History gambling/probability

1. Gambling &
the Development of the Theory of
Probability

2. Overview
1. Ancient origins
2. 16th & 17th century
3. 18th & 19th century
4. Modern probability

3. 1. Ancient Origins
• Dice throwing appeared independently all across the world thousands
of years Before Christ:
- Greece, The Roman Empire, China, Iran, India, etc.
• Since at least 3500 BC:
Astragali/Tali (“αστράγαλοι” = knucklebones):
- ankle bones usually taken from goats or sheep
- most commonly found in Greece and the Roman Empire
- Ancient precursor for dice
- were used in games of chance & rituals
(Larsen & Marx, 2001)
Image 1. Image of Astragali. (Kemp, 2010)

4. 2. 16th & 17th century
• During the 16th and 17th century
a great deal of attention was given to games of chance,
such as: tossing coins, throwing dice, playing cards
• Gerolamo Cardano (1501- 1576)
- famous Italian physician, mathematician and gambler
- wrote a short manual in 1524 which contained the first
mathematical treatment of probability – published in
almost a century later in 1633
(Debnath & Basu, 2014; Packel, 2014)
Image 2. Gerolamo Cardano
(Kuetting & Sauer, 2010)

5. ‘Liber de Ludo Aleae’
(Games of Chance)
• First mathematical treatment of
probability
• Proposed idea:
probability p between 0 and 1 to an event whose
outcome is random, and then applied this idea to
games of chance
• Anticipated (but never explicitly stated):
the law of large numbers: when the probability of
an event is p, and n a large number of trials, then
the number of times it will occur is close to np
(Debnath & Basu, 2014; Packel, 2014)

6. • Some believe Gerolamo Cardano’s contributions to
the field of probability did not provide any real
development of probability theory. However, to
others he is still the…
real father of probability!
(Gorrochurn, 2012; Debnath & Basu, 2014)

7. …but if not him who else would be considered to be
the father of probability
???

8. • Chevalier de Mere (1610–1685)
A French nobleman and gambling expert, had consistently been
losing money in a game with dice. He proposed mathematical
questions to:
• Blaise Pascal (1623–1662)
A French mathematician, physicist and philosopher, who
then communicated these questions to another French
mathematician:
• Pierre de Fermat (1623–1662)
(Debnath & Basu, 2014; Packel, 2014)

9. From 1654
Blaise Pascal & Pierre de Fermat
then began a lively correspondence about
• problems & questions dealing with games of chance
• Their famous correspondence introduced:
- the concept of probability
- expected value
- conditional probability
It is therefore often regarded as the mark of the birth of
CLASSICAL PROBABILITY THEORY
(Kuetting & Sauer, 2008; Debnath & Basu, 2014)

10. • Christian Huygens (1629–1695):
• Dutch mathematical physicist & astronomer
• Inspired by famous correspondence between
Pascal & Fermat
• published his first treatise in 1657 on
probability theory:
De Rationiis in Ludo Aleae
(On Reasoning in Games of Dice)
(Kuetting & Sauer, 2008; Debnath & Basu, 2014)

11. • Huygens introduced the definition of probability of an event as:
a quotient of favourable cases/ all possible cases
• He reintroduced concept
of mathematical expectation
• elaborated Cardano’s idea of
expected value of random variables
• Huygen’s treatise remained the
best book on probability theory until the publication
of Jacob Bernoulli’s (1654–1705) … 
(Kuetting & Sauer, 2008; Debnath & Basu, 2014)
Image 3. Christian Huygens
(Kuetting & Sauer, 2010)

12. 3. 18th and 19th century
• Jacob Bernoulli’s (1654–1705)
path-breaking book:
Ars Conjectandi
(TheArt of Prediction)
1713
Image 4. Ars Conjectandi (Stoudt, 2015)
(Kuetting & Sauer, 2008; Debnath & Basu, 2014)

13. • He formulated a fundamental principle of
probability theory:
The law of large numbers
(Bernoulli’s theorem)
It included:
• Applications of probability theory are discussed in great detail
• the problem of repeated experiments in
which a particular outcome is either a success or a failure
• & he formulated the Bernoulli’s binomial distribution
(random variable represents the number of r successes
in n trials)
• His work, together with Pierre Simon Laplace’s work
(later discussed in this presentation), is defined as the
classical definition of probability! (Jaynes, 2003)
Image 5. Jacob Bernoulli
(Kuetting & Sauer, 2010)

14. • Abraham de Moivre (1667- 1754):
• Published
The Doctrine of Chances
Image 6. The doctrine of chances (Moivre, 2000)

15. • made important contributions using examples of:
• Dice games
• Lottery draws
• he also defined the quotient for calculating classical
probability
• The Moivre-Laplace theorem was one of his most
important contributions
= first version of the Central Limit Theorem of the
probability theorem
Image 7. The doctrine of chances example 1 (Moivre,
2000)
Image 8. The doctrine of chances problem XXIII (Moivre,
2000)

16. Example of a problem
Image 9. The doctrine of chances pages 44 and 45 (Moivre, 2000)

17. Other important mathematicians from
the 16th and 17th century…
• Pierre Simon Laplace (1749–1827)
• Thomas Bayes (1702–1761)
• Joseph Luis Lagrange (1736–1813)
• Leonhard Euler (1707–1783)
• A.M. Legendre (1752–1831)
• Karl Friedrich Gauss (1777–1858)
• John Venn (1834–1923)
• Richard von Mises (1883–1953),
• Augustin De Morgan (1866–1871)
• Emile Borel (1871–1956)
• P.L. Chebyshev (1821–1894).
(Debnath & Basu, 2014)

18. • Pierre Simon Laplace (1749–1827):
• French physicist and mathematician
• First one to make major attempts to develop the theory of probability
as a new area beyond the theory of games of chance
• Published (1812): Theorie Analytique des
Probabilities (Analytical Theory of Probablity):
- Remarkable contributions which led to the definition of
the classical probability (Jaynes, 2003)
- applications to mathematical and social sciences, such as
mortality rate, etc.
Important contributor in STATISTICS
(Debnath & Basu, 2014)
Image 10. Theorie Analytique
des Probabilities (Camille
Sourget Librairie, 2017)

19. 4. Modern Probability
Alexander Nikolajewitsch Kolmogoroff (1903-1987)
- Russian mathematician
- Major contributions in 20th century
- Often seen as world’s leading expert
- Published book:
Foundations of the Theory of Probability
(1933)
laying the modern axiomatic foundations of
probability theory
(Kuetting & Sauer, 2008)
Image 11. Alexander N.
Kolmogoroff
(Kuetting & Sauer, 2008)

20. Kolmogoroff’s axioms –
a small set of basic rules which modern
probability is build on
• Axiom 1
• Axiom 2
• Axiom 3
If Axion 1 & 2 are mutually exclusive!
(Kuetting & Sauer, 2008)

21. References
• Larsen, R. J., & Marx, M. L. (2001). An Introduction to Mathematical Statistics and its Applications
(3rd ed.). London: Prentice-Hall.
• de Moivre, A. (2000). The Doctrine of Chances, or, a Method of Calculating the Probabilities of
Events in Play, (3rd ed.). New York: Chelsea. (Original work published 1716)
• Jaynes, E. T. (2003). Probability Theory: the Logic of Science. UK: Cambridge University Press.
• Kuetting, H., & Sauer, M. J. (2008). Elementare Stochastik (2nd ed.). Germany: Springer-Verlag.
• Gorroochurn,P. (2012). Classic Problems of Probability. US: Wiley NJ.
• Camille Sourget Librairie (2017). Theorie Analytique des Probabilities. Retrieved from
https://camillesourget.com/wp-content/uploads/2008/12/laplace-2_eclairci_redimensionne-
218x300.jpg
• Kemp, R. (2010). Image of Astragali. Retrieved from
http://www.aerobiologicalengineering.com/wxk116/Roman/BoardGames/astra.gif
• Stoudt, G. S. (2015). Ars conjectandi. Retrieved from
http://www.people.iup.edu/gsstoudt/history/images/arsconj.gif