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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 n1 n – Stated in this symbolism, the property of the numbers n on which the triangle is based r n n n1 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 nr 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! nn 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