Rouge et noir
Rouge et noir , more rarely Banque rouge et noir , also R & N , German. Red and Black is a game of chance with playing cards ; it is said to have been introduced to the French court by Cardinal Mazarin . By adding two more opportunities, the Trente et quarante emerged .
The game
Rouge et noir is played with six full packs of French playing cards , i.e. 312 sheets, at a table, on the ceiling ( tapis ) of which a tableau with a red and a black insert field is printed. The number of punchers who play against the banker is unlimited.
Early in the game, the banker offers the cards right the players to mix, then he mixes himself again and can stand out . When the punchers have placed their bets, the banker takes as many cards from the pile as he can comfortably hold in his left hand and draws with the shout: “ Le jeu est fait! “Take a card and place it face up in the middle of the table. Now he draws further cards one by one and puts them on the table next to the first one in a row, silently adding the values of the individual cards until the sum reaches a value greater than or equal to 31 and less than or equal to 40, and states the total. Here the figures are ten, the aces one and the other cards according to their eyes.
The row laid out in this way applies to rouge . Then the banker places a second row for noir in the same way .
The chance whose row shows the lower total wins; the banker only ever names the chance rouge and never noir , i. e. he says either « Le rouge perd! "Or" Le rouge gagne! ».
| rouge |
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33 points | |
|---|---|---|---|---|---|---|---|
| Noir |
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37 points | ||
| Le rouge gagne! | |||||||
The banker now collects all losing sets and pays out the sets in the winning field at a ratio of 1: 1.
If both rows count the same number of eyes, the banker says “ Après ”: The coup is a tie and the rows must be laid again, unless the value of both rows is 31 ( Refait de trente-un ) in this one In this case, the banker collects half of all the sentences in both fields.
The used cards are set aside, and new cards are drawn from the banker's pile for the next coup.
analysis
To determine the bank's chances at Rouge et noir , one can rely on the following consideration:
- The total 40 can only appear if one of the four values K, Q, J or 10 is the last card in the corresponding row.
- The total number 39 can only appear if one of the five values K, Q, J, 10 or 9 is the last card in the corresponding row.
- The total 38 can only appear if one of the six values K, Q, J, 10, 9 or 8 is the last card in the corresponding row; etc.
- Finally, the total number 31 can be obtained if one of the thirteen possible values falls as the last card in the corresponding row.
If one now assumes, in a simplified manner, that the probability of occurrence of the numbers 31, 32, ..., 39, 40 is in the ratio 13: 12: ...: 5: 4, one obtains that the refait on average is approximately once in 39 valid (!) coups occurs ( après- coups are not counted).
The following applies: The probability for a party to get just 31 points is - according to this approximation - just 13 / (13 + 12 + 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4) = 13/85.
Since the two scores for rouge and noir can (at least approximately) be viewed as independent , the probability that both series have 31 points can be calculated by multiplication: (13/85) · (13/85) = (13 · 13 ) / (85 * 85) = 169/7225.
If you lay 7225 times two rows, there are on average 169 cases in which both rows have just 31 points. There are also 144 + 121 + 100 + 81 + 64 + 49 + 36 + 25 + 16 = 636 times an après ; the remaining 6,420 cases are divided equally between rouge and noir .
However, since the après- coups (as invalid games) are not (may (!)) Taken into account when calculating the bank advantage , and the bank only collects half of the stakes in the event of a refait, the bank advantage is obtained: (1/2) (169 / (7225 - 636)) = (1/2) (169/6589) = 1.28% = approx. (1/2) (1/39)
A slightly more precise result is obtained if, instead of the above classic calculation method, the probability of occurrence of the numbers 31, 32, ..., 40 is determined under the assumption of an infinitely large deck of playing cards - the bank advantage is then the slightly smaller value of 1.20%.
The bank advantage is therefore slightly smaller than with the bet on red or black in roulette , this is 1.35%.
literature
- PN Huyn: La Théorie des jeux de Hasard . 1788 (Classical analysis, contains the approximate calculation given in the article).
- Siméon Denis Poisson : Mémoire sur l'avantage du banquier au jeu de trente et quarante . In: Annales de Mathématiques Pures et Appliquées , 16, 1825, pp. 173-208.
Individual evidence
- ↑ Rouge-et-noir . In: Meyers Großes Konversations-Lexikon . 6th edition. Volume 17, Bibliographisches Institut, Leipzig / Vienna 1909, p. 195 .