of cubes by induction!
               nn  1 n n  1
          n               2    2       2

         j  2   4
         j 1
              3

                       
         Christian Huygens
           (1629-1695)
• Taught by tutors at home
• Went to University of Leiden
• By 1666, he was known as a physicist,
  astronomer, and mathematician
• Admired Newton
• Wrote Traité de Lumiére describing his radically
  new wave theory of light
• Invented pendulum clock
         Christian Huygens
           (1629-1695)
• Came up with formula T  2
                                  l
                                  g
  relating to oscillating period T of a simple
  pendulum undergoing small swings with it’s
  length l. This afforded a practical way of
  measuring g, the acceleration due to gravity.
• Wrote De Ratiociniis in Ludo Aleae (On
  Mathematical Probability)
• Most important discovery ws the important
  concept of mathematical expectation (or as he
  called it the value of chance of winning a game)
      James (Jacques, Jacob)
            Bernouilli
• Became a minister and studied math and
  astronomy against his father’s wishes.
• He was the first to achieve full understanding of
  Leibniz’s differential calculus
• He taught these to his younger brother John.
      James (Jacques, Jacob)
            Bernouilli
• Bernouilli’s contributions to mathematics is Ars
  Conjectandi (the Art of Conjecture)
  – First part: A reproduction of Huygen’s De Ratiociniis
    with commentary.
  – Second part: all the standard results on permutations
    and combinations.
  – Third part: Consists of 24 problems relating to the
    various games of chance that were popular in
    Bernouilli’s day.
  – Fourth part (Most important): Bernouilli’s theorem:
    if p is the probability of an event, if k is the actual
    number of times the event occurs in n trials, if e > 0 is
    an arbitrarily small number, and if P is the probability
                            k
    that the inequality n  p  e is satisfied, then P
    increases to 1 as n grows with out bound.
        John (Jean, Johann)
            Bernouilli
• Studied medicine and was tutored by his
  brother in mathematics privately
• Followed in James’s footsteps by leading
  exponents of the calculus
• Antagonism devolved and the two brothers
  became rivals
• Came up with a formula for a curve along which
  a body only affected by gravity would fall with
  constant velocity:

   dy b y  a  dx a
        2     2        3
        John (Jean, Johann)
            Bernouilli
• John Bernouilli tutored L’Hospital
• L’Hospital published various discoveries given to
  him by Bernouilli
• Bernouilli fought back after L’Hospital died
• L’Hospital’s work was proven to be Bernouilli’s
                De Moivre
• Wrote Doctrine of Chances: or a Method of
 Calculating the Probability of Events in Play
  (1718)
• Supported himself by solving problems
  proposed to him by wealth patrons who wanted
  to know what stakes to offer in games of
  chance.
• His book contained numerous problems on
  throwing dice and other probabilities
                  De Moivre
• Became lethargic and predicted his own death
  – He was sleeping 20 hours a day and finding he was
    sleeping a quarter of an hour longer than on the
    preceding day
  – He calculated he would die in his sleep on the very
    day in which he slept up to the limit of 24 hours.
  – He died at the age of 87; the cause of his death was
    recorded as “somnolence”
    (i.e. sleepiness)
 Pierre Laplace (1749 – 1827)
• Educated in school between ages of 7 and 16
• Went to University of Caen at 16
• Intended study theology then decided his true
  vocation lay in mathematics.
• Greatest achievement was Traité de Mécanique
 Céleste
  – Which is designed to solve the great and mechanical
    problems of the solar system and to bring theory to
    coincide so closely with observation that empirical
    equations should no longer be needed.
 Pierre Laplace (1749 – 1827)
• Frequently unable to reconstruct the details in
  his chains of reasoning Laplace would say, “It is
  easy to see that…” and give the result with out
  any further explanation.
  – One astronomer observed, “I never came across one
    of Laplace’s ‘Thus it plainly appears,’ without feeling
    sure that I had hours of hard work before me to fill up
    the chasm and find out and show how it plainly
    appears.”
       Paradox by Nicholas
            Bernouilli
• Lucky Luke walks into Cæsar’s Palace in
  Vegas and sees a new game called only
  “The Coin.” At this game, Luke is given a
  coin to toss. As long as tails show, the
  game continues; once heads shows the
  game is over and Luke collects his
  winnings. If heads shows on the 1st toss,
  Luke receives $1, on the 2nd toss he
  receives $2, 3rd $4, 4th $8, and so on.
    Formula for Expectation
• E = a1p1 + a2p2 + … + anpn
       Paradox by Nicholas
            Bernouilli
• Lucky Luke walks into Cæsar’s Palace in
  Vegas and sees a new game called only
  “The Coin.” At this game, Luke is given a
  coin to toss. As long as tails show, the
  game continues; once heads shows the
  game is over and Luke collects his
  winnings. If heads shows on the 1st toss,
  Luke receives $1, on the 2nd toss he
  receives $2, 3rd $4, 4th $8, and so on.
      Lucky Luke’s Paradox
• Luke is not called “Lucky” for nothing, but
  he isn’t sure he has enough money to play
  the game. Assuming this is a fair game
  (i.e. cost to play = expected winnings),
  how much does the game cost to play?
                  Laplace
• Wrote a book to aquaint a broader circle
  of readers with the fundamentals of
  prbability theory and its applications
  without resorting to higher mathematics.
• Gives definition of the probability of an
  event as:
  – Pr[event] = # of favorable outcomes / total #
    of outcomes
     Mutually Exclusive and
      Independent Events
• 1) If A and B are mutually exclusive events
  (they cannot both happen at the same
  time) then,
  – Pr [A or B] = Pr[A] + Pr[B]
• 2) If A and B are two independent events
  (the occurance of one doesn’t affect the
  probability of the other) then,
  – Pr[A and B] = Pr[A] Pr[B]
        Bernoulli’s Formula
• If the probability of successes on a single
  trial is denoted by p and the probability of
  failure by q = 1-p, then p and q remain
  constant from trial to trial.
• Bernoulli showed that the probability of
  observing exactly r successes in n trials
  was expressed by the rth term of the
  expansion for (p+ q)r:
  – Pr[r successes and n-r failures] = (nCr) pr qn-r

				
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posted:2/14/2012
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