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