Probability Theory

Bonnie Hand

Jeff Becker

Early Games, but No Probability

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

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

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

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

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

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

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

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

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

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)

Background– 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

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?

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…

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

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

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)

Background– 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?

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)

Ars Conjecture: Divided into four parts– 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?

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…

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

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

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…

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

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

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

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

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

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:– 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