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Lecture 8 Probability Theory Mathematics E-320, Spring, 2012 Lecture 8 Probability Theory Oliver Knill Harvard university Monday, March 26, 12 What is Probability ? Probability is the Science of Chance and Randomness Monday, March 26, 12 Butterfly Dreaming Monday, March 26, 12 Gaming Historically, Probability Emerged from Gaming. Geronimo Cardano: Liber de Ludo Aleae (1526) Monday, March 26, 12 Experiments and Events Probability Laboratory Ω 2 3 4 5 elements are 6 1 experiments, Ω subsets are events P[A] probability, usually |A|/|Ω| Monday, March 26, 12 Dice The chance to throw an 2 3 even 4 5 number 6 1 with a dice is 3/6 Monday, March 26, 12 Two Dice What is the chance that the sum of two dice is bigger than 8? X[(a,b)] = a+b random variable X. Monday, March 26, 12 Two Dice, Sum > 8 11 12 13 14 15 16 21 22 23 24 25 26 Probability: 31 32 33 34 35 36 41 42 43 44 45 46 10/36 51 52 53 54 55 56 61 62 63 64 65 66 Monday, March 26, 12 52 Cards, How many Shuffles? Monday, March 26, 12 A card Problem What is the chance that a randomly shuffled Deck of cards You will pick up 4 Kings at first? Monday, March 26, 12 Monday, March 26, 12 What is the Laboratory? Monday, March 26, 12 What is the good event? How many experiments are there in the good event? Monday, March 26, 12 What is the probability? = 4! 48!/52! = 1/270’725 Monday, March 26, 12 What is the probability? P[A] = |A|/|Ω| = 4! 48!/52! = 1/270’725 Monday, March 26, 12 We can see this differently: The laboratory consists of all possible ways to choose 4 cards without looking at the order. The probability of a one member is: Laplace First 1/ Principle |Ω| Monday, March 26, 12 Four formulas n! permutations n choosing k from n ( k ) without repetition order not matter k choose k from n n n with repetition k!( k ) choose k from n order matters Monday, March 26, 12 Problem 1: Scrabble How many words? STORY Monday, March 26, 12 Problem 2: Hound of Basekervilles 338787 letters How big is the chance to type it by random? Monday, March 26, 12 Problem 3: Hunger Games choose 3 from 12 Monday, March 26, 12 Problem 4: Combination Lock choose 3 from 40. Order matters Monday, March 26, 12 Petersburg Casino Daniel Bernoulli 1738 Monday, March 26, 12 How much do we expect to win? Experiment! Monday, March 26, 12 Hangover Monday, March 26, 12 Martingale Betting System 18th century France Double the Entrance Fee, if Losing. Stop Playing and walk out if you win. Without Limit on Cash and Betting Limit, you always Win. Paradox: There is a mathematical theorem which assures that with a time limit, the expected win is zero. Monday, March 26, 12 Let's try it out. Monday, March 26, 12 Doubling Strategy Monday, March 26, 12 Can one Compute the Motion? Monday, March 26, 12 Predictability VI Using a cellular phone and a computer, a group of people were able to predict the Roulette numbers in London: they won1.3 Millon Pounds until they got arrested. Monday, March 26, 12 Predictability IX Monday, March 26, 12 The Monty Hall Problem 1975 bacame a Sensation in 1991 Monday, March 26, 12 Lets Make a Deal Monday, March 26, 12 Monday, March 26, 12 1 2 3 Monday, March 26, 12 1 2 3 Monday, March 26, 12 1 2 3 Monday, March 26, 12 1 2 3 Monday, March 26, 12 Marilyn vos Savant 1991 Parade Column Monday, March 26, 12 Monday, March 26, 12 Monday, March 26, 12 Even more surprise "Dave has two children, one of whom is a boy born at night. What is the probability that Dave has two boys". The information of "being born at night" makes the probability bigger than 1/3. Due to Gary Foshee: Monday, March 26, 12 Bertrand Paradox Joseph Bertrand, 1888 Monday, March 26, 12 Monday, March 26, 12 First Solution Monday, March 26, 12 Second Solution Monday, March 26, 12 Third Solution Monday, March 26, 12 Birthday Paradox Given a class of 23 students. What is the probability that two have the same birthday? Monday, March 26, 12 Adding Random Variables S n= X 1+ X2 + .....+Xn How fast does it grow? Monday, March 26, 12 In one dimensions Random Walk Monday, March 26, 12 in two dimensions Random Walk Monday, March 26, 12 in three dimensions Random Walk Monday, March 26, 12 Many runs go to infinity in general. Monday, March 26, 12 The Law of Large Numbers Sn /n m the average = expectation = Mean Monday, March 26, 12 Monday, March 26, 12 The Central Limit Theorem X n has zero mean Sn /√ n Monday, March 26, 12 Law of Iterated Logarithm X n has zero mean Sn /√ 2n log log n accumulates in bounded interval. Monday, March 26, 12 Monday, March 26, 12 Is π random? 3.1415926535897932384626433832795028841971693993751058209749445923078164062862\ 089986280348253421170679821480865132823066470938446095505822317253594081284811\ 174502841027019385211055596446229489549303819644288109756659334461284756482337\ 867831652712019091456485669234603486104543266482133936072602491412737245870066\ 063155881748815209209628292540917153643678925903600113305305488204665213841469\ 519415116094330572703657595919530921861173819326117931051185480744623799627495\ 673518857527248912279381830119491298336733624406566430860213949463952247371907\ 021798609437027705392171762931767523846748184676694051320005681271452635608277\ 857713427577896091736371787214684409012249534301465495853710507922796892589235\ 420199561121290219608640344181598136297747713099605187072113499999983729780499\ 510597317328160963185950244594553469083026425223082533446850352619311881710100\ 031378387528865875332083814206171776691473035982534904287554687311595628638823\ 53787593751957781857780532171226806613001927876611195909216420199 Monday, March 26, 12 Monday, March 26, 12 More History: Cirolamo Cardano 1501-1576 Blaise Pascal and Pierre Fermat Pierre Laplace (1814) Andrey Kolmogorov Monday, March 26, 12 Do you remember Cardano? Monday, March 26, 12 3 2 X +aX+bX+c=0 Solving the Cubic Depressed version x=X+a/3 3 Scipione del Ferro x +px+q=0 1465 -1526 With x=u-p/(3u), we get the Niccolò Fontana quadratic equation: Tartaglia 1500-1557 u 6+qu-p/27=03 Fiore-Tartaglia Challenge Gerolamo Cardano 1501-1576 Monday, March 26, 12 Pascal and Fermat Correspondence 1654 PIerre Fermat, Blaise Pascal: 1601-1665 1623-1663 Monday, March 26, 12 Laplace Kolmogorov Pierre Andrey Laplace Kolmogorov 1749-1827 1903-1987 Monday, March 26, 12 Modern questions Monday, March 26, 12 Quantium Mechanics Monday, March 26, 12 Other random variables Monday, March 26, 12 Statistics Monday, March 26, 12 Literature Ivars Ekeland: The broken Dice Ivars Peterson The jungles of Randomness. Richard Isaac: The Pleasures of Probability Edward Beltrami: What is Random? Wolff: Breakthroughs in Mathematics Monday, March 26, 12.
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