Roulette

In 2004, California legalized a form of roulette known as California Roulette.[8] By law, the game must use cards and not slots on the roulette wheel to pick the winning number.

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

1 to 18 (Low or Manque), or 19 to 36 (High or Passe)

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

Red or black (Rouge ou Noir)

A bet that the number will be the chosen color.

Even or odd (Pair ou Impair)

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

Dozen bet

A bet that the number will be in the chosen dozen: first (1-12, Première douzaine or P12), second (13-24, Moyenne douzaine or M12), or third (25-36, Dernière douzaine or D12).

Column bet

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.

Snake Bet

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.

For a roulette wheel with ${\displaystyle n}$ green numbers and 36 other unique numbers, the chance of the ball landing on a given number is ${\displaystyle {\frac {1}{(36+n)}}}$. For a betting option with ${\displaystyle p}$ numbers defining a win, the chance of winning a bet is ${\displaystyle {\frac {p}{(36+n)}}}$

For example, if a player bets on red, there are 18 red numbers, ${\displaystyle p=18}$, so the chance of winning is ${\displaystyle {\frac {18}{(36+n)}}}$.

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 by ${\displaystyle {\frac {36}{p}}}$.

For example, betting on 1-12 there are 12 numbers that define a win, ${\displaystyle p=12}$, the payout is ${\displaystyle {\frac {36}{12}}=3}$, so the bettor wins 3 times their bet.

The average return on a player's bet is given by ${\displaystyle {\frac {p}{(36+n)}}\times {\frac {36}{p}}={\frac {36}{(36+n)}}}$

For ${\displaystyle n>0}$, the average return is always lower than 1, so on average a player will lose money.

With 1 green number, ${\displaystyle n=1}$, the average return is ${\displaystyle {\frac {36}{37}}}$, that is, after a bet the player will on average have ${\displaystyle {\frac {36}{37}}}$ of their original bet returned to them. With 2 green numbers, ${\displaystyle n=2}$, the average return is ${\displaystyle {\frac {36}{38}}}$. With 3 green numbers, ${\displaystyle n=3}$, the average return is ${\displaystyle {\frac {36}{39}}}$.

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

This is a name, more accurately "grands voisins du zéro", for the 17 numbers that 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 (Spiel is 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 12 numbers that 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 used, most 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 12 numbers (placed as 6 splits), which is as close to 1⁄3 of 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, 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 11m 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 8 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 as maximum 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 $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:

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.

• 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 (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 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 4 stations - 2 different corners, 1 split and 1 six-line. The dealer takes the number 4, multiplies it by 30 and adds the remaining 8 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.

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, engineers instead attempt to overcome the house edge through predicting the mechanical performance of the wheel, most notably by Joseph Jagger at Monte Carlo in 1873. These schemes work by determining that the ball is more likely to fall at certain numbers. If effective, they 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) 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 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 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.

In the early 1990s, Gonzalo Garcia-Pelayo believed that casino roulette wheels were not perfectly random, and that by recording the results and analysing them with a computer, he could gain an edge on the house by predicting that certain numbers were more likely to occur next than the 1-in-36 odds offered by the house suggested. He did this at the Casino de Madrid in Madrid, Spain, winning 600,000 euros in a single day, and one million euros in total. Legal action against him by the casino was unsuccessful, being ruled that the casino should fix its wheel.[15][16]

To defend against exploits like these, many casinos use tracking software, use wheels with new designs, rotate wheel heads, and randomly rotate pocket rings.[17]

At the Ritz London casino in March 2004, two Serbs and a Hungarian used a laser scanner 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.[18] 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.[19]

The numerous even-money bets in roulette have inspired many players over the years to attempt to beat the game by using one or more variations of a martingale betting strategy, wherein the gambler doubles the bet after every loss, so that the first win would recover all previous losses, plus win a profit equal to the original bet. The problem with this strategy is that, remembering that past results do not affect the future, it is possible for the player to lose so many times in a row, that the player, doubling and redoubling their bets, either runs out of money or hits the table limit. A large financial loss is certain in the long term if the player continued to employ this strategy. Another strategy is the Fibonacci system, where bets are calculated according to the Fibonacci sequence. Regardless of the specific progression, no such strategy can statistically overcome the casino's advantage, since the expected value of each allowed bet is negative.

Betting systems in roulette can be divided in to two main categories:

Negative progression system (e.g. Martingale)

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 system (e.g. Paroli)

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.[20]

The Reverse Martingale system, also known as the Paroli system, follows the idea of the martingale betting strategy, but reversed. Instead of doubling a bet after a loss the gambler doubles the bet after every win. The system creates a false feeling of eliminating the risk of betting more when losing, but, in reality, it has the same problem as the martingale strategy. By doubling bets after every win, one keeps betting everything they have won until they either stop playing, or lose it all.

The Labouchère System is a progression betting strategy like the martingale but does not require the gambler to risk their stake as quickly with dramatic double-ups. The Labouchere System involves using a series of numbers in a line to determine the bet amount, following a win or a loss. Typically, the player adds the numbers at the front and end of the line to determine the size of the next bet. If the player wins, they cross out numbers and continue working on the smaller line. If the player loses, then they add their previous bet to the end of the line and continue to work on the longer line. This is a much more flexible progression betting system and there is much room for the player to design their initial line to their own playing preference.

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.

The system, also called montant et demontant (from French, meaning upwards and downwards), is often called a pyramid system. It is based on a mathematical equilibrium theory devised by a French mathematician of the same name. Like the martingale, this system is mainly applied to the even-money outside bets, and is favored by players who want to keep the amount of their bets and losses to a minimum. The betting progression is very simple: After each loss, one unit is added to the next bet, and after each win, one unit is deducted from the next bet. Starting with an initial bet of, say, 1 unit, a loss would raise the next bet to 2 units. If this is followed by a win, the next bet would be 1 units.

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.

There are numerous other betting systems that rely on this fallacy, or that attempt to follow 'streaks' (looking for patterns in randomness), varying bet size accordingly.

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.