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Roulette (named after the French language word meaning "little wheel") is a casino game which was likely developed from the Italy game Biribi. In the game, a player may choose to place a bet on a single number, various groupings of numbers, the color red or black, whether the number is odd or even, or if the number is high or low.
To determine the winning number, a croupier spins a wheel in one direction, then spins a ball in the opposite direction around a tilted circular track running around the outer edge of the wheel. The ball eventually loses momentum, passes through an area of deflectors, and falls onto the wheel and into one of the colored and numbered pockets on the wheel. The winnings are then paid to anyone who has placed a successful bet.
The game has been played in its present form since as early as 1796 in Paris. An early description of the roulette game in its current form is found in a French novel La Roulette, ou le Jour by Jaques Lablee, which describes a roulette wheel in the Palais Royal in Paris in 1796. The description included the house pockets: "There are exactly two slots reserved for the bank, whence it derives its sole mathematical advantage." It then goes on to describe the layout with "two betting spaces containing the bank's two numbers, zero and double zero". The book was published in 1801. An even earlier reference to a game of this name was published in regulations for New France (Québec) in 1758, which banned the games of "dice, hoca, faro, and roulette".Roulette Wheel Study, Ron Shelley, (1988)
The roulette wheels used in the casinos of Paris in the late 1790s had red for the single zero and black for the double zero. To avoid confusion, the color green was selected for the zeros in roulette wheels starting in the 1800s.
In 1843, in the German spa casino town of Bad Homburg, fellow Frenchmen François and Louis Blanc introduced the single 0 style roulette wheel in order to compete against other casinos offering the traditional wheel with single and double zero house pockets.
In some forms of early American roulette wheels, there were numbers 1 to 28, plus a single zero, a double zero, and an American Eagle. The Eagle slot, which was a symbol of American liberty, was a house slot that brought the casino an extra edge. Soon, the tradition vanished and since then the wheel features only numbered slots. According to Hoyle "the single 0, the double 0, and the eagle are never bars; but when the ball falls into either of them, the banker sweeps every thing upon the table, except what may happen to be bet on either one of them, when he pays twenty-seven for one, which is the amount paid for all sums bet upon any single figure".9781167231667 ISBN 9781167231667
In the 19th century, roulette spread all over Europe and the US, becoming one of the most popular casino games. When the German government abolished gambling in the 1860s, the Blanc family moved to the last legal remaining casino operation in Europe at Monte Carlo, where they established a gambling mecca for the elite of Europe. It was here that the single zero roulette wheel became the premier game, and over the years was exported around the world, except in the United States where the double zero wheel remained dominant.
In the United States, the French double zero wheel made its way up the Mississippi from New Orleans, and then westward. It was here, because of rampant cheating by both operators and gamblers, that the wheel was eventually placed on top of the table to prevent devices from being hidden in the table or wheel, and the betting layout was simplified. This eventually evolved into the American-style roulette game. The American game was developed in the gambling dens across the new territories where makeshift games had been set up, whereas the French game evolved with style and leisure in Monte Carlo.
During the first part of the 20th century, the only casino towns of note were Monte Carlo with the traditional single zero French wheel, and Las Vegas with the American double zero wheel. In the 1970s, casinos began to flourish around the world. In 1996 the first online casino, generally believed to be InterCasino, made it possible to play roulette online. (2025). 9781414448619, Gale. ISBN 9781414448619 By 2008, there were several hundred casinos worldwide offering roulette games. The double zero wheel is found in the United States, Canada, South America, and the Caribbean, while the single zero wheel is predominant elsewhere.
The roulette table usually imposes minimum and maximum bets, and these rules usually apply separately for all of a player's inside and outside bets for each spin. For inside bets at roulette tables, some casinos may use separate roulette table chips of various colors to distinguish players at the table. Players can continue to place bets as the ball spins around the wheel until the dealer announces "no more bets" or "rien ne va plus".
When a winning number and color is determined by the roulette wheel, the dealer will place a marker, also known as a dolly, on that number on the roulette table layout. When the dolly is on the table, no players may place bets, collect bets or remove any bets from the table. The dealer will then sweep away all losing bets either by hand or by rake, and determine the payouts for the remaining inside and outside winning bets. When the dealer is finished making payouts, the dolly is removed from the board and players may collect their winnings and make new bets. Winning chips remain on the board until picked up by a player.
In number ranges from 1 to 10 and 19 to 28, odd numbers are red and even are black. In ranges from 11 to 18 and 29 to 36, odd numbers are black and even are red.
There is a green pocket numbered 0 (zero). In American roulette, there is a second green pocket marked 00. Pocket number order on the roulette wheel adheres to the following clockwise sequence in most casinos:
The European-style layout has a single zero, and the American style layout is usually a double-zero. The American-style roulette table with a wheel at one end is now used in most casinos because it has a higher house edge compared to a European layout.
The French style table with a wheel in the centre and a layout on either side is rarely found outside of Monte Carlo.
| Straight/single | Bet on a single number | Entirely within the square for the chosen number |
| Split | Bet on two vertically/horizontally adjacent numbers (e.g. 14-17 or 8–9) | On the edge shared by the numbers |
| Street | Bet on three consecutive numbers in a horizontal line (e.g. 7-8-9) | On the outer edge of the number at either end of the line |
| Corner/square | Bet on four numbers that meet at one corner (e.g. 10-11-13-14) | On the common corner |
| Six line/double street | Bet on six consecutive numbers that form two horizontal lines (e.g. 31-32-33-34-35-36) | On the outer corner shared by the two leftmost or the two rightmost numbers |
| Trio/basket | A three-number bet that involves at least one zero: 0-1-2 (either layout); 0-2-3 (single-zero only); 0-00-2 and 00-2-3 (double-zero only) | On the corner shared by the three chosen numbers |
| First four | Bet on 0-1-2-3 (single-zero layout only) | On the outer corner shared by 0-1 or 0-3 |
| Top line | Bet on 0-00-1-2-3 (double-zero layout only) | On the outer corner shared by 0-1 or 00-3 |
In some casinos, the farthest outside bets (low/high, red/black, even/odd) result in the player losing only half of their bet if a zero comes up. This is known as the "halfback" or la partage rule. Others may use an "imprisonment" or en prison rule, wherein a loss on such an outside bet causes the player's money to be "imprisoned", and if the next spin would be a win for that bet, the player's wager is returned.
where n is the number of pockets in the wheel.
The initial bet is returned in addition to the mentioned payout: it can be easily demonstrated that this payout formula would lead to a zero expected value of profit if there were only 36 numbers (that is, the casino would break even). Having 37 or more numbers gives the casino its edge.
| 0 | 0 | 35 to 1 | 36 to 1 | −$0.027 | 37 to 1 | −$0.053 |
| 00 | 00 | 35 to 1 | 37 to 1 | −$0.053 | ||
| Straight up | Any single number | 35 to 1 | 36 to 1 | −$0.027 | 37 to 1 | −$0.053 |
| Row | 0, 00 | 17 to 1 | 18 to 1 | −$0.053 | ||
| Split | Any two adjoining numbers vertical or horizontal | 17 to 1 | to 1 | −$0.027 | 18 to 1 | −$0.053 |
| Street | Any three numbers horizontal (1, 2, 3 or 4, 5, 6, etc.) | 11 to 1 | to 1 | −$0.027 | to 1 | −$0.053 |
| Corner | Any four adjoining numbers in a block (1, 2, 4, 5 or 17, 18, 20, 21, etc.) | 8 to 1 | to 1 | −$0.027 | to 1 | −$0.053 |
| Top line (US) | 0, 00, 1, 2, 3 | 6 to 1 | to 1 | −$0.079 | ||
| Top line (European) | 0, 1, 2, 3 | 8 to 1 | to 1 | −$0.027 | ||
| Double street | Any six numbers from two horizontal rows (1, 2, 3, 4, 5, 6 or 28, 29, 30, 31, 32, 33 etc.) | 5 to 1 | to 1 | −$0.027 | to 1 | −$0.053 |
| 1st column | 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34 | 2 to 1 | to 1 | −$0.027 | to 1 | −$0.053 |
| 2nd column | 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35 | 2 to 1 | to 1 | −$0.027 | to 1 | −$0.053 |
| 3rd column | 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36 | 2 to 1 | to 1 | −$0.027 | to 1 | −$0.053 |
| 1st dozen | 1 through 12 | 2 to 1 | to 1 | −$0.027 | to 1 | −$0.053 |
| 2nd dozen | 13 through 24 | 2 to 1 | to 1 | −$0.027 | to 1 | −$0.053 |
| 3rd dozen | 25 through 36 | 2 to 1 | to 1 | −$0.027 | to 1 | −$0.053 |
| Odd | 1, 3, 5, ..., 35 | 1 to 1 | to 1 | −$0.027 | to 1 | −$0.053 |
| Even | 2, 4, 6, ..., 36 | 1 to 1 | to 1 | −$0.027 | to 1 | −$0.053 |
| Red | 32, 19, 21, 25, 34, 27, 36, 30, 23, 5, 16, 1, 14, 9, 18, 7, 12, 3 | 1 to 1 | to 1 | −$0.027 | to 1 | −$0.053 |
| Black | 15, 4, 2, 17, 6, 13, 11, 8, 10, 24, 33, 20, 31, 22, 29, 28, 35, 26 | 1 to 1 | to 1 | −$0.027 | to 1 | −$0.053 |
| 1 to 18 | 1, 2, 3, ..., 18 | 1 to 1 | to 1 | −$0.027 | to 1 | −$0.053 |
| 19 to 36 | 19, 20, 21, ..., 36 | 1 to 1 | to 1 | −$0.027 | to 1 | −$0.053 |
Top line (0, 00, 1, 2, 3) has a different expected value because of approximation of the correct -to-1 payout obtained by the formula to 6-to-1.
If en prison rules are used, the house edge on the 1:1 bets is reduced to 1.35%-1.38% on a single-zero wheel, and to 2.63% on a double-zero wheel.
For European roulette, a single number wins and loses :
For triple-zero wheels, a single number wins and loses :
Call the bet a triple , where is the set of chosen numbers, is the size of the bet, and determines the return of the bet. (2025). 9789738752078, Infarom. ISBN 9789738752078
The rules of European roulette have 10 types of bets. First the 'Straight Up' bet can be imagined. In this case, , for some , and is determined by
-r, &\omega \ne \omega_0\\
35 \cdot r, &\omega = \omega_0
\end{cases}
The bet's expected net return, or profitability, is equal to
\frac{1}{37} \sum_{\omega \in \Omega} \xi(\omega) =
\frac{1}{37} \left(\xi(\omega_0) +
\sum_{\omega \ne \omega_0} \xi(\omega)\right) =
\frac{1}{37} \left(35 \cdot r - 36 \cdot r \right) =
-\frac{r}{37} \approx -0.027r.
Without details, for a bet, black (or red), the rule is determined as
-r, &\omega \text{ is red} \\
-r, &\omega = 0 \\
r, &\omega \text{ is black}
\end{cases},
and the profitability
For similar reasons it is simple to see that the profitability is also equal for all remaining types of bets. ., en.wikibooks.org
In reality this means that, the more bets a player makes, the more they are going to lose independent of the strategies (combinations of bet types or size of bets) that they employ:
The odds for the player in American roulette are even worse, as the bet profitability is at worst , and never better than .
For example, if a player bets on red, there are 18 red numbers, , so the chance of winning is .
The payout given by the casino for a win is based on the roulette wheel having 36 outcomes, and the payout for a bet is given by .
For example, betting on 1-12 there are 12 numbers that define a win, , the payout is , so the bettor wins 3 times their bet.
The average return on a player's bet is given by
For , the average return is always lower than 1, so on average a player will lose money.
With 1 green number, , the average return is , that is, after a bet the player will on average have of their original bet returned to them. With 2 green numbers, , the average return is . With 3 green numbers, , the average return is .
This shows that the expected return is independent of the choice of bet.
There are different number series in roulette that have special names attached to them. Most commonly these bets are known as "the French bets" and each covers a section of the wheel. For the sake of accuracy, zero spiel, although explained below, is not a French bet, it is more accurately "the German bet". Players at a table may bet a set amount per series (or multiples of that amount). The series are based on the way certain numbers lie next to each other on the roulette wheel. Not all casinos offer these bets, and some may offer additional bets or variations on these.
Nine chips or multiples thereof are bet. Two chips are placed on the 0-2-3 trio; one on the 4–7 split; one on 12–15; one on 18–21; one on 19–22; two on the 25-26-28-29 corner; and one on 32–35.
The bet consists of four chips or multiples thereof. Three chips are bet on splits and one chip straight-up: one chip on 0–3 split, one on 12–15 split, one on 32–35 split and one straight-up on number 26.
This type of bet is popular in Germany and many European casinos. It is also offered as a 5-chip bet in many Eastern European casinos. As a 5-chip bet, it is known as "zero spiel naca" and includes, in addition to the chips placed as noted above, a straight-up on number 19.
Very popular in British casinos, tiers bets outnumber voisins and orphelins bets by a massive margin.
Six chips or multiples thereof are bet. One chip is placed on each of the following splits: 5–8, 10–11, 13–16, 23–24, 27–30, and 33–36.
The tiers bet is also called the "small series" and in some casinos (most notably in South Africa) "series 5-8".
A variant known as "tiers 5-8-10-11" has an additional chip placed straight up on 5, 8, 10, and 11 and so is a 10-piece bet. In some places the variant is called "gioco Ferrari" with a straight up on 8, 11, 23 and 30, the bet is marked with a red G on the racetrack.
Five chips or multiples thereof are bet on four splits and a straight-up: one chip is placed straight-up on 1 and one chip on each of the splits: 6–9, 14–17, 17–20, and 31–34.
Any of the above bets may be combined; e.g. "orphelins by 1 and zero and the neighbors by 1". The "...and the neighbors" is often assumed by the croupier.
Final 4, for example, is a 4-chip bet and consists of one chip placed on each of the numbers ending in 4, that is 4, 14, 24, and 34. Final 7 is a 3-chip bet, one chip each on 7, 17, and 27. Final bets from final 0 (zero) to final 6 cost four chips. Final bets 7, 8 and 9 cost three chips.
Some casinos also offer split-final bets, for example final 5-8 would be a 4-chip bet, one chip each on the splits 5–8, 15–18, 25–28, and one on 35.
The maximum amount allowed to be wagered on a single bet in European roulette is based on a progressive betting model. If the casino allows a maximum bet of $1,000 on a 35-to-1 straight-up, then on each 17-to-1 split connected to that straight-up, $2,000 may be wagered. Each 8-to-1 corner that covers four numbers) may have $4,000 wagered on it. Each 11-to-1 street that covers three numbers may have $3,000 wagered on it. Each 5-to-1 six-line may have $6,000 wagered on it. Each $1,000 incremental bet would be represented by a marker that is used to specifically identify the player and the amount bet.
For instance, if a patron wished to place a full complete bet on 17, the player would call "17 to the maximum". This bet would require a total of 40 chips, or $40,000. To manually place the same wager, the player would need to bet:
| + 17 to the maximum ! Bet type ! Number(s) bet on ! Chips ! Amount waged |
| $1,000 |
| $2,000 |
| $2,000 |
| $2,000 |
| $2,000 |
| $3,000 |
| $4,000 |
| $4,000 |
| $4,000 |
| $4,000 |
| $6,000 |
| $6,000 |
The player calls their bet to the croupier (most often after the ball has been spun) and places enough chips to cover the bet on the table within reach of the croupier. The croupier will immediately announce the bet (repeat what the player has just said), ensure that the correct monetary amount has been given while simultaneously placing a matching marker on the number on the table and the amount wagered.
The payout for this bet if the chosen number wins is 392 chips, in the case of a $1000 straight-up maximum, $40,000 bet, a payout of $392,000. The player's wagered 40 chips, as with all winning bets in roulette, are still their property and in the absence of a request to the contrary are left up to possibly win again on the next spin.
Based on the location of the numbers on the layout, the number of chips required to "complete" a number can be determined.
Most typically (Mayfair casinos in London and other top-class European casinos) with these maximum or full complete bets, nothing (except the aforementioned maximum button) is ever placed on the layout even in the case of a win. Experienced gaming staff, and the type of customers playing such bets, are fully aware of the payouts and so the croupier simply makes up the correct payout, announces its value to the table inspector (floor person in the U.S.) and the customer, and then passes it to the customer, but only after a verbal authorization from the inspector has been received.
Also typically at this level of play (house rules allowing) the experienced croupier caters to the needs of the customer and will most often add the customer's winning bet to the payout, as the type of player playing these bets very rarely bets the same number two spins in succession. For example, the winning 40-chip / $40,000 bet on "17 to the maximum" pays 392 chips / $392,000. The experienced croupier would pay the player 432 chips / $432,000, that is 392 + 40, with the announcement that the payout "is with your bet down".
There are also several methods to determine the payout when a number adjacent to a chosen number is the winner, for example, player bets "23 full complete" and number 26 is the winning number. The most notable method is known as the "station" method. When paying in stations, the dealer counts the number of ways or stations that the winning number hits the complete bet. In the example above, 26 hits 4 stations - 2 different corners, 1 split and 1 six-line. The dealer takes the number 4, multiplies it by 36, making 144 with the players bet down.
In some casinos, a player may bet full complete for less than the table straight-up maximum, for example, "number 17 full complete by $25" would cost $1000, that is 40 chips each at $25 value.
All betting systems that rely on patterns, when employed on casino edge games will result, on average, in the player losing money. In practice, players employing betting systems may win, and may indeed win very large sums of money, but the losses (which, depending on the design of the betting system, may occur quite rarely) will outweigh the wins. Certain systems, such as the Martingale, described below, are extremely risky, because the worst-case scenario (which is mathematically certain to happen, at some point) may see the player chasing losses with ever-bigger bets until they run out of money.
The American mathematician Patrick Billingsley said that no betting system can convert a subfair game into a profitable enterprise. At least in the 1930s, some professional gamblers were able to consistently gain an edge in roulette by seeking out rigged wheels (not difficult to find at that time) and betting opposite the largest bets.
Edward O. Thorp (the developer of card counting and an early hedge-fund pioneer) and Claude Shannon (a mathematician and electronic engineer best known for his contributions to information theory) built the first wearable computer to predict the landing of the ball in 1961. This system worked by timing the ball and wheel, and using the information obtained to calculate the most likely octant where the ball would fall. Ironically, this technique works best with an unbiased wheel though it could still be countered quite easily by simply closing the table for betting before beginning the spin.
In 1982, several casinos in Britain began to lose large sums of money at their roulette tables to teams of gamblers from the US. Upon investigation by the police, it was discovered they were using a legal system of biased wheel-section betting. As a result of this, the British roulette wheel manufacturer John Huxley manufactured a roulette wheel to counteract the problem.
The new wheel, designed by George Melas, was called "low profile" because the pockets had been drastically reduced in depth, and various other design modifications caused the ball to descend in a gradual approach to the pocket area. In 1986, when a professional gambling team headed by Billy Walters won $3.8 million using the system on an old wheel at the Golden Nugget in Atlantic City, every casino in the world took notice, and within one year had switched to the new low-profile wheel.
Thomas Bass, in his book The Eudaemonic Pie (1985) (published as The Newtonian Casino in Britain), has claimed to be able to predict wheel performance in real time. The book describes the exploits of a group of University of California Santa Cruz students, who called themselves the Eudaemons, who in the late 1970s used computers in their shoes to win at roulette. This is an updated and improved version of Edward O. Thorp's approach, where Newtonian laws of motion are applied to track the roulette ball's deceleration; hence the British title.
To defend against exploits like these, many casinos use tracking software, use wheels with new designs, rotate wheel heads, and randomly rotate pocket rings.
At the Ritz London casino in March 2004, two Serbs and a Hungarian used a laser scanning hidden inside a mobile phone linked to a computer to predict the sector of the wheel where the ball was most likely to drop. They netted £1.3m in two nights. The sting: did gang really use a laser, phone and a computer to take the Ritz for £1.3m? | Science | The Guardian, guardian.co.uk They were arrested and kept on police bail for nine months, but eventually released and allowed to keep their winnings as they had not interfered with the casino equipment. (2025). 9780230113848, Palgrave Macmillan. ISBN 9780230113848
Negative progression systems involve increasing the size of one's bet when they lose. This is the most common type of betting system. The goal of this system is to recoup losses faster so that one can return to a winning position more quickly after a losing streak. The typical shape of these systems is small but consistent wins followed by occasional catastrophic losses. Examples of negative progression systems include the Martingale system, the Fibonacci system, the Labouchère system, and the d'Alembert system.
Positive progression systems involve increasing the size of one's bet when one wins. The goal of these systems is to either exacerbate the effects of winning streaks (e.g. the Paroli system) or to take advantage of changes in luck to recover more quickly from previous losses (e.g. Oscar's grind). The shape of these systems is typically small but consistent losses followed by occasional big wins. However, over the long run these wins do not compensate for the losses incurred in between.
This system is one that is designed so that when the player has won over a third of their bets (less than the expected 18/38), they will win. Whereas the martingale will cause ruin in the event of a long sequence of successive losses, the Labouchère system will cause bet size to grow quickly even where a losing sequence is broken by wins. This occurs because as the player loses, the average bet size in the line increases.
As with all other betting systems, the average value of this system is negative.
This betting system relies on the gambler's fallacy—that the player is more likely to lose following a win, and more likely to win following a loss.
Many betting systems are sold online and purport to enable the player to 'beat' the odds. One such system was advertised by Jason Gillon of Rotherham, UK, who claimed one could 'earn £200 daily' by following his betting system, described as a 'loophole'. As the system was advertised in the UK press, it was subject to Advertising Standards Authority regulation, and following a complaint, it was ruled by the ASA that Mr. Gillon had failed to support his claims, and that he had failed to show that there was any loophole.
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