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Introduction

Trente et Quarante is French for 'thirty and forty.' It is an old European casino game still found in Monte Carlo and other major French casinos. When I went to Monte Carlo, I didn't knowingly see the game, but I wasn't specifically looking for it either.

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The game can be most closely compared to baccarat. There are two hands, and players bet on which one will have the better score. In this case, the lower the score the better.

Each player in turn, beginning on the dealer's left, places any bet up to the limit, and names 'red' or 'black.' The dealer gives the player five cards face up. If three or more are of the color named, the dealer pays the bet; if three or more are of the opposite color, the dealer collects the bet. Roulette is a casino game named after the French word meaning little wheel.In the game, players may choose to place bets on either a single number, various groupings of numbers, the colors red or black, whether the number is odd or even, or if the numbers are high (19–36) or low (1–18).

Rules

1. Six 52-card decks are used.
2. Aces are valued as one point, 2s to 10s according to pip value, and face cards as 10 points.
3. There are four bets available: Black, Red, Color, and Inverse. In addition, an Insurance wager is available on each of these four bets.
4. After players have made their wagers, the dealer will deal cards to determine the value of the 'black' hand. A running total will start at 0 and go up as the cards are dealt from the shoe. The dealer will stop when the total points is 31 or more. The maximum possible score is 40.
5. Next, the dealer will do the same thing but for the 'red' hand.
6. The Black wager shall win if the black hand has fewer points than the red hand. If the red hand has fewer points, then the result is a loss. If there is a tie at 32 to 40 points, then the result is a push. If there is a tie at 31 points, then the player may choose to lose half, or 'imprison' his bet, which I explain below.
7. The Red wager is the opposite of the Black wager, winning if the red hand is less and losing if it is more. The tie rules are the same.
8. The Color wager acts like a Black wager if the first card dealt to the black hand is black, and like a Red wager if that first card is red.
9. The Inverse wager is the opposite of the Color wager, acting like a Black wager if the first card is red, and a Red wager if the first card is black.
10. The player may take out Insurance on the Black, Red, Color, or Inverse wagers. The Insurance wager must equal 1% of the primary wager and pays the loss if the outcome is a 31 tie. If the primary wager wins or loses the full amount, then the Insurance wager loses. A 32-40 tie results in an Insurance tie. In other words, Insurance pays 49 to 1 on a 31 tie, pushes on any other tie, and otherwise loses.
11. If the player imprisons a wager after a 31 tie, then it is blocked off somehow. Then the next hand is played out. If that next hand results in a win for the blocked off wager, then it is returned to the player. If the next hand results in a loss for the blocked off wager, then it is lost. Any tie, including on 31, results in the blocked off wager remaining on the table until there is a win or loss to resolve it.

Odds

The following table shows the probability of each number of points for each hand.

Points Distribution

Points	Probability
31	0.148177
32	0.137900
33	0.127618
34	0.116876
35	0.106136
36	0.094963
37	0.083814
38	0.072255
39	0.060751
40	0.051510
Total	1.000000

The next table shows the probability of each possible outcome between the Black and Red hands.

Points Distribution

Event	Probability
Black wins	0.445184
Red wins	0.445184
Tie (32-40)	0.087705
Tie (31)	0.021927
Total	1.000000

The next table shows the return table for the Black, Red, Color, and Inverse bets. The lower right cell shows a house edge of 1.10%.

Return Table for Black, Red, Color, and Inverse

Event	Pays	Probability	Return
Win	1	0.445184	0.445184
Push	0	0.087705	0.000000
Lose half	-0.5	0.021927	-0.010964
Lose all	-1	0.445184	-0.445184
Total	1.000000	-0.010964

The next table shows the return table for Insurance, assuming the player chooses to lose half on a 31 tie. The lower right cell shows a player advantage of 18.41%!.

Return Table for Insurance

Event	Pays	Probability	Return
Win	49	0.021927	1.074436
Lose	-1	0.890367	-0.890367
Push	0	0.087705	0.000000
Total	1.000000	0.184069

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The above Insurance table shows that Insurance is indeed a great bet. Aside from some Super Bowl proposition wagers, the best bet I've ever known. Unfortunately, as stated in the rules, Insurance is limited to 1% of the primary wager. The next table shows the combined effect of betting 100 units on Black, Red, Color, or Inverse and 1 unit on Insurance. The lower right cell shows an expected loss of 0.912295 units. The combined expected value between the two bets is thus -0.912295/101 = -0.00903262.

Combined Return Table on 101 Units Wagered

Primary Wager Event	Primary Wager Pays	Insurance Pays	Total Win	Probability	Return
Win	100	-1	99	0.445184	44.073184
Push	0	0	0	0.087705	0.000000
Lose half	-50	49	-1	0.021927	-0.021927
Lose all	-100	-1	-101	0.445184	-44.963551
Total	1.000000	-0.912295

Strategy

• It doesn't matter what you bet on between Black, Red, Color, and Inverse.
• Always take insurance.
• The odds between losing half and imprisoning a bet after a 31 tie are the same.

In other words, do whatever you want, as long as you take Insurance.

Methodology

This analysis was based on a random simulation of over 26 billion hands. A cut card was used in the simulation, placed after 271 cards.

Links

• montecarlocasinos.com has a page on the game rules.
• The Doctrine of Chances: Probabilistic Aspects of Gambling by Stewart N. Ethier. This book has a chapter on Trente et Quarante. My house edge for the game is 0.000018 higher than Ethier's. I believe this is due to cut card effect, which would not have played a role in Ethier's method of an exact combinatorial analysis.

Written by:Michael Shackleford

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