Transcript Probability - New Mexico State University

Lecture 4, MATH 210G.03, Spring 2013
(Modern) History of Probability
Part II. Reasoning with
Uncertainty: Probability and
Statistics
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
• 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.
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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 likemodern-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
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 (16671754).
• 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
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.