- Let's say I'm betting on a 37 number roulette wheel (18 red 18 black and one green 0)?' On the surface, the best probability for the roulette player to be ahead is in one trial (spin): 48.6% to win (versus 51.4% to lose), as far as even-money betting is concerned.
- The roulette odds on green are different because this only applies to 0 (or 00 if you are playing the American version). Roulette color odds won’t differ between red and black. = = = How to Play Roulette. Playing roulette can be intimidating even for experienced gamblers.
- The popularity of betting on double zero and single green odds could just be because these zeros are clear indications of house advantage, and so the roulette player sees them as particularly significant squares to bet on! Choosing concrete strategies like these gives you an obvious method for playing roulette and stops you from feeling that.
The American roulette wheel features the green 0 and 00, while the European roulette wheel features just the single green 0. Let's first cover the green payouts for the American roulette wheel. If you place a bet on 0, you would receive a payout of 35 to 1 if your bet hits.
Introduction
Roulette is one of the easiest games to play and understand in the casino. As usual the easier a game is to understand the greater the house edge, and roulette is no exception. If you are looking for a easy to understand and slow paced table game, and are willing to sacrifice on the house edge, then you may like roulette. If you want something more stimulating or with a decent return I would suggest looking at other games.
One difference between roulette and all other table games is that roulette chips have no value denomination printed on them. This actually is the true definition of a chip, one which indicates its value is technically called a check. The roulette table comes with six to eight sets of different colored chips, each set consisting of 300 chips. When a player purchases chips he gets his own color and the value of each chip is the buy-in divided by the number of chips received. The dealer will place a token on top of the dealer's stack of that color of chips to indicate the value.
American Rules
A U.S. roulette wheel consists of 38 numbered slots: numbers 1 to 36, a zero, and a double zero. The betting layout consists of every individual number as well as a host of 'outside' bets on combinations of numbers. After the players make their bets, the dealer spins the wheel and a ball and after several seconds the ball will land in one of the numbered slots.
The following table displays the available bets, the win (on a 'to one' basis), and the probability of winning under U.S. rules. All casinos in the U.S. follow these rules except for in Atlantic City. The house edge on all bets is 1/19, or 5.26%, except for the 0-00-1-2-3 combination, which carries a house edge of 7.89%.
Double-Zero Roulette
Bet | Pays | Probability Win | House Edge |
---|---|---|---|
Red | 1 | 47.37% | 5.26% |
Black | 1 | 47.37% | 5.26% |
Odd | 1 | 47.37% | 5.26% |
Even | 1 | 47.37% | 5.26% |
1 to 18 | 1 | 47.37% | 5.26% |
19 to 36 | 1 | 47.37% | 5.26% |
1 to 12 | 2 | 31.58% | 5.26% |
13 to 24 | 2 | 31.58% | 5.26% |
25 to 36 | 2 | 31.58% | 5.26% |
Six line (6 numbers) | 5 | 15.79% | 5.26% |
First five (5 numbers) | 6 | 13.16% | 7.89% |
Corner (4 numbers) | 8 | 10.53% | 5.26% |
Street (3 numbers) | 11 | 7.89% | 5.26% |
Split (2 numbers) | 17 | 5.26% | 5.26% |
Any one number | 35 | 2.63% | 5.26% |
European Rules
What most call 'European roulette' is roulette played on a 37-number wheel, with one zero only. The pays are the same as in American roulette. Best craps tables in vegas. However, with one zero the house edge is cut to 1/37, or 2.70%, on all bets.
The term European roulette is somewhat of a misnomer because you can find single-zero roulette all over the world. In fact, it is the standard outside the United States. Even in the United States you can find single-zero roulette in some high-limit rooms.
Atlantic City Rules
In Atlantic City, any all even money bets (red, black,odd, even, 1-18, 19-36) follow a variation of the European half-back rule (see below). If the ball lands in 0 or 00, then the player will lose only half of any even money bet. This lowers the house edge to 2.63% on these bets. This rule does not apply on single zero wheels.
French Rules
French roulette is played on a single wheel and also features a favorable 'en prison' or half-back rule. Under the 'half-back' rule, if the player makes any even money bet (red, black, odd, even, 1-18, 19-36), and the ball lands in zero, then the player gets half the bet back, known as 'la partage' in French.
The term French roulette is also somewhat of a misnomer because the 'la partage' rule can sometimes be found in casinos outside of France, including some high-limit rooms in Las Vegas.
Rather than lose half, the player sometimes may also choose to imprison the bet. If an imprisoned bet wins on the next spin it is released and the player gets it back, without winnings. What is subject to casino rules is what happens to an imprisoned bet if the ball lands in zero again on the next spin. At some casinos the bet loses, and at others it would become double imprisoned. If a double-imprisoned bet bet won on the next spin, it would move up a level, and become single-imprisoned again. If it lost, then if would become triple-imprisoned if the casino allowed it, otherwise it would lose.
The following table shows the probability of a win, push, and loss under every imprisonment variation I'm aware of.
Imprisonment Odds
Imprisonment | Win | Push | Loss | Expected Value |
---|---|---|---|---|
Single | 0.48648649 | 0.01314828 | 0.50036523 | -0.01387874 |
Double | 0.48648649 | 0.01332346 | 0.50019005 | -0.01370356 |
Triple | 0.48648649 | 0.01332583 | 0.50018768 | -0.01370120 |
Infinite | 0.48648649 | 0.01332586 | 0.50018765 | -0.01370117 |
French | 0.48648649 | 0.01351351 | 0.50000000 | -0.01351351 |
When given the choice to lose half or face imprisonment, the expected value is the same under French rules. Otherwise the expected loss is less losing half.
Triple-Zero Roulette
Triple-Zero Roulette is roulette played on a 39-number wheel. The wheel has the same numbers and colors as a double-zero wheel, plus a 39th place on the wheel. This 39th place will always be green and often the Logo of the casino where the game is located, for example an apple at the New York New York. The game premiered at the Venetian in Las Vegas in September, 2016.
Much like zero and double-zero, all even money bets lose if the ball lands in green. The order of the numbers 1 to 36 is the same as in single-zero roulette. The 0, 00, and Logo are all consecutive, with the Logo in the middle.
There is an added bet on any green, which pays 11 to 1. The house edge on every bet is 1/13 or 7.69%.
Short Pays
I've noticed some electronic roulette games don't pay the full 35 to 1 on single-numbers bets, and/or 17 to 1 on double-number bets. Invariably, any game that short pays the player like this is going to be on a double-zero wheel as well. The following two tables show the house edge on single- and double-number bets, according to the win, assuming a double-zero wheel, and based on 'to one' odds. Be careful that on electronic games wins are often given on a 'for one' basis, meaning the original wager is not returned on a win. To convert a 'for one' odds to 'to one' odds, subtract one.
Single-Number Bets
Win | House Edge |
---|---|
35 to 1 | 5.26% |
34 to 1 | 7.89% |
33 to 1 | 10.53% |
32 to 1 | 13.16% |
31 to 1 | 15.79% |
30 to 1 | 18.42% |
Double-Number Bets
Win | House Edge |
---|---|
17 to 1 | 5.26% |
16 to 1 | 10.53% |
15 to 1 | 15.79% |
Best Roulette in the West
Best in Nevada: The Bellegio, Mandalay Bay, MGM Grand, Mirage, Rio, and Wynn all have European roulette, with a house edge of 1.35%. Usually such wheels are in the high-limit rooms, but the MGM and Mirage also have a European wheel in the main casino at a $25 minimum.
Best in California: It is my understanding that the only single-zero roulette in California is at the Barona casino, with a house edge of 2.70%.
Las Vegas Rules Survey
I am proud to present my Las Vegas roulette survey at my companion site Wizard of Vegas.
Number Placement
To the casual observer, it would appear that the numbers on the wheel are not organized and seem to be distributed randomly. The only obvious patterns are that red and black numbers alternate and that usually two odd numbers alternate with two even numbers. However the distribution of numbers was carefully arranged so that the sum of the numbers for any given section of the wheel would be roughly equal to any other section of equal size. Most numbers are part of a pair, with one number between them. These pairs add to either 37 or 39.
For what it is worth, the sum of all the numbers in roulette is 666.
Wheel Tracking
Some people and books claim that roulette wheels are biased, with a heavy side and a light side. Gravity causes the numbers in the heavy side to hit more often. This, I believe, used to be true when the quality of the equipment was poor. However, modern roulette wheels are much better and very rigorously tested. Only in a casino using a very dated wheel, may you be able to find a biased one by testing many thousands of spins. Based on stories I have heard, your odds of finding such wheels are probably best in Europe.
Betting Systems
Ugh. If you think you can beat roulette with a betting system please read my section debunking betting systems. If you don't believe what I say there, here is what the Encyclopedia Britannica says under the subject of roulette:
The oldest and most common betting system is the Martingale or 'doubling-up' system,in which bets are doubled progressively. This probably dates back to the invention of the Roulette wheel, but every day of the week some gambler somewhere reinvents it, or some variation of it, and believes he has something new. Over the years hundreds of 'sure-fire' winning systems have been dreamed up, but regardless of what system is used, in the long run it cannot overcome the house's advantage of the 0, or 0 and 00. This house advantage is the only system that consistently wins in the long run.
'No one can possibly win at roulette unless he steals money from the table while the croupier isn't looking.' — Albert Einstein
San Diego Area Roulette
The California Constitution prohibits a ball and wheel alone to determine the outcome in roulette. In the greater San Diego area the casinos have thought of some creative ways to offer roulette anyway. Visit my page onSan Diego County roulette to see how they do it, and who offers the best odds.Ball Steering
I get asked a lot about whether I believe dealers can influence the spin to a certain area of the wheel. I used to be very skeptical of it. However, after considering the articles How to Win at Roulette — Part I and Part II by Arnold Snyder as well as other evidence, I think it would be easy for a colluding dealer could do a late slow spin to benefit a wheel clocking player. An extremely skillful dealer may even be able to influence the ball to a certain sector of the wheel. So, for now, I'm still straddling the fence on this one (ow!).
Organic Roulette Experiment
In March 2017, I concluded an analysis of an electronic roulette game by Interblock for ball steering away from perceived wheel clockers. For more information please see my page on the Organic Roulette Experiment.
Play for Free
Play roulette for free with this Flash game. I finally found a Flash game for my site which downloads quickly.
Written by: Michael Shackleford
By Ion Saliu, Founder of Roulette Mathematics
This question was posted in mathematical newsgroups (alt.math.recreational, alt.math.undergrad, alt.sci.math.probability): 'Winning and Quitting on Red/Black in Roulette'.
- 'Obviously in roulette betting on in the long run you are going to lose your money but at some point chances are you'll be in profit. To take an extreme example if you had $1000 you could reasonably expect to be up $1 at some point. Is it possible to generalize this? I want to win W dollars at which point I will quit. How much cash C would I need to have probability P of succeeding? Let's say I'm betting on a 37 number roulette wheel (18 red 18 black and one green 0)?'
On the surface, the best probability for the roulette player to be ahead is in one trial (spin): 48.6% to win (versus 51.4% to lose), as far as even-money betting is concerned. I don't agree that it is the best strategy (betting all your bankroll on one spin).
Odds Of Winning At Roulette
Theoretically, no bankroll will put a player ahead guaranteed, IF flat-betting and playing very long consecutive sessions. There are moments, however, when the roulette player can be ahead by at least one betting unit. Even in even-money bets, the player has a good chance to be ahead by at least one unit after 5, or 10, or even 100 spins. But more than 20 spins are NOT recommended; the probability (odds) to lose go(es) above 50%! Think about it!
Online live roulette casino game. The main thing, mathematically, is the number of player's wins in N trials. To be ahead, means the player has won at least one more roulette spin (number of successes) than the number of losses in N trials. The question then becomes:
'What are the probabilities for the player to be ahead in various numbers of trials?'
Everybody can use my probability software SuperFormula, option L: At Least M successes in N trials.
Winning probability: p = 18/37; M must be at least (N/2) + 1.
Here is a number of cases from the player's perspective.
Winning probability: p = 18/37; M must be at least (N/2) + 1.
Here is a number of cases from the player's perspective.
The figures are applicable to all even-money roulette bets: black or red; even or odd; low or high (1-18 or 19-36).
Odds Of Landing On Green In Roulette
1 trial (spin)
- probability (odds) to win: 48.6%; odds = 1 in 2.05
- probability (odds) to lose: 51.4%; odds = 1 in 1.95
(the probability to lose is 19/37; adding zero to unfavorable cases).
- probability (odds) to win: 48.6%; odds = 1 in 2.05
- probability (odds) to lose: 51.4%; odds = 1 in 1.95
(the probability to lose is 19/37; adding zero to unfavorable cases).
2 trials (spins)
- probability (odds) to win 2 of 2: 23.7% (1 of 2 doesn't mean 'being ahead')
- probability (odds) to lose 1 of 2: 76.3%
- probability (odds) to win 2 of 2: 23.7% (1 of 2 doesn't mean 'being ahead')
- probability (odds) to lose 1 of 2: 76.3%
3 spins
- probability (odds) to win at least 2 of 3: 48%
- probability (odds) to lose at least 2 of 3: 52%
- probability (odds) to win at least 2 of 3: 48%
- probability (odds) to lose at least 2 of 3: 52%
10 spins
- probability (odds) to win at least 6 of 10: 34.4%; odds = 1 in 2.91
- probability (odds) to lose at least 6 of 10: 41.1%; odds = 1 in 2.43
- probability (odds) to win at least 6 of 10: 34.4%; odds = 1 in 2.91
- probability (odds) to lose at least 6 of 10: 41.1%; odds = 1 in 2.43
20 spins
- probability (odds) to win at least 11 of 20: 36.5%
- probability (odds) to lose at least 11 of 20: 46.2%
- probability (odds) to win at least 11 of 20: 36.5%
- probability (odds) to lose at least 11 of 20: 46.2%
100 spins
- probability (odds) to win at least 51 of 100: 35.5%; odds = 1 in 2.82
- probability (odds) to lose at least 51 of 100: 56.8%; odds = 1 in 1.76.
It's getting worse for the player..
- probability (odds) to win at least 51 of 100: 35.5%; odds = 1 in 2.82
- probability (odds) to lose at least 51 of 100: 56.8%; odds = 1 in 1.76.
It's getting worse for the player..
The roulette strategy (or system) is a totally different ball game! But there are professional gamblers out there, including roulette players! They must have strategies, some roulette systems deduced from some figures like the ones above! The player can be ahead at any point in the game. If so, maybe it's time to move to another (or casino) table: It improves the odds of winning!
Always keep track of the losing and winning streaks. Be strong and put an end to a winning streak. You are ahead, you quit the roulette table. Go to another table and wait until you are ahead. The bankroll is of the essence: It must assure going through long losing streaks. Divide the streaks in 10 spins or 20 spins. Never fight aggressively short or mid-term losing streaks. This is the best approach for those who do not know Ion Saliu's casino gambling systems. A good approach to gambling is the next best thing to a good gambling system! Applicable to blackjack and baccarat, too!
Axiomatic one, everybody knows that the casinos have an edge or house advantage (HA) in all the games they offer, roulette including. The house advantage is created by the payouts in rapport to total possibilities for the respective bet. We can apply this simple formula based on units paid UP over total possibilities TP:
(always expressed as a percentage.)
For example, in single-zero roulette, the one-number (straight-up) bet has payout of 35 to 1. The to qualifier is very important: the casino pays you 35 units and they give you back the unit you bet; thus, you get 36 units. There are 37 possibilities in single-zero roulette: 36 numbers from 1 to 36 plus the 0 number. Therefore, HA = 1 – (UP / TP) = 1 – (36 / 37) = 1 – 0.973 = 0.027 = 2.7%.
Let's calculate HA for the 1 to 1 bets: black/red, even/odd, low/high. HA = 1 – (UP / TP) = 1 – (2 / 2.055) = 1 – 0.973 = 0.027 = 2.7%. There are little differences among bets depending on how many decimal points we work with in our calculations.
The point is, the casinos have an advantage, or the players have a disadvantage. Nonetheless, the players' disadvantage is far better than what they face in state-run lotteries. Yet, most casino gamblers lose big, including at roulette tables. They do not have sufficient bankrolls to withstand long losing streaks.
However, around 45% of the roulette numbers lead the gamblers to profits in a few thousand spins. That is, with a sufficient bankroll, a player has a pretty good chance to make a profit, even if playing a random roulette number, or a favorite number. I analyzed about 8000 roulette spins from Hamburg Spielbank (casino). Quite a few numbers ended up making a profit: roulette systems, magic numbers.
By contrast, the more lottery drawings a player plays, the higher the degree of certainty of a loss. Let's make a comparative analysis to the roulette long series above (spins: total roulette numbers, 37, multiplied by 200). If playing the pick-3 lottery for some 100,000 drawings, it is guaranteed that all pick-3 straight sets will be losers. Some numbers will hit up to 3% to 5% above the norm — but that is not nearly enough to assure a profit. A frequency of 3% to 5% above the norm leads to profits in roulette, however.
Ion Saliu's Paradox and Roulette
Ion Saliu's Paradox of N Trials is presented in detail at saliu.com, especially the probability theory page and the mathematics of gambling formula. If p = 1 / N, we can discover an interesting relation between the degree of certainty DC and the number of trials N. The degree of certainty has a limit, when N tends to infinity. That limit is 1 — 1/e, or approximately 0.632..If you play 1 roulette number for the next 38 spins, common belief was that you expected to win once. Gambling disorder definition. Not! Non! Only if you play 38 numbers in 1 spin, your chance to hit the winning number is 100%. Here is an interesting table, which includes also The Free Roulette System #1 presented at the main roulette site.
The maximum gain comes when playing 38 numbers in one spin: 36.3%. Obviously, it makes no sense to play that way because of the house advantage. On the other hand, a so-called wise gambler is more than happy to play one number at a time. What he does is simply losing slowly! Not only that, but losing slowly is accompanied by losing more. That cautious type of gambling is like a placebo. A roulette system such as Free System #1 scares most gamblers. 'Play 34 or 33 numbers in one shot? I'll have a heart attack!' In reality, the Free Roulette System #1 offers a 28.8% advantage over playing singular numbers in long sessions. That's mathematics, and there is no heart to worry about, axiomatic one.
Roulette Odds Green Payout
You can also use SuperFormula to calculate all kinds of probabilities and advantage percentages. The option L — At least M successes in N trials is a very useful gambling instrument. If you play 19 numbers in one spin, the probability to win is 50%. If you play 19 numbers in 2 consecutive spins, the probability to win at least once is 75%.
Editor's note
• In an apparent change of heart, the Hamburg casino (Spielbank) offers online roulette results for free. The new link is (for the time being!):
• In an apparent change of heart, the Hamburg casino (Spielbank) offers online roulette results for free. The new link is (for the time being!):
www.spielbank-hamburg.de/spielsaal/permanenzen.php4
• • Real-life roulette spins are also available from the Wiesbaden, Germany, Casino (Spielbank)
Free Practice Roulette Wheel
www.spielbank-wiesbaden.de/DE/621/Permanenzen2.php: Wiesbaden Spielbank Permanenzen
00 Roulette Odds
Roulette: Software, Systems, Super Strategy
See a comprehensive directory of the pages and materials on the subject of roulette, software, systems, and the Super Strategy.- Theory, Mathematics of Roulette Systems, Strategies, Software.
- The Best-Ever Roulette Strategy, Systems based on mathematics of progressions and free-for-all.
- An inventory of free and outrageously pricedroulette systems.
- Software, Systems: Roulette Wheel Positioning, Sectors, Birthday Paradox.
- 'Roulette System that Won Millions!'
~ Ion Saliu's roulette theory, systems, and software were pirated by an Australian gambling group. - James Bond Roulette System for Dozens, Columns.
- Roulette Systems, Threats from Casino Chairman.
- Anti-Gambler Advice from John Patrick, Casino Moleextraordinaire and conspirator.
- Wizard of Odds Had High Praise For Ion Saliu's Gambling Theory.
- Casinos pay troubled individuals to intimidate intelligent gambling system authors.
- Download roulette, blackjack, casino gambling, systems, software.
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