Ancient History Astragali: six sided bones. Not symmetrical. Excavation finds: sides numbered or engraved. primary mechanism through which oracles solicited the opinions of their gods. In Asia Minor: 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 Burnt City 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.
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 were consummate gamblers, as were the early Christians. 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
“Calculus of probabilities”: incompatible with Greek philosophy and early Christian theology. Greeks not inclined to quantify random events in any useful fashion. reconciling mathematically what did happen with what should have = an improper juxtaposition of the “earthly plane” with the “heavenly plane.” Greeks accepted “chance”, 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 deliberate 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…”
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
The Problem of Points The date cited by many historians as the beginning of probability is 1654. 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.
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
Probability theory soon became popular... major contributors included Jakob Bernoulli (1654-1705) and Abraham de Moivre (1667-1754). In 1812 Pierre de Laplace (1749-1827” 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.
Now 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 KolmogorovVon Mises Markov
[Dice are “descendents” of bones] A.True B.False
[Mathematical theory of probability was initiated by Pascal and Fermat] A.True B.False
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
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?
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 4 times. At that point the school bell rings and they have to stop and divide the winnings. How much should each player get?
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 they each of the albums is equally valuable, how many of them should each player get? Round to the closest whole number.