Transcript Probability: History

Lecture 5, MATH 210G.2, Fall 2016
(Modern) History of Probability
Part II. Reasoning with
Uncertainty: Probability and
Statistics
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.
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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 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 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.
• 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 (354-430): “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 (1501-1576): 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.
Antoine Gombaud - Chevalier de Mere (16071684) asked Blaise Pascal (1623-1662), 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 (1607-1665)
• famous Pascal-Fermat
correspondence ensues
• foundation for more general results.
• …Others got involved including
Christiaan Huygens (1629-1695).
• 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.
• 20th 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
Kolmogorov
Markov
Von Mises
21st Century
• High dimensional data
• Data compression
• Weak signals
Frequentist (Fisher 1890-1962) vs Bayesian
(1702-1761)
[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
The problem of points: advanced theory
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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.
Exercises (solutions follow)
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 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
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?
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.
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.
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.
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?
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
The following exercises were not covered in class and will not be
reflected in exams. Solutions will be written eventually
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?
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?
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?