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Roulette

Introduction

One of the big four casino table games (the others being Blackjack, Craps and Baccarat) Roulette is the only significantly popular wheel game widely available today.

The game is played on a circular wheel whose circumference is surrounded by pockets. If playing at a European game the pockets are numbered 1 to 36, alternately coloured red and black with the final pocket numbered 0 and coloured green. If playing at an American game the pockets are numbered 1 to 36 again alternately coloured red and black and the final two pockets are numbered 0 and 00 and both coloured green.

When playing the game the players place wagers – normally in the form of special coloured casino chips specific to the individual player, though standard casino chips can be used as well – on a felt layout. Unlike standard casino chips Roulette chips do not have a predefined value. When the player first ‘buys-in’ a chip colour is assigned to them. The value if these chips are determined by the buy-in divided by the number of chips bought. This will generally be recorded by the Croupier (dealer) where that specific colour of chips is racked.

Typically the layout has 3 columns with the numbers in order starting from the top left. The 0 and 00 are represented at the very top of the felt (closest to the wheel) directly above the 3 columns. This allows the player to bet in a variety of manners;

·         ‘Half’ – The numbers 1-18 or 19-36 (sometimes called ‘Manque’ and ‘Passe’)

·         ‘Even’ or ‘Odd’ – the Even or Odd numbers (sometimes called ‘Rouge’and ‘Noir’)

·         ‘Red’ or ‘Black’ – the Red or Black numbers (sometimes called ‘Pair’ and ‘Impair’)

·         ‘Column’ – 12 numbers in a vertical line on the layout

·         ‘Thirds’ – The numbers 1-12, 13-24 or 25-36

·         ‘Six Line’ or ‘Double Street’ – 2 adjacent horizontal lines of numbers on the layout (a total of 6 numbers)

·          ‘First 5’ – The numbers 1, 2 and 3 alongside the 0 and 00 (only available on American wheels)

·         ‘Corner’ or ‘Square’ – 4 numbers forming a square

·         ‘Basket’ or ‘First 4’ – The numbers 0, 1, 2 and 3 (only available on European wheels)

·         ‘Street’ – 3 numbers in a horizontal line on the layout

·         ‘Basket’ – The numbers 0, 1, 2 or 0, 00, 2 or 00, 2, 3 (only available on American wheels)

·         ‘Trio’ – The numbers 0, 1, 2 or 0, 2, 3 (only available on European wheels)

·         ‘Split’ – 2 adjacent numbers

·         ‘Straight Up’ – a bet on a single number

 

 

Some casinos – though we are not aware of any online – will offer what is referred to as the ‘Snake’ bet. This bet consists of the numbers 1, 5, 9, 12, 14, 16, 19, 23, 27, 30, 32 and 34. Looking at the layouts above you’ll be able to see that this bet ‘snakes’ from left to right on the layout following connected Red numbers.
Once the players have placed their bets the Croupier spins the Roulette wheel in one direction before spinning a small ball round the inclined track on the outside of the wheel in the opposite directions. As the ball loses momentum it drops off of the inclined track, bounces off of the deflectors intended to randomize the final location off the ball, then comes to rest in one of the numbered pockets.

Unlike the layout, the numbers surrounding the wheel are not in sequential order, though by and large they do follow a standard sequence in the majority of casinos.

For a single 0 Roulette wheel (European) the numbers appear in the following order clockwise round the wheel;

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

For a double 00 Roulette wheel (American) the numbers appear in the following order clockwise round the wheel;

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

The numbers on the Roulette wheel sum to 666 which has lead to various outlandish legends about the game’s association with the number of the beast and the devil.

RaceTracks

While not always present, Roulette layouts sometimes include a ‘Racetrack’ which displays the numbers in the same order as the Roulette wheel and allows the player to bet on sequences of numbers as they appear on the wheel. Typically these Racketrack layouts are found on European Roulette games with specific number groupings given French names;

Les Vousins de Zero (‘The Neighbours of Zero’ also sometime called ‘Serie 0/2/3’) – When the player places this bet 9 chips are placed at once; 2 on the 0, 2, 3 combination, and 1 chip on each of the following Split bets 4/7, 12/15, 18/21, 19/22, 25/28, 26/29 and 32/35. These bets combined cover the wheel clockwise from 22 to 25. The 0, 2, 3 combination is paid at 11 to 1 while any of the Split bets is paid at 17 to 1.

Jeu Zero (the ‘Zero Game’ also known as ‘Zero Spiel’ or simply ‘Zero’) – When the player places this bet 4 chips are placed at once; 1 on each of the following Split bets 0/3, 12/15 and 32/35 and 1 chip Straight Up on 26. These bets combined cover the wheel clockwise from 12 to 15. The split bets are paid at 17 to 1 while the Straight Up bet is paid at 35 to 1.

Le tiers du Cylindre (‘Thirds of the Wheel’ also sometimes called ‘Serie 5/8’) – When the player places this bet 6 chips are placed at once; 1 on each of the following Split bets, 10/11, 13/16, 23/24, 27/30 and 33/36. This Split bets are paid at 17 to 1.

Orphelins (‘Orphens’ sometimes called ‘Oprh’) – When the player places this bet 5 chips are placed at once; 1 on each of the following Split bets 6/9, 14/17, 17/20 and 31/34 and 1 chip is bet Straight up on the number 1. These combinations represent the sections of the wheel in between the Les Vousins de Zero and Le tiers du Cylindre bets. The Split bets are paid at 17 to 1 while the Straight Up bet is paid at 35 to 1.

Neighbours – When the player places this bet 5 chips are placed at once. The player selects a number and that number and the 2 directly to either side of it on the Racetrack all receive a single chip Straight Up bet. The Straight Up bets are paid at 35 to 1.

Numbers and Colours

On both the European and American wheels the number groups 1-10 and 19-28 match odd numbers with red and even numbers with black. The number groups 11-18 and 29-36 match odd numbers with black and even numbers with red.

Called Bets

Online, when playing standard Random Number Generated (RNG) versions of Roulette, all betting takes place prior to the ball being spun. Offline and potentially online where a ‘Live’ version of the game is selected, betting can continue after the Croupier releases the ball up until the Croupier shouts ‘No more bets’. The Called bets are usually placed on a Racetrack layout system and it’s worth noting that Croupiers will regularly assume that a Called number in intended as a Neighbours bet.

Payouts

The payouts for Roulette are standard across both the European and American versions of the game resulting in the European version having a far lower house edge.

Bet CombinationPayout Odds ProbabilityHouse Edge
Halfs1 to 118/380.473684215.26%
Even/Odd1 to 118/380.473684215.26%
Red/Black1 to 118/360.473684215.26%
Column/Thirds/Snake2 to 112/380.315789475.26%
Double Street5 to 16/380.157894735.26%
First 56 to 15/380.131578957.89%
Corner8 to 14/380.105263165.26%
Street/Basket11 to 13/380.078947375.26%
Split17 to 12/380.052631585.26%
Straight Up35 to 11/380.026315795.26%
Bet CombinationPayout Odds ProbabilityHouse Edge
Halfs1 to 118/370.486486492.70%
Even/Odd1 to 118/370.486486492.70%
Red/Black1 to 118/370.486486492.70%
Column/Thirds/Snake2 to 112/370.324324322.70%
Double Street5 to 16/370.162162162.70%
First 4/Corner8 to 14/370.108108112.70%
Street/Trio11 to 13/370.081081082.70%
Split17 to 12/370.054054052.70%
Straight Up35 to 11/370.027027032.70%

En Prison

Some, but not all, European Roulette games offer the player favourable ‘En Prison’ rule. Where this rule is in force, when the player bets on one of the even money bets (Halfs, Even/Odd or Red/Black) and the ball lands in the 0 slot one of two things will happened depending on the casino’s enforcement of the rule’

1) Half the players bet is returned (called ‘La Partage’ )

2) The player’s bet is placed ‘in prison’. This means that the bet rides on the result of the next round; if the bet would win on the next spin the full bet is returned to the player, if the bet would lose on the next spin the full bet is lost.

In the first scenario the House Edge of the game is as follows;

Bet CombinationPayout Odds ProbabilityHouse Edge
Halfs1 to 118/370.486486491.35%
Even/Odd1 to 118/370.486486491.35%
Red/Black1 to 118/370.486486491.35%

The second scenario is slightly more complicated depending on how the casino chooses to enforce the En Prison rule. Specifically the issue arises where the subsequent spin results in a further 0.

In this situation there are 3 possible options;

1) The casino ignore the zero and the first non-zero spin decides whether the bet is returned to the player or not. This rule results in the same House Edge as the La Partage rule.

2) The casino keep the bet on a subsequent zero.

3) The casino create additional ‘in prison’ restrictions on the bet (i.e. if 2 zeros have come up, the player would require 2 winning spins on the next two rounds to release their bet).

In the situation where the casino keeps the additional bet on a subsequent zero the House Edge for the various bets is as follows;

Bet CombinationPayout Odds ProbabilityHouse Edge
Halfs1 to 118/370.486486491.39%
Even/Odd1 to 118/370.486486491.39%
Red/Black1 to 118/370.486486491.39%

The variation that allows multiple levels of imprisonment makes only a fractional difference to the standard En Prison rule where the player loses if a second 0 come up. This is due to the rapidly diminishing probability of multiple 0s coming up (presuming one 37 has already occurred a 2nd zero will occur 1 in 37 times, a 3rd zero will occur 1369 times, a 4th zero will occur 1 in 50653 times, a 5th zero will occur 1 in 1874161 times and so on).

Progressive Side Bets

Microgaming currently offer a version of Roulette called ‘Roulette Royale’. This version of the game uses a European wheel (single zero) and automatically places a one credit bet on a progressive jackpot side bet. The side bet is one if the same number comes up 5 times in a row (regardless of whether the player one or lost on the main bets of those spins).

The probability of winning this side bet is (1/37)^4 = 0.000000533 or 1 in 1,874,161. The reason that the power is 4 rather than 5 is that it does not matter which number starts the sequence, so the first spin is irrelevant.

The House Edge for this bet is calculated as such;

House Edge = (0.000000533*[VALUE OF JACKPOT])-(1-0.000000533)
House Edge = (0.000000533*[VALUE OF JACKPOT])-0.999999467

We are aware of the jackpot values for this game being as low as 60000 and as high as 977403. This gives a range of House Edge values of 96.80% to 47.90%. Even at its highest value this is one of the worst bets you can make in the casino.

The break even value for this side bet would be 1876171.61.

The overall House Edge of the combined bets once the bets on the main game are included is going to vary substantially depending on the amount wagered on the main game. The Jackpot side bet, as it has a substantially higher House Edge than the main game, will always be a penalty on the average House Edge, but the more wagered on the main game the more diluted the penalty will become.

The best strategy would simply be to stick with the standard Microgaming European Roulette game and avoid Roulette Royale altogether.

Optimal Strategy

Roulette does not really offer the opportunity for strategy decisions to make a difference to the cost of the game. The only decision really open to the player is the rules of the wheel they play at. The player should always choose a single 0 (European) Roulette wheel where available and if possible one that uses the En Prison rule. This means that the player will only bet on the even money bets (Halfs, Even/Odd or Red/Black).

Other than the above, the difference between the different available bets is variance rather than cost (the cost of playing a straight up number is identical to that of any other bet on the wheel, but the chance of big wins or losses change. The only exception to this is the First 5 bet on a double 0 (American) wheel, which has a far higher cost to the player and should be avoided.

Bet Combination Calculator

While it’s relatively trivial to calculate the probability of success with individual wagers on the Roulette wheel, when different combinations of bets are placed it becomes a more complex calculation. Below we’ve provided a calculator that will allow you to quickly calculate the range of possible results and the Standard Deviation depending on various bets placed.

We’ve also included a calculator to allow you to check the fairness of your results based on the bets you’ve placed;

History

As with the history of many gambling games, poor recording makes it hard to define exactly where and when Roulette was developed. The first known reference to the game in its current form comes from the novel La Roulette, ou le Jour by French author Jaques Lablee which was published in 1796 who describes a game with the played with a wheel that had two numbers – 0 and 00 – that the player could not bet on, meaning that if the ball landed on these numbers the house automatically won.

Early American Roulette wheel often had slots numbered 1 to 28 alongside slots with the 0 and 00 and a slot with an American Eagle symbol. The American Eagle slot was considered the house slot while the player could wager on every other number.

Reportedly it was Frenchmen Francois and Louis Blanc who originally introduced the single 0 Roulette wheel in Germany in 1843 in an attempt to undercut competing casinos. In the 1860’s when the German government outlawed gambling the Blanc family moved to Monte Carlo and continued to offer their single 0 version of Roulette.

Betting Systems

Roulette is a very popular game amongst gamblers for deploying progression betting systems. These systems raise or lower bets based on the recent results in the game and are normally deployed on the even money bets at the Roulette table. The most popular of these is widely considered to be the Martingale system. This is what’s referred to as a Negative Betting Progression and sees the player double their bet after every losing result.

To emphasize the dangers of using this system, below is a table displaying the number of losses, the amount of the next bet in the progression and the odds of a loss of this size occurring (this assumes a minimum bet of $1, where higher minimums are in force this progression will escalate substantially more rapidly);

# of lossesSize of next wagerOdds of occurrence (European)Odds of occurrence (American)
121 in 1.951 in 1.85
241 in 3.791 in 3.61
381 in 7.381 in 6.86
4161 in 14.381 in 13.03
5321 in 28.011 in 24.76
6641 in 54.541 in 47.05
71281 in 106.201 in 89.39
82561 in 206.821 in 169.84
95121 in 402.751 in 322.69
1010241 in 784.301 in 613.11
1120481 in 1527.321 in 1164.90
1240961 in 2974.251 in 2213.31

The above table shows how quickly a player using the Martingale system can end up moving from very low wagers to very high wagers. In most instances the player will ultimately end up being ahead by one unit (in this case dollar) for every progression. The Martingale progression is good at creating frequent small wins. However the rare occurrence (which isn’t actually all that rare when you consider a Royal Flush in Jacks or Better Video Poker is due to occur approximately one in every 40k hands) will result in a catastrophic loss for the player.

Alongside these rare huge losses it has to be considered that the vast majority of casinos will not offer betting limits that allow this progression to move past 9 losses meaning that 1 in 400 progressions will result in a loss that cannot be recuperated on subsequent spins.

It’s also worth noting the substantial difference playing on a European single 0 wheel makes – a solid reason to always look for a European wheel even though this particular issue isn’t specific to progression betting systems.

Regardless of which system is in play, progression betting systems rely on what is known as the Gambler’s Fallacy. This is the reasoning that past results will affect future ones. For example, if Red has come up 10 times in a row on a Roulette wheel many gamblers mistakenly believe that – as Black is equally likely to occur – Black must come out soon to ‘rebalance’ the odds. This is fundamentally untrue. While it is true that in the long run the game will tend towards an equal number of Red and Black results, a bias to one colour or the other does not have to be balanced out by an equal bias of the other result and instead will be faded to insignificance simply by the large number of normal results.

James Grosjean gives a very straight-forward treatment of this principal in this article debunking all “due” theories – http://beyondnumbers.lvablog.com/2013/12/17/the-denominator-where-due-happens/.

While playing with the various betting systems may be fun, changing bets at the Roulette table will never by itself be enough to give the player any advantage over the house. You’re still playing a losing game.

Winning Strategies for Roulette

While there are some legitimate playing strategies that can give the player the edge while playing the game of Roulette, the vast majority of the systems out there that purport to do this are completely bogus and devoid of any worth. Below are notes on some of the systems that can gain the player a mathematical advantage.

It should be noted that while the strategies discussed below are legal, the vast majority of casinos will restrict your access to their Roulette tables/promotions if not outright ban you from their casino if they think that you’re applying them.

Offline

Please be aware that the following strategies are only valid where the game being played is not shuffled after every hand, or depend on a weak dealer providing the players more information than they should have. As such they cannot gain an advantage where the player is using Random Number Generator (RNG) games – the computerized versions of Blackjack found at online casinos. There are live dealer games available online where some of the techniques described below may be applicable, but in my opinion the game conditions available online are not conducive to gaining any sustainable level of advantage.

Regarding the legality of Card Counting and Shuffle Tracking– the strategies discussed below are legal within the US and UK as long the player can perform them without use of external aids (computers/calculators). The moment that a computer is used to keep track of the cards or perform calculations, this becomes a violation of gaming law. We strongly discourage anyone from engaging in any form of illegal activity.

Comp Counting

Comp Counting is the practice of using the complimentary benefits that casinos reward players with to gain an edge over a game. This works if the value of the returned comps is greater than the cost of playing the game and is detailed in Max Rubin’s book ‘Comp City’. This book is well worth a read for anyone who holidays frequently in any of the gambling meccas.

The basic rules that Rubin suggests for those looking to gain an advantage from use of comps are;

1) Play at a busy table (this will make your bets harder to track)
2) Play at a slow table (this will decrease the amount you are betting)
3) Look for a table where there are a good number of players using standard casino chips rather than the coloured wheel chips and play with standard chips rather than wheel chips (this will make it harder to track your bets)
4) Place a large initial bet when the game supervisor is rating you and if possible remove some of the bet when the supervisor looks away, but before the ball has been spun.

The idea is to convince the game supervisor that you are betting more than you actually are resulting in comps being awarded at a higher than accurate level. The above is a very stripped down version of what Rubin advises and can be fine-tuned substantially if you are playing regularly.

It should also be possible for two players to bet on opposite results while playing for comps on the same wheel reducing the variance involved in obtaining the comps, however if the casino becomes aware of players working together in this fashion the players are likely to find their play and privileges restricted.

Bias Wheel

One of the earliest methods that could legitimately gain an edge over the game of Roulette was based on the identification of slight biases in individual wheels. This method requires a substantial amount of scouting time due to the slow rate of play and the substantial amount of data that needs to be collected on each wheel to successfully identify biases towards certain numbers/sections of the wheel.

With more primitive Roulette wheels this may still be possible, but due to sizable wins by players as late as the 1990s, modern casinos track normally track the results of their own wheels and rebalance the wheels regularly to ensure that biases do not occur.

While there may be occasional situations where a bias wheel could be identified and exploited the time involved in the identification of such opportunities alongside the likelihood that any bias will be short-lived makes the Bias Wheel effect primarily a play of the past.

Visual Ballistics and Acoustic Prediction

Several serious players – most specifically Laurance Scott – have claimed that it is possible, with extensive practice to predict where the ball will land by observing the speed of rotation of both the wheel and the ball.  This technique relies on the procedures that allow players to continue wagering for a short period of time after the Croupier has spun the ball. Other methods that are often packaged with Visual Ballistics include Acoustic Prediction which involves listening to the sound the ball makes while it travels round the wheel and using the change in pitch to judge when the ball is likely to drop from the track onto the wheel.

These techniques do not rely on precision when predicting where the ball will land, but rather look to exclude just small sections of the wheel where the ball won’t land. On a European Roulette wheel, excluding just three numbers from the possible results will give the player an advantage (a European Roulette wheel has 37, if it can be confirmed that the ball will not land on 2 of those numbers there are only 35 possibilities and given the straight up payout of 35 to 1, meaning the player gets 36x their bet returned, the game now returns a player advantage).

[Interesting side note – Historically this premise has been used successfully by several teams deploying computers to track the ball and rotor speed to predict the section of the wheel that it will fall in, including one run by the Godfather of Card Counting Edward O. Thorp, but as the use of computer aides within the casino environment is now illegal in most (all?) major jurisdictions use of these computers would now constitute cheating.]

While the theory of these techniques is absolutely mathematically sound, the successful implementation of these techniques is far more controversial. Unlike other techniques to gain an edge over the house, like Card Counting at Blackjack, practicing for success cannot be confirmed by any means other than analysis of results. When Card Counting at home the practitioner can stop and check the count on the remaining cards to confirm that they’ve been accurate in their assessment of the shoe and confirm they have an advantage. With Visual Ballistics, no checks are reasonably possible and the successful or unsuccessful prediction of the result does not necessarily confirm or refute the player’s ability. Large sampling is required to confirm a player has achieved a working advantage, though the more precise the prediction (more number confirmed that will not come up) the smaller the sample needed to verify an advantage.

On top of this, while a deck of cards contains the same distribution everywhere, the specifics of Roulette wheels change from wheel to wheel and the environment in a casino (people and noise) can substantially alter the playing conditions, creating a situation where even someone who could successfully deploy Visual Ballistics at home may find their accuracy substantially diminished in the real casino environment.

All in all Visual Ballistics and Acoustic Prediction are difficult techniques to verify and even if they can be deployed successfully are likely to involve many years of practice to gain the necessary skills.

Ball Steering

A cheating technique rather than an Advantage Play technique, the premise of Ball Steering is that a Croupier can learn to spin a Roulette ball in such a way as to ensure the ball lands on a pre-determined sector of the wheel. This would allow the Croupier to review the bets placed before the spin and determine where to land the ball to either benefit the players or the casinos.

While there may be rare Croupiers that can achieve such accuracy, it’s highly unlikely that many players would ever come across a game where Ball Steering was being deployed and certainly not in a specific attempt to benefit the casino. There are two reasons for this;

1) A Croupier who could hit certain sectors of the wheel would quickly realise that their skills could be deployed far more profitably for themselves than for the house. By working with an accomplice playing at the table the Croupier could target the ball to allow their accomplice to win, then ensure that the table figures (profit made for the casino) remain consistent but causing other players to lose more frequently. A Croupier cheating for an accomplice is far more likely than one cheating for the house.

2) If a Croupier deploying Ball Steering was to be encountered, their skills could quickly be turned against them. By waiting until the ball was spun and subsequently betting on the sectors of the wheel that did not have substantial volumes of player bets the smart player could gain an edge themselves and as the player is waiting until after the Croupier releases the ball, there’s little the Croupier could do to prevent this other than targeting sectors of the wheel with other player bets.

Personally I feel it’s highly unlikely that Ball Steering represents any substantial threat to players, though several notable players including Arnold Snyder disagree.

Online

There are several methods of legitimately gaining an advantage over the online Roulette games. By and large these involve the use of promotions, bonuses and comp point systems as the games by themselves – assuming the game is functioning in a correct fashion – will not provide any additional information (for instance, bias towards a particular section of the wheel) as may be found offline.

It should be noted that online advantage play is not specific to Roulette and in recent years the wagering requirements for player’s choosing to play only Roulette has inflated to the point that Roulette is now often a sub-optimal choice for the player. Alongside this Roulette is often entirely excluded while playing with a bonus due to the possibility to engage in low risk wagering (betting on all, or close to all, possible outcomes). For more information on beating online casinos see BeatingBonuses.com.

References

‘Comp City’ – Max Rubin