is acasinogame named after theFrenchword meaning. 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 (1936) or low (118).

To determine the winning number and color, acroupierspins a wheel in one direction, then spins a ball in the opposite direction around a tilted circular track running around thecircumferenceof the wheel. The ball eventually losesmomentum, passes through an area of deflectors, and falls onto the wheel and into one of 37 (in French/European style roulette) or 38 (in American style roulette) colored and numbered pockets on the wheel.

The first form of roulette was devised in 18th centuryFrance. Many historians believeBlaise Pascalintroduced a primitive form of roulette in the 17th century in his search for aperpetual motion machine.[1]The roulette mechanism is a hybrid of a gaming wheel invented in 1720 and the Italian gameBiribi.[2]

The game has been played in its present form since as early as 1796 inParis. An early description of the roulette game in its current form is found in a French novelLa Roulette, ou le Jourby Jaques Lablee, which describes a roulette wheel in thePalais Royalin 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, o betting spaces containing the banks 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 forNew FranceQubec) in 1758, which banned the games of dice, hoca, faro, and roulette.[3]

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 ofBad Homburg, fellow FrenchmenFrançois and Louis Blancintroduced the single0style roulette wheel in order to compete against other casinos offering the traditional wheel with single and double zero house pockets.[citation needed]

In some forms of early American roulette wheels, there were numbers 1 through 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 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 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.[4]

In the 19th century, roulette spread all over Europe and the US, becoming one of the most famous and 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 atMonte 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 had remained dominant.

In the United States, the French double zero wheel made its way up theMississippifromNew 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 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, andLas Vegaswith the American double zero wheel. In the 1970s, casinos began to flourish around the world. By 2008 there were several hundred casinos worldwide offering roulette games. The double zero wheel is found in the U.S., Canada, South America, and the Caribbean, while the single zero wheel is predominant elsewhere.

The sum of all the numbers on the roulette wheel (from 0 to 36) is666, which is theNumber of the Beast.[5]One legend says thatFrançois Blancmade a deal with the devil to obtain the secrets of roulette.[citation needed]

Roulette players have a variety of betting options. Placing inside bets is either selecting the exact number of the pocket the ball will land in, or a small range of pockets based on their proximity on the layout. Players wishing to bet on the outside will select bets on larger positional groupings of pockets, the pocket color, or whether the winning number is odd or even.[6]The payout odds for each type of bet are based on itsprobability.

The roulette table usually imposes minimum and maximum bets, and these rules usually apply separately for all of a players 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 announcesno more betsorrien 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 winning 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 other losing bets either by hand or rake, and determine all of the payouts to the remaining inside and outside winning bets. When the dealer is finished making payouts, the marker is removed from the board where players collect their winnings and make new bets. The winning chips remain on the board.

In 2004, California legalized a form of roulette known as California Roulette.[7]By law, the game must use cards and not slots on the roulette wheel to pick the winning number.

The pockets of the roulette wheel are numbered from 0 to 36.

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:[citation needed]

0-32-15-19-4-21-2-25-17-34-6-27-13-36-11-30-8-23-10-5-24-16-33-1-20-14-31-9-22-18-29-7-28-12-35-3-26

0-28-9-26-30-11-7-20-32-17-5-22-34-15-3-24-36-13-1-00-27-10-25-29-12-8-19-31-18-6-21-33-16-4-23-35-14-2

French style layout, French single zero wheel

The cloth covered betting area on a roulette table is known as thelayout. The layout is either single zero or double zero. 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. The French style table with a wheel in the centre and a layout on either side is rarely found outside of Monte Carlo.[citation needed]

In roulette, bets can either be inside or outside bets.

Entirely within the square for the chosen number

Bet on two vertically/horizontally adjacent numbers (e.g. 14-17 or 8-9)

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

Bet on four numbers that meet at one corner (e.g. 10-11-13-14)

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

A three-number bet that involves at least one zero: 0-1-2 (either layout); 0-2-3 (single-zero only); 0-00-2 or 00-2-3 (double-zero only)

On the corner shared by the three chosen numbers

Outside bets typically have smaller payouts with better odds at winning. Except as noted, all of these bets lose if a zero comes up.

A bet that the number will be in the chosen range.

A bet that the number will be the chosen color.

A bet that the number will be of the chosen type.

A bet that the number will be in the chosen dozen: first (1-12,

A bet that the number will be in the chosen vertical column of 12 numbers, such as 1-4-7-10 on down to 34. The chip is placed on the space below the final number in this sequence.

A special bet that covers the numbers 1, 5, 9, 12, 14, 16, 19, 23, 27, 30, 32, and 34. It has the same payout as the dozen bet and takes its name from the zigzagging, snakelike pattern traced out by these numbers. The snake bet is not available in all casinos; when it is allowed, the chip is placed on the lower corner of the 34 square that borders the 19-36 betting box. Some layouts mark the bet with a two-headed snake that winds from 1 to 34, and the bet can be placed on the head at either end of the body.

In the United Kingdom, the farthest outside bets (low/high, red/black, even/odd) result in the player losing only half of his/her bet if a zero comes up.

The payout (except for the special case of Top line bets), for American and European roulette, can be calculated by:

wherenis the number of squares the player is betting on. The initial bet is returned in addition to the mentioned payout. It can be easily demonstrated that this payout formula would lead to a zeroexpected valueof profit if there were only 36 numbers. Having 37 or 38 numbers gives the casino its edge.

(on a $1 bet) (French)Oddsagainst winning (American)Expected value

any two adjoining numbers vertical or horizontal

any three numbers horizontal (1, 2, 3 or 4, 5, 6, etc.)

any four adjoining numbers in a block (1, 2, 4, 5 or 17, 18, 20, 21, etc.)

any six numbers from two horizontal rows (1, 2, 3, 4, 5, 6 or 28, 29, 30, 31, 32, 33 etc.)

Note that Top line (0, 00, 1, 2, 3) has a different expected value because ofapproximationof the correct61⁄5-to-1 payout obtained by the formula to 6-to-1. Note also that 0 and 00 are not odd or even, or high or low.

En prisonrules, when used, reduce the house advantage.

Thehouse averageorhouse edge or house advantage(also called theexpected value) is the amount the player loses relative for any bet made, on average. If a player bets on a single number in the American game there is a probability of1⁄38that the player wins 35 times the bet, and a37⁄38chance that the player loses his bet. The expected value is:

For European roulette, a single number wins1⁄37and loses36⁄37:

The presence of the green squares on the roulette wheel and on the table is technically the only house edge. Outside bets will always lose when a single or double zero comes up. However, the house also has an edge on inside bets because the pay outs are always set at 35 to 1 when you mathematically have a 1 out of 38 (1 out of 37 for French/European roulette) chance at winning a straight bet on a single number. To demonstrate the house edge on inside bets, imagine placing straight $1 wagers on all inside numbers (including 0 and 00) to assure a win: you would only get back $36, having spent $38. The only exceptions are the five numbers bet where the house edge is considerably higher (7.89% on an American wheel), and the even money bets in some European games (French Roulette) where the house edge is halved because only half the stake is lost when a zero comes up. This is commonly called the la partage rule, and it is considered being the main difference of European and French roulette. There is also modification of this rule, which is calleden prisonrule. These rules cut the house edge into half (1.35%) in French roulette, when playing even-money bets, as half of the even-money bets are given back for player if the zero is draw in the wheel.

The house edge should not be confused with the hold. The hold is the average percentage of the money originally brought to the table that the player loses before he leavesthe actual win amount for the casino. The Casino Control Commission in Atlantic City releases a monthly report showing the win/hold amounts for each casino. The average win/hold for double zero wheels is between 2130%, significantly more than the 5.26% house edge. This reflects the fact that the player is churning the same money over and over again. A 23.6% hold, for example, would imply that, on average, the player bets the total he brought to the table five times, as 23.6% is approximately equal to100% − (100% − 5.26%)5. For example, a player with $100 making $10 bets on red (which has a near 50/50 chance of winning) is highly unlikely to lose all his money after only 10 bets, and will most likely continue to bet until he has lost all of his money or decides to leave. A player making $10 bets on a single number (with only 1/38 chance of success) with a $100 bankroll is far more likely to lose all of his money after only 10 bets.

In the early frontier gambling saloons, the house would set the odds on roulette tables at 27 for 1. This meant that on a $1 bet you would get $27 and the house would keep your initial dollar. Today most casino odds are set by law, and they have to be either 34 to 1 or 35 to 1. This means that the house pays you $34 or $35 and you get to keep your original $1 bet.

As an example, we can examine the European roulette model, that is,roulette with only one zero. Since this roulette has 37 cells with equal odds of hitting, this is a final model of field probability(,2,P)\displaystyle (\Omega ,2^\Omega ,\mathbb P ), where=0,,36\displaystyle \Omega =\0,\ldots ,36\,P(A)=A37\displaystyle \mathbb P (A)=\frac A37for allA2\displaystyle A\in 2^\Omega .

Call the betS\displaystyle Sa triple(A,r,)\displaystyle (A,r,\xi ), whereA\displaystyle Ais the set of chosen numbers,rR+\displaystyle r\in \mathbb R _+is the size of the bet, and, and:R\displaystyle \xi :\Omega \to \mathbb R determines the return of the bet.

The rules of European roulette have 10 types of bets. First we can examine the Straight Up bet. In this case,S=(0,r,)\displaystyle S=(\\omega _0\,r,\xi ), for some0\displaystyle \omega _0\in \Omega , and\displaystyle \xi is determined by

The bets expected net return, or profitability, is equal to

\displaystyle M[\xi ]=\frac 137\sum _\omega \in \Omega \xi (\omega )=\frac 137\left(\xi (\omega ^\prime )+\sum _\omega \neq \omega ^\prime \xi (\omega )\right)=\frac 137\left(35\cdot r-36\cdot r\right)=-\frac r37\approx -0.027r.

Without details, for a bet, black (or red), the rule is determined as

For similar reasons it is simple to see that the profitability is also equal for all remaining types of bets.r37\displaystyle -\frac r37.[8]

In reality this means that, the more bets a player makes, the more he is going to lose independent of the strategies (combinations of bet types or size of bets) that he employs:

Here, the profit margin for the roulette owner is equal to approximately 2.7%. Nevertheless, several roulette strategy systems have been developed despite the losing odds. These systems can not change the odds of the game in favor of the player.

It is worth noting that the odds for the player in American roulette are even worse, as the bet profitability is at worst338r0.0789r\displaystyle -\frac 338r\approx -0.0789r, and never better thanr190.0526r\displaystyle -\frac r19\approx -0.0526r.

For a roulette wheel withn\displaystyle ngreen numbers and 36 other unique numbers the chance of the ball landing on a given number is1(36+n)\displaystyle \frac 1(36+n). For a betting option withp\displaystyle pnumbers that define a win, the chance of winning a bet isp(36+n)\displaystyle \frac p(36+n)

For example, betting on red, there are 18 red numbers,p=18\displaystyle p=18, the chance of winning is18(36+n)\displaystyle \frac 18(36+n).

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 by36p\displaystyle \frac 36p.

For example, betting on 1-12 there are 12 numbers that define a win,p=12\displaystyle p=12, the payout is3612=3\displaystyle \frac 3612=3, so the better wins 3 times their bet.

The average return on a players bet is given byp(36+n)36p=36(36+n)\displaystyle \frac p(36+n)\times \frac 36p=\frac 36(36+n)

Forn0\displaystyle n0the average return is always lower than 1 so on average a player will lose money. With 1 green numbern=1\displaystyle n=1the average return is3637\displaystyle \frac 3637, that is, after a bet the player will on average have3637\displaystyle \frac 3637of their original bet returned to them. With 2 green numbersn=2\displaystyle n=2the average return is3638\displaystyle \frac 3638.

This shows that the expected return is independent of the choice of bet.

Although most often named call bets technically these bets are more accurately referred to as announced bets. The legal distinction between a call bet and an announced bet is that a call bet is a bet called by the player without him placing any money on the table to cover the cost of the bet. In many jurisdictions (most notably theUnited Kingdom) this is considered gambling on credit and is illegal in some jurisdictions around the world. An announced bet is a bet called by the player for which he immediately places enough money to cover the amount of the bet on the table, prior to the outcome of the spin or hand in progress being known.

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.[citation needed]

This is a name, more accurately grands voisins du zro, for the seventeen numbers which lie between 22 and 25 on the wheel including 22 and 25 themselves. The series is 22-18-29-7-28-12-35-3-26-0-32-15-19-4-21-2-25 (on a single zero wheel).

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.

Zero game, also known as zero spiel (Spielis German for game or play), is the name for the numbers closest to zero. All numbers in the zero game are included in the voisins, but are placed differently. The numbers bet on are 12-35-3-26-0-32-15.

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.

This is the name for the twelve numbers which lie on the opposite side of the wheel between 27 and 33 including 27 and 33 themselves. On a single-zero wheel, the series is 27-13-36-11-30-8-23-10-5-24-16-33. The full name (although very rarely usedmost players refer to it as tiers) for this bet is le tiers du cylindre (translated from French into English meaning one third of the wheel) because it covers twelve numbers (placed as six splits), which is as close to1⁄3of the wheel as one can get.

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; 33-36.

The tiers bet is also called the small series and in some casinos (most notably inSouth 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.

These numbers make up the two slices of the wheel outside the tiers and voisins. They contain a total of eight numbers, comprising 17-34-6 and 1-20-14-31-9.

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.

A number may be backed along with the two numbers on the either side of it in a 5-chip bet. For example, 0 and the neighbors is a 5-chip bet with one piece straight-up on 3, 26, 0, 32 and 15. Neighbors bets are often put on in combinations, for example 1, 9, 14 and the neighbors is a 15-chip bet covering 18, 22, 33, 16 with one chip; 9, 31, 20, 1 with two chips and 14 with three chips.

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.

Another bet offered on the single zero game is final, finale or finals.

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.

A complete bet places all of the inside bets on a certain number. Full complete bets are most often bet by high rollers asmaximum bets.

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 $1000 on a 35-to-1 straight-up, then on each 17-to-1 split connected to that straight-up, $2000 may be wagered. Each 8-to-1 corner that covers four numbers) may have $4000 wagered on it. Each 11-to-1 street that covers three numbers may have $3000 wagered on it. Each 5-to-1 six-line may have $6000 wagered on it. Each $1000 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:

The player calls his 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 players wagered 40 chips, as with all winning bets in roulette, are still his 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.

Zero costs 17 chips to complete and pays 235 chips.

Number 1 and number 3 each cost 27 chips and pay 297 chips.

Number 2 is a 36-chip bet and pays 396 chips.

1st column numbers 4 to 31 and 3rd column numbers 6 to 33, cost 30 chips each to complete. The payout for a win on these 30-chip bets is 294 chips.

2nd column numbers 5 to 32 cost 40 chips each to complete. The payout for a win on these numbers is 392 chips.

Numbers 34 and 36 each cost 18 chips and pay 198 chips.

Number 35 is a 24-chip bet which pays 264 chips.

Most typically (Mayfaircasinos inLondonand other top class European casinos) with thesemaximumorfull completebets, 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 customers 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 should a number adjacent to a chosen number be the winner; for example, player bets 40 chips on 23 to the maximum and number 26 is the winning number. The most notable method is known as the station system or 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 four stations two different corners, one split and one six-line. The dealer takes the number four, multiplies it by 30 and adds the remaining eight to the payout. 4 30 = 120, 120 + 8 = 128. If calculated as stations they would just multiply 4 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.

Over the years, many people have tried to beat the casino, and turn roulettea game designed to turn a profit for the houseinto one on which the player expects to win. Most of the time this comes down to the use of betting systems, strategies which say that the house edge can be beaten by simply employing a special pattern of bets, often relying on theGamblers fallacy, the idea that past results are any guide to the future (for example, if a roulette wheel has come up 10 times in a row on red, that red on the next spin is any more or less likely than if the last spin was black).

All betting systems that rely on patterns, when employed on casino edge games will result, on average, in the player losing money.[9]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 he runs out of money.

The American mathematician Patrick Billingsley said[10]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.

Whereas betting systems are essentially an attempt to beat the fact that a geometric series with initial value of 0.95 (American roulette) or 0.97 (European roulette) will inevitably over time tend to zero,engineersinstead attempt to overcome the house edge through predicting the mechanical performance of the wheel, most notably byJoseph JaggeratMonte Carloin 1873. These schemes work by determining that the ball is more likely to fall at certain numbers, and if sufficiently good will raise the return of the game above 100%, defeating the betting system problem.

Edward O. Thorp(the developer of card counting and an early hedge-fund pioneer) andClaude Shannon(a mathematician and electronic engineer best known for his contributions toinformation theory) built the firstwearable computerto 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 likelyoctantwhere 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 USA. 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