Transcript Probability Theory

Probability Theory
Bonnie Hand
Jeff Becker
Early Games, but No Probability
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Archaeological digs throughout the ancient
world consistently turn up a curious
overabundance of astragali, the heel bones
of sheep and other vertebrates.
The bones were used for religious
ceremonies and for gambling.
Astragali
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Astragali have six sides but are not
symmetrical. Those found in excavations
typically have their sides numbered or
engraved.
For many ancient civilizations, astragali were
the primary mechanism through which
oracles solicited the opinions of their gods.
Egyptians and Greeks
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Pottery dice have been found in Egyptian
tombs built before 2000 B.C, and by the time
Greek civilizations were in full flower, dice
were everywhere.
Loaded dice have also been found from
antiquity. While mastering the mathematics of
probability would prove to be a formidable
task for our ancestors, they quickly learned
how to cheat.
The Start of Probabilistic Ideas
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Probability got off to a rocky start because of its
incompatibility with two of the most dominant forces
in the evolution of our Western culture, Greek
philosophy and early Christian theology.
The Greeks were comfortable with the notion of
chance, but it went against their nature to suppose
that random events could be quantified in any useful
fashion.
For the early Christians, though, there was no such
thing as chance. Every event, no matter how trivial,
was perceived to be a direct manifestation of God’s
deliberate intervention.
A Thought of Pascal
“The excitement that a gambler feels when
making a bet is equal to the amount he might
win times the probability of winning it.”
-Pascal
ACTIVITY!!!
Imagine that you were living in the seventeenth
century as a nobleman. One day your friend
Chevalier de Méré was visiting and
challenged you to a game of chance. You
agreed to play the game with him. He said, "I
can get a sum of 8 and a sum of 6 rolling two
dice before you can get two sums of 7’s."
Would you continue to play the game?
Cardano
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In 1494, Fra Luca Paccioli wrote the first printed work
addressing probability, Summa de arithmetica, geometria,
proportioni e proportionalita.
In 1550, Geronimo Cardano inspired by the Summa
wrote a book about games of chance called Liber de
Ludo Aleae which means A Book on Games of Chance.
Some consider Cardano’s Liber de Ludo Aleae (1565),
which was first published in 1663 and Galilei’s work,
Sopra le Scoperte dei Dadi (1620), which was first
published in 1718 to be the start of probability theory, but
there is a consensus that it all began with some
questions on gambling posed by Antoine Gombaud,
Chevalier de Méré and Damien Mitton to Pascal in 1654.
Cardano
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The first recorded evidence of probability theory can
be found as early as 1550 in the work of Cardano. In
1550, Cardano wrote a manuscript in which he
addressed the probability of certain outcomes in rolls
of dice, the problem of points, and presented a crude
definition of probability. Had this manuscript not been
lost, Cardano would have certainly been accredited
with the onset of probability theory. However, the
manuscript was not discovered until 1576 and
printed in 1663, leaving the door open for
independent discovery.
The Birth of Probability Theory
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Chevalier de Méré gambled frequently to increase his wealth.
He bet on a roll of a die that at least one 6 would appear during
a total of four rolls. From past experience, he knew that he was
more successful than not with this game of chance. Tired of his
approach, he decided to change the game. He bet that he
would get a total of 12, or a double 6, on twenty-four rolls of two
dice. Soon he realized that his old approach to the game
resulted in more money. He asked his friend Blaise Pascal why
his new approach was not as profitable. Pascal worked through
the problem and found that the probability of winning using the
new approach was only 49.1 percent compared to 51.8 percent
using the old approach.
Pascal and Fermat
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The Pascal-Fermat correspondence started over a
question by a gambler, Chevalier De Mere. His
question is called "The Problem of Points." In
modern language, the problem is this: Two players
(A and B) of equal skill play a game (think of tossing
a fair coin). The first one to win a fixed number of
games (say 6) wins the whole stake. The game is
interrupted when Player A needs a to win and Player
B needs b to win. How should the stake be divided?
Clearly, if a = b, even division is called for. But what if
a = 1 and b = 5. What's fair?
Pascal and Fermat
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Pascal and Pierre de Fermat continued to
exchange their thoughts on mathematical
principles and problems through a series of
letters. Historians think that the first letters
written were associated with the above
problem and other problems dealing with
probability theory. Therefore, Pascal and
Fermat are the mathematicians credited with
the founding of probability theory.
Pascal
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Pascal later (in the Pensées) used a
probabilistic argument, Pascal's Wager, to
justify belief in God and a virtuous life. The
work done by Fermat and Pascal into the
calculus of probabilities laid important
groundwork for Leibniz's formulation of the
infinitesimal calculus.
Christianus Huygens
(1629-1695)
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Background
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Born into a family of wealth and position
Lord of Zelem and of Zuylichem
Expected to study in order to fit into his position
Tried really hard until age of 25 with his two
articles Theorems on the quadrature of
hyperbolae, ellipses and circles (age 18) and New
Inventions concerning the magnitude of the circle
(age 25) but were of little interest
Thoughts of Huygens
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Huygens realized it was not all fun and
games when he wrote:
“Though I would like to believe that if
someone studies these things a little more
closely, then he will almost certainly come to
the conclusion that it is not just a game,
which has been treated here, but that the
principles and the foundations are laid of a
very nice and deep speculation.”
So who was he in the probability field?
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Scientist who first put forward in a systematic
way the new propositions evoked by the
problems set to Pascal and Fermat.
Heard about the problem of points but was
not aware of the solutions of Fermat and of
Pascal.
Important work finally…
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Fermat posed more difficult questions to
Huygens which were eventually used in his
(Huygens) Exercises
1657 wrote the treatise De Ratiociniis in
Aleae Ludo or On Reasoning in a Dice
Game
Getting to the Good Stuff
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Huygens’s approach started from the idea of
“equally likely” outcomes.
Used the idea of expectation or “expected
outcome”
Ex. You are offered one chance to throw a
single die. If 6 comes up you get $10; if 3
comes up you get $5; otherwise you get
nothing. What is a fair price to pay for
playing this game?
Answer
 1/6
x $10 + 1/6 x $5 + 4/6 x $0 =
$2.50
What else is expectation useful for?
 Fundamental to the way insurance
companies assess their risks when they
underwrite policies
Probability in English
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In 1692, John Arbuthnot's translation of Huygens' De
Ratiociniis in Ludo Aleae becomes the first
publication on probability in the English language.
It is titled Of the Laws of Chance, or, a method of
Calculation of the Hazards of Game, Plainly
demonstrated, And applied to Games as present
most in Use.
The preface contains the following observations:
“It is impossible for a Die, with such determin'd force
and direction, not to fall on such determin'd side,
only I don't know the force and direction which
makes it fall on such determin'd side, and therefore I
call it Chance, wich is nothing but the want of art;...”
Jakob Bernoulli
(1654-1705)
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Background
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destined to be a minister of the Reformed Church
by his parents
studied theology and humanities
Age 18 interested in astronomy
Age 22 went to Geneva and taught math to a
blind girl
Jakob and Christianus?
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Influenced by Huygens’ book De Ratiociniis
in Aleae Ludo
1685 started working on games of chance.
Examined the relationship between
theoretical probability and its relevance to
various practical situations
Ars Conjecture (1713)
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Ars Conjecture: Divided into four parts
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1. treatise of Huygens’ “Reasoning on Games of
Chance” with notes and the first notion on the
art of conjecture
2. theory of permutations and combinations
3. solutions of games of chance
4. principles developed to civil, moral and
economic affairs
Jakob and Cardano?
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Bernoulli sharpened Cardano’s idea of the
Law of Large Numbers.
If a repeatable experiment has a theoretical
probability p of turning out in a favorable way
then when you do the experiment a large
number of times, your outcome will fall within
a specified margin of error close to p*n,
where n is the number of trials
Real World and Math…
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Because we believe in the law of large numbers,
selling a lot of life insurance policies is a good thing
for insurance companies.
Companies will make a profit if they know the
expected death rates. That information is gained
through the selling of life insurance policies.
In the 18th century they thought it was undoubtingly
dangerous.
Bills of Mortality
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English preoccupied with concrete facts so
probability was not of interest to them.
The Church of England was to prepare
parish registers in 1538. It was a register of
all weddings, christenings and burials.
Before this, the Bills of Mortality were printed.
These Bills compared the number of plague
deaths to the other sickness.
John Graunt
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Many of the bills became missing and many of the
church registers were destroyed in the Great Fire of
1666. John Graunt wrote a book called Natural and
Political Observations on the Bills of Mortality in
1662. This and the complete collection of such
material existed up to 1668 plus all the yearly bills
issued after that date compiled by an unknown
author were the only sources of what was known
about diseases causing deaths in that century.
The results are in…
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In the reproduction of the Bill of 1665, the deaths are
divided by causes. The causes were assessed by
searchers or an ancient matron of low intellectual
caliber. Needless to say, there was a large source of
error in the uncertainty of diagnosis.
1 in 4 of the estimated population at risk in 1665 died
of plague and since this population undoubtedly
decreased rapidly as the plague deaths increased,
the actual rate would be much higher.
DeMoivre
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Probability theory continued to grow with
Abraham DeMoivre’s Doctrine of Chances:
or, a Method of Calculating the Probability of
Events in Play, published in 1718.
Pierre Simon Laplace
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French mathematician
Used his mathematics to focus on the workings of
the solar system
1809 went to probability to analyze probable error in
scientific data gathering
1812 published The Analytical Theory of Probability
Wrote a second edition in order to reach others
called Philosophical Essay on Probabilities in 1814
which contained very little mathematical symbols
and formulas
Wrap Up
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The first major accomplishment in the development
of probability theory was the realization that one
could actually predict to a certain degree of accuracy
events which were yet to come. The second
accomplishment, which was primarily addressed in
the 1800's, was the idea that probability and
statistics could converge to form a well defined,
firmly grounded science, which seemingly has
limitless applications and possibilities.
Probability Today
Timeline
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2000(BC) - Games of chance played in ancient civilizations
1494 - Fra Luca Paccioli wrote the first printed work addressing probability
called Summa de arithmetica, geometria, proportioni e proportionalita.
1550 - Geronimo Cardano wrote a book about games of chance called Liber
de Ludo Aleae
1654 - Chevalier de Méré asks Pascal gambling question and 7 letters are
exchanged between Pascal and Fermat in a mere 4 month span
1657 – Christanus Huygens wrote the treatise De Ratiociniis in Aleae Ludo
1662 - John Graunt writes Observations on the Bills of Mortality
1692 - John Arbuthnot's translation of Huygens' De Ratiociniis in Ludo Aleae
becomes the first publication on probability in the English language
1713 – Jakob Bernoulli writes Ars Conjecture
1718 - Abraham DeMoivre’s Doctrine of Chances: or, a Method of Calculating
the Probability of Events in Play is published
1812 – Pierre Simon Laplace publishes The Analytical Theory of Probability
1814 – Laplace writes a second edition in order to reach others called
Philosophical Essay on Probabilities
Resources
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Hald, Anders. “History of probability and statistics and their applications
before 1750.” New York : Wiley, 1990.
David, F. N. “Games, gods and gambling; the origins and history of
probability and statistical ideas from the earliest times to the Newtonian era.”
New York, Hafner Pub. Co., 1962.
Todhunter, Isaac. “A history of the mathematical theory of probability from
the time of Pascal to that of Laplace.” New York, Chelsea Pub. Co., 1949.
And the following websites:
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http://mathforum.org/workshops/usi/pascal/pascal_probability.html
http://teacherlink.org/content/math/interactive/probability/history/briefhistory/home.
html
http://www.stat.stanford.edu/~cgates/PERSI/courses/stat_121/lectures/lecture2/
http://en.wikipedia.org/wiki/Blaise_Pascal
http://www.leidenuniv.nl/fsw/verduin/stathist/sh_17.htm
http://en.wikipedia.org/wiki/Christiaan_Huygens
http://www.math.utsa.edu/~leung/probabilityandstatistics/beg.html