Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T07:17:04.887Z Has data issue: false hasContentIssue false
  • English
  • Français

On the fundamental theorem of card counting, with application to the game of trente et quarante

Part of: Stochastic processes Combinatorial probability

Published online by Cambridge University Press:  01 July 2016

S. N. Ethier  and
David A. Levin
S. N. Ethier*
Affiliation:
University of Utah
David A. Levin*
Affiliation:
University of Utah
*
∗ Postal address: Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 84112, USA.
∗ Postal address: Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 84112, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A simplified proof of Thorp and Walden's fundamental theorem of card counting is presented, and a corresponding central limit theorem is established. Results are applied to the casino game of trente et quarante, which was studied by Poisson and De Morgan.

Keywords

Sampling without replacementexchangeabilitymartingaleU-statisticcentral limit theorem

MSC classification

Primary: 60G09: Exchangeability
Secondary: 60G35: Signal detection and filtering 60C05: Combinatorial probability
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 1 , March 2005 , pp. 90 - 107
Copyright
Copyright © Applied Probability Trust 2005 

References

Bertrand, J. (1888). Calcul des Probabilités. Gauthier-Villars, Paris.Google Scholar
Boll, M. (1936). La Chance et les Jeux de Hasard. Librairie Larousse, Paris.Google Scholar
Boll, M. (1945). Le Trente et Quarante. Le Triboulet, Monaco.Google Scholar
De Morgan, A. (1838). An Essay on Probabilities, and on Their Application to Life Contingencies and Insurance Offices. Longman, London.Google Scholar
Gall, M. (1883). La Roulette et le Trente-et-Quarante. Delarue, Paris.Google Scholar
Grégoire, G. (1853). Traité du Trente-Quarante. Comptoir des Imprimeurs-Unis, Paris.Google Scholar
Griffin, P. (1976). The rate of gain in player expectation for card games characterized by sampling without replacement and an evaluation of card counting systems. In Gambling and Society: Interdisciplinary Studies on the Subject of Gambling, ed. Eadington, W. R., Thomas, Springfield, IL, pp. 429–442.Google Scholar
Griffin, P. A. (1999). The Theory of Blackjack, 6th edn. Huntington Press, Las Vegas, NV.Google Scholar
Huyn, P. N. (1788). La Théorie des Jeux de Hasard, ou Analyse du Krabs, du Passe-Dix, de la Roulette, du Trente & Quarante, du Pharaon, du Biribi & du Lotto. Unknown publisher, Paris.Google Scholar
Lee, A. J. (1990). U-Statistics: Theory and Practice. Marcel Dekker, New York.Google Scholar
May, J. (2004). Trente et quarante. Available at http://ourworld.compuserve.com/homepages/greenbaize21/trenteet.htm Google Scholar
Nandi, H. K. and Sen, P. K. (1963). On the properties of U-statistics when the observations are not independent. II. Unbiased estimation of the parameters of a finite population. Calcutta Statist. Assoc. Bull. 12, 124–148.CrossRefGoogle Scholar
Poisson, S.-D. (1825). Mémoire sur l' avantage du banquier au Jeu de trente et quarante. Ann. Math. Pures Appl. 16, 173–208.Google Scholar
Polovtsoff, Gen. P. (1937). Monte Carlo Casino. Stanley Paul, London.Google Scholar
Scarne, J. (1974). Scarne's New Complete Guide to Gambling. Simon & Schuster, New York.Google Scholar
Scrutator, (1924). The Odds at Monte Carlo. John Murray, London.Google Scholar
Silberer, V. (1910). The Games of Roulette and Trente et Quarante as Played at Monte Carlo. (Reprint of the technical chapters from Vom grünen Tisch in Monte Carlo.) Harrison & Sons, London.Google Scholar
Székely, G. J. (2003). Problem corner. Chance 16 (4), 52–53.Google Scholar
Thorp, E. O. and Walden, W. E. (1973). The fundamental theorem of card counting with applications to trente-et-quarante and baccarat. Internat. J. Game Theory 2, 109–119.Google Scholar
Todhunter, I. (1865). A History of the Mathematical Theory of Probability. Macmillan, Cambridge.Google Scholar