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The Fundamental Theorem of Card Counting with applications to Trente-et-Quarante and Baccarat


The Fundamental Theorem of Card Counting is a unifying principle for the analysis of card games of chance which are characterized by sampling without replacement. The Theorem says (roughly) the “spread” in distribution of player expectations for partially depleted card packs increases with depletion of the card pack. Furthermore, average player expectation is non-decreasing (increasing under suitable hypotheses) with increasing depletion.

The Theorem is used to prove that significant favorable strategies based on card counting do not exist for Trente-et-Quarante or for “tie” bets in Nevada Baccarat. This is in sharp contrast with previous results for Blackjack and for Nevada Baccarat side bets on natural eight and natural nine.

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This paper was supported in part by Air Force Grant AFOSR 70-1870 A, B, C.

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Thorp, E.O., Walden, W.E. The Fundamental Theorem of Card Counting with applications to Trente-et-Quarante and Baccarat. Int J Game Theory 2, 109–119 (1973).

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  • Economic Theory
  • Game Theory
  • Sharp Contrast
  • Fundamental Theorem
  • Card Game