Transcript Probability

Lecture 5, MATH 210G.03 Spring 2014
(Modern) History of Probability
Part II. Reasoning with
Uncertainty: Probability and
Statistics
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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 southeastern Iran
Each possible configuration was associated
with the name of a god and carried the soughtafter 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.
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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 14th
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:
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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
• 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
Kolmogorov
Markov
Von Mises
[Dice are “descendents” of
bones]
A. True
B. False
[Mathematical theory of probability was initiated
by Pascal and Fermat]
A. True
B. False
Just for fun…and 10 points
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
The latest on RPS
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.