Thirty and Forty

Introduction

The name 'Trente et Quarante' translates from French to 'thirty and forty.' This traditional European casino game is still played in Monte Carlo and various prominent French casinos. I visited Monte Carlo but didn't actively seek out this game during my trip.

This game is quite similar to baccarat, where players place bets on two competing hands to determine which will score higher. Interestingly, in this game, a lower score is actually more favorable.

Rules

1. Six 52-card decks are used.
2. In scoring, Aces are counted as one point, while cards numbered 2 through 10 hold their face value. All face cards are considered worth 10 points.
3. There are four different betting options to choose from: Black, Red, Color, and Inverse. Additionally, players have the chance to make an Insurance bet for any of these four.
4. Once players have placed their bets, the dealer will distribute cards to figure out the total for the 'black' hand. The points begin at zero and accumulate as cards are drawn from the shoe until the total reaches 31 or more. The highest achievable score is 40.
5. Following this, the dealer will repeat the process to calculate the 'red' hand's total.
6. The Black bet wins if the black hand has a lower score compared to the red hand. If the red hand wins, the black wager is lost. A tie occurs between 32 and 40 points, which results in a push. However, if both hands tie at 31 points, the player has the choice to either forfeit half their bet or keep it 'imprisoned' for the next round, which I'll explain further.
7. The Red wager functions in opposition to the Black wager—winning when the red hand has a lower score and losing when it has a higher score. The rules for ties remain identical.
8. The Color bet mirrors a Black wager if the first card shown to the black hand is black, and it functions as a Red wager if the card is red.
9. Conversely, the Inverse wager operates inversely to the Color bet; it acts as a Black wager if the initial card is red and behaves like a Red wager if the first card is black.
10. Players can opt for Insurance on their Black, Red, Color, or Inverse bets. The Insurance amount must be equivalent to 1% of the main wager, and it will cover losses in the event of a tie at 31. If the primary bet wins or loses outright, the Insurance bet is forfeited. A tie at 32-40 leads to an Insurance tie. To summarize, Insurance pays out 49 to 1 for a 31 tie, pushes on any other ties, and otherwise results in a loss.
11. If a player decides to imprison their wager after a 31-point tie, their bet is temporarily set aside. The next round is played, and if it results in a win for the imprisoned wager, the bet is returned to the player. If the outcome is a loss, the bet is lost. Any ties—including a tie at 31—keep the wager in detention until a win or loss resolves its status.

Odds

The following table illustrates the probability distribution for each potential score for both hands.

The subsequent table outlines the chances of various outcomes when comparing the Black and Red hands.

Another table presents the return rates for the Black, Red, Color, and Inverse bets, highlighting a house edge of 1.10% in the bottom right corner.

In the next table, you can find the expected return for Insurance bets, particularly if the player chooses to lose half their wager on a 31 tie. This table indicates a remarkable player advantage of 18.41%.

The Insurance table above demonstrates that this bet is particularly lucrative—likely the best I've encountered, aside from some niche Super Bowl proposition bets. Unfortunately, as outlined in the rules, the maximum allowable Insurance bet is capped at 1% of the primary wager. A subsequent table reveals the cumulative effect of betting 100 units on Black, Red, Color, or Inverse alongside 1 unit for Insurance, indicating a projected loss of 0.912295 units. Therefore, the combined expected value across both bets equates to -0.912295/101 = -0.00903262.

Strategy

• Your choice of betting on Black, Red, Color, or Inverse has no bearing on your strategy.
• Always take insurance.
• The likelihood of losing half your bet versus opting to imprison it after a 31 tie is identical.

Put simply, feel free to make your own choice, but remember to take out Insurance.

Methodology

This analysis was conducted with the help of a random simulation encompassing over 26 billion hands, utilizing a cut card that was placed after every 271 cards.

Links

• montecarlocasinos.com has a page on the game rules.
• The Doctrine of Chances: Examining the Probabilistic Nature of Gambling by Stewart N. Ethier. In this work, the author dedicates a chapter to Trente et Quarante. My personal house edge for this game is 0.000018 greater than what Ethier calculated, likely due to the influence of the cut card, which may not have been factored into Ethier's combinatorial analysis.

Written by: Michael Shackleford