November 10, 2004 On the fundamental theorem of card counting with application to the game of trente et quarante S. N. Ethier1 and David A. Levin1 1Postal address: Department of Mathematics, University of Utah, 155 S. 1400 E., Salt Lake City, UT 84112, USA. E-mail:
[email protected],
[email protected] Abstract. A simplified proof of Thorp and Walden’s fundamental theorem of card count- ing 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. Key words. Sampling without replacement, exchangeability, martingale, U-statistic, cen- tral limit theorem. MSC: Primary 60G09. Secondary 60G35, 60C05. Short title: Card counting and trente et quarante. 1. Introduction. Thorp and Walden (1973) proved, assuming a fixed player strategy, that “the ‘spread’ in the distribution of player expectations for partially depleted card packs increases with the depletion of the card pack,” and they termed this result the fundamental theorem of card counting. Their proof relies on the theory of convex contractions of measures and some combinatorial analysis. Our aim here is to provide a simpler proof, one that depends only on exchangeability and Jensen’s inequality. This simplification not only makes the theorem easier to understand, it allows us to investigate the case of a variable player strategy. It also leads to a central limit theorem in the fixed-strategy case, which in turn permits an analysis of the card-counting potential of the casino game of trente et quarante, without the need for Monte Carlo simulation.