Trente et Quarante
Trente et quarante (“thirty and forty” in French) is an elegant card game that dates at least as far back as the 17th century and is still played in Monte Carlo. It has been studied by Poisson, De Morgan, Bertrand, and Thorp, among others. In Section 20.1
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Trente et Quarante
There came, at a time when the chief Lenoir was at Paris, and the reins of government were in the hands of his younger brother, a company of adventurers from Belgium, with a capital of three hundred thousand francs, and an infallible system for playing rouge et noir, and they boldly challenged the bank of Lenoir, and sat down before his croupiers, and defied him. William Makepeace Thackeray, The Kickleburys on the Rhine
Trente et quarante (“thirty and forty” in French) is an elegant card game that dates at least as far back as the 17th century and is still played in Monte Carlo. It has been studied by Poisson, De Morgan, Bertrand, and Thorp, among others. In Section 20.1 we describe the four betting opportunities and evaluate the associated probabilities and house advantages. In Section 20.2 we analyze the card-counting potential of trente et quarante.
20.1 Red, Black, Color, Inverse Trente et quarante, also known as rouge et noir (“red and black” in French), is played with a six-deck shoe comprising 312 cards. Suits do not matter except insofar as they determine color: Diamonds and hearts are red, whereas clubs and spades are black. Aces have value one, court cards have value 10, and every other card has value equal to its nominal value. Two rows of cards are dealt. In the first row, called Black, cards are dealt until the total value is 31 or greater. In the second row, called Red, the process is repeated. Thus, each row has associated with it a total between 31 and 40 inclusive. Four even-money bets are available, called red, black, color, and inverse. (To avoid confusion, we use upper case when referring to the row, e.g., Red, and lower case when referring to the wager, e.g., red.) Let us define the winning row (Red or Black) to be the one with the smaller total if the two S.N. Ethier, The Doctrine of Chances, Probability and its Applications, DOI 10.1007/978-3-540-78783-9 20, © Springer-Verlag Berlin Heidelberg 2010
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totals are different; of course, the other row is the losing row . A bet on red wins (resp., loses) if Red is the winning (resp., losing) row. A bet on black wins (resp., loses) if Black is the winning (resp., losing) row. A bet on color wins (resp., loses) if the color of the first card dealt to Black is the same as the color of the winning (resp., losing) row. A bet on inverse wins (resp., loses) if the color of the first card dealt to Black is different from the color of the winning (resp., losing row). For all four bets, a push occurs if the Red and Black totals are equal and greater than 31. If the Red and Black totals are both equal to 31, half the amount of the bet is lost; this is called a refait and is the source of the casino’s advantage. For all four bets, if the Red and Black totals are both equal to 31, there is typically an en prison option similar to that in roulette. We will not consider this option in what follows (but see Problem 20.8 on p. 636). Associated with each of the four even-money bets is an insurance bet for 1 percent of
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