Lecture 5, MATH 210G.M03, Fall (Modern) History of Probability

Transcription

1 Lecture 5, MATH 210G.M03, Fall 2017 (Modern) History of Probability

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3 Part II. Reasoning with Uncertainty: Probability and Statistics

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6 Ancient History: Greece and Asia Minor Astragali: six sided bones. Excavation finds: sides numbered or engraved. primary mechanism through which oracles solicited the opinions of their gods. divination rites involved casting five astragali. the oldest known dice were excavated as part of a 5000-year-old backgammon set at the Burnt City, an archeological site in south-eastern Iran Each possible configuration was associated with the name of a god and carried the sought-after advice. An outcome of (1,3,3,4,4), for instance, was said to be the throw of the savior Zeus, and was taken as a sign of encouragement. A (4,4,4,6,6), on the other hand, the throw of the child-eating Cronos, would send everyone scurrying for cover.

7 Astragali were eventually replaced by dice Pottery dice have been found in Egyptian tombs built before 2000 B.C Loaded dice have also been found from antiquity. The Greeks and Romans and early Christians were gamblers. The most popular dice game of the middle ages: hazard Arabic al zhar means a die. brought to Europe by soldiers returning from the Crusades, Rules much like modern-day craps. Cards introduced 14 th Primero: early form of poker. Backgammon etc were also popular during this period. The first instance of anyone conceptualizing probability in terms of a mathematical model occurred in the sixteenth century

8 Calculus of probabilities : incompatible with Greek philosophy and early Christian theology. Greeks not inclined to quantify random events in any useful fashion. Greeks accepted chance as whimsy of gods, but were not empiricists. Plato s influence: knowledge was not something derived by experimentation. stochastic from stochos : target, aim, guess Early Christians: every event, no matter how trivial, was perceived to be a direct manifestation of God s intervention St. Augustine ( ): We say that those causes that are said to be by chance are not nonexistent but are hidden, and we attribute them to the will of the true God

9 Cardano ( ): trained in medicine, addicted to gambling Sought a mathematical model to describe abstractly outcome of a random event. Formalized the classical definition of probability: If the total number of possible outcomes, all equally likely, associated with some actions is n and if m of those n result in the occurrence of some given event, then the probability of that event is m/n. EX: a fair die roll has n= 6 possible outcomes. If the event outcome is greater than or equal to 5 is the one in which we are interested, then m = 2 the outcomes 5 and 6 and the probability of an even number showing is 2/6, or 1/3. Cardano wrote a book in 1525, but it was not published until 1663

10 The Problem of Points The date cited by many historians as the beginning of probability is Antoine Gombaud - Chevalier de Mere ( ) asked Blaise Pascal ( ), and others: Two people, A and B, agree to play a series of fair games until one person has won six games. They each have wagered the same amount of money, the intention being that the winner will be awarded the entire pot. But suppose, for whatever reason, the series is prematurely terminated, at which point A has won five games and B three. How should the stakes be divided? The correct answer is that A should receive seven-eights of the total amount wagered.

11 Pascal corresponds with Pierre Fermat ( ) famous Pascal-Fermat correspondence ensues foundation for more general results. Others got involved including Christiaan Huygens ( ). In 1657 Huygens published De Ratiociniis in Aleae Ludo (Calculations in Games of Chance) What Huygens actually wrote was a set of 14 Propositions bearing little resemblance to modern probability but it was a start

12 Probability theory soon became popular... major contributors included Jakob Bernoulli ( ) and Abraham de Moivre ( ). In 1812 Pierre de Laplace ( ) ThéorieAnalytique des Probabilités. Before Laplace: mathematical analysis of games of chance. Laplace applied probabilistic ideas to many scientific and practical problems: Theory of errors, actuarial mathematics, and statistical mechanics etc l9th century.

13 20 th Century applications: applications of probability extend to Mathematical statistics genetics, psychology, economics, engineering, Main contributors: Chebyshev, Markov, von Mises, and Kolmogorov. The search for a widely acceptable definition of probability took nearly three centuries and was marked by much controversy. A. Kolmogorov (1933): axiomatic approach Foundations of Probability now part of a more general discipline known as measure theory." Chebychev Markov Kolmogorov Von Mises

14 21 st Century High dimensional data/machine learning Data compression Weak signals

15 Frequentist (Fisher ) vs Bayesian ( )

16 [Dice are descendents of bones] A. True B. False

17 [Mathematical theory of probability was initiated by Pascal A. True B. False and Fermat]

18 Just for fun and 10 points

19 Ick Ack Ock Rules: each symbol beats another according to this schema ick, the stone breaks ack the scissors ack the scissors cut ock the paper ock the paper catch ick the stone Points: you earn a point each time you win a single match Scope: to win the number of matches decided at the beginning, 1 on 1, 2 out of 3, 3 out of 5

20 The latest on RPS

21 The problem of points: advanced theory Suppose Jack and Jill flip a fair coin until someone wins time times. After 15 rounds, Jack has won 7 times and Jill has won 8 times. If they continue to play, what is the probability that Jill will be the first to win 10 times? Analysis: someone is guaranteed to win at the end of four or fewer more games. WHY? Suppose that Jack calls heads each time. We can analyze each of 32 possible outcomes and see how many of them are unfavorable to Jill. This means Tails comes up zero or 1 times. The possible unfavorable sequences are: HHHH THHH HTHH HHTH HHHT Each occurs with probability 1/16 and there are five of them so the probability that Jack will win is 5/16 and the probability Jill will win is 7/16. If we wish to analyze more challenging examples then we should develop a theory.

22 Exercises (solutions follow)

23 Exercise 1 Suppose that Leila and Tofu each wagered $1 in a best of seven tourney of Rock, Paper, Scissors. First to win four times takes all. After five rounds Tofu has three wins and Leila has one win. At this point Leila s mom calls her home for dinner. What is the fairest way to divide the $2 that has been wagered?

24 Exercise 1 Suppose that Leila and Tofu each wagered $1 in a best of seven tourney of Rock, Paper, Scissors. First to win four times takes all. After five rounds Tofu has three wins and Leila has one win. At this point Leila s mom calls her home for dinner. What is the fairest way to divide the $2 that has been wagered? Solution: Leila needs 3 straight wins. Chance of each win is ½. Chance of three straight is 1/8. So Leila should get 25cents and Tofu should bet $1.75

25 Exercise 2 Optimus and Thor are playing a game of coin flip. Each time Thor flips the coin, Optimus guesses heads or tails. If Optimus guesses right he wins that round. Otherwise Thor wins the round. Each wagers $1 and the first with 10 wins gets the $2. After 12 rounds Optimus has won 8 times and Thor 6 times. At that point the school bell rings and they have to stop and divide the winnings. How much should each player get?

26 Exercise 2 How much should each player get? Solution: Optimus has 8 winds and Thor 6. After 5 more games either optimus will have won at least two more or Thor will have won at least 4 more. There are 32 possible outcomes of 5 games. The number of these in which Thor wins at least four is 1+5=6 so the odds of Thor winning if play continues is 6/32=3/16 and of Optimus winning is 13/16.

27 Exercise 3 Dweezil and Moon Unit each own half of the 94 different Frank Zappa albums. They agree to play a series of checkers games until one of them wins ten times, in which case the winner gets all the albums. After 15 rounds Dweezil has won 7 times and Moon Unit eight times. At this point they have to stop because Moon Unit has to go to tuba lessons. Assuming each of the albums is equally valuable, how many of them should each player get? Round to the closest whole number.

28 Exercise 3 Solution: Dweezil needs three more wins and Moon unit needs two more. At most four more games are needed. Of the 16 possible outcomes of the four games, 11 of them are favorable to MU and 5 are favorable to Dweezil. So MU should get 94x11/16 which rounds up to 65 and Dweezil should get the remaining 29 FZ albums.

29 Exercise 4. Ick and Ock and the problem of points Your math teacher gives you five minutes at the end of class to play ick ack ock with your classmate. First to win ten times gets 100 points. After 15 rounds, discarding ties, you have won 8 times and your classmate has 7 wins. How should the 100 points for the assignment be divided between the two of you?

30 Exercise 4. Ick and Ock and the problem of points Solution: you need tow more wins; your classmate needs three. At most 4 more games are needed since after four either you will have won two or more or your classmate will have won three or more. There are 16 possible outcomes. Of these, six have you win twice, 4 have you win three and one has you win all four, so 11 favorable outcomes out of 16. You should get 11/16 of the 100 dollars, or $68.75 and your classmate should get the remaining $31.25

31 The following exercises were not covered in class and will not be reflected in exams. Solutions will be written eventually

32 Balls in an Urn Suppose that there are 100 balls in an urn, 30 of them black, 30 red, and 40 other. What is the probability that a ball, drawn from the urn at random, will be black or red? What is the probability that it will not be black?

33 Balls in an urn 2 Suppose that there are 50 red balls and 50 black balls in an urn. What is the probability that the first two balls drawn randomly from the urn will be black?

34 Balls in an urn 3 Suppose that there are 60 red balls and 40 black balls in an urn. Suppose that 50 balls are drawn randomly from the urn. What is the probability that the next ball drawn will be black?