European Roulette Probability Calculator
Probability in Roulette Games. As mentioned earlier, we are basing this on European Roulette, which consists of 37 fields. We can easily calculate the probability of hitting for example “black” in a game in the following manner: Thus, the probability of hitting black is 48.65%. The expected value is actually slightly better than this; we rounded a bit there, and the house edge for the casino in American roulette is actually closer to 5.26%. Once again, this is exactly equivalent to 2/38 — the number of zeros divided by the total number of pockets on the wheel. This is almost double the house edge as European roulette.
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. 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
Since roulette is a game of chance, there is no way for players to predict with absolute certainty where the little white ball will land on the next spin. Yet, the main goal of all roulette players is to win and get the most from the game with their bets when they join the roulette table. One easy way to achieve this is to manage your bankroll efficiently.
There are several aspects of adequate bankroll management. The most important thing is to determine the size of your overall bankroll, or the amount of money you intend to set aside for a given betting session. The next step is setting a base betting unit for yourself or the amount of money you put on stake per spin.
Less experienced players, who have trouble deciding on the size of their base betting unit in relation to their overall bankroll, can resort to using the so-called roulette betting calculators. The said calculators are available online and often can be used free of charge. The use of these calculators can be especially helpful to players, who are looking to incorporate a specific betting system into their roulette session in order to generate more substantial profits.
| Outcome Probability = | 181212346 Numbers |
| 37 Total Numbers |
| Success Percentage = | 181212346 | x 100 = | 48.64%32.43%2.70%5.41%8.11%10.81%16.2% |
| 37 |
You Win =
Expected Return on your Bet =
How Does a Roulette Betting Calculator Work?
Using the roulette betting calculators available online is quite straightforward. The calculators aim at indicating what the most suitable base betting unit for a given player will be on the basis of their bankroll size. The base betting unit is the sum players put on stake on each spin of the ball.
European Roulette Probability Calculator Solver
It is important to remember that discipline plays a crucial role at the roulette table – the size of the base betting unit is meant to be a fixed constant that players stick to throughout their betting session. Sticking to the fixed unit ensures players are spending the optimum amount of time at the table while still being able to turn a decent profit whenever they experience a longer winning streak.
This is how the online roulette betting calculators work. All players are required to do is enter the exact amount of their bankroll in the respective field and click on the Calculate button. For example, let’s assume you have set aside £200 for a single roulette betting session. The calculator will compute three different betting units using this sum, namely a safe unit, a recommended unit and a high-risk unit. Which one you choose to adopt depends on the type of roulette player you are, more risk-averse player or a high roller.
Many roulette experts recommend players to use a base unit that amounts to 2% of the total bankroll and caution them to steer clear of using a unit that exceeds 5% of the sum, they are planning to spend on a single betting session. Inexperienced roulette players are often advised to stick to base units amounting to 1% of their bankrolls, at least until they figure out what betting strategy works best for them.
| Betting Units for £200 stake | |
|---|---|
| Safe (1%) | £2 |
| Recommended (2%) | £4 |
| High-Risk (4%) | £8 |
Safe Betting Units
If you enter £200 as the size of your overall bankroll, the calculator will compute that your safe betting unit amounts to £2. This unit corresponds to 1% of your overall bankroll. Sticking to a safe betting unit might enable players to prolong the time they spend at the roulette table. It is recommended for players, who are more cautious with their money and are satisfied with putting relatively small amounts on stake.
However, players who choose a safe betting unit should bear in mind there are maximum and minimum stake limits at landbased roulette tables. Often, it might be impossible to bet only £1 per spin since in landbased casinos, the minimum roulette bets usually start at £5. Nevertheless, this issue can easily be resolved by playing the game online. Many web-based gambling operators offer low stakes roulette variations where it is possible to make bets of £1 or even £0.50. So, if you decide to stick to a safe betting unit, you should consider joining a low stake roulette game from the comfort of your home.
Online Roulette Variations
Playtech Roulette
Microgaming Roulette
NetEnt Roulette
Betting Calculator
Recommended Betting Units
The second unit the calculator will compute is the recommended betting unit. You may notice that the recommended unit always corresponds to 2% of your overall bankroll, regardless of the sum you have initially entered. In our case, the recommended betting unit for a bankroll of £200 would amount to more than £4. This is, in fact, the ideal base unit since it allows players to optimise the time they spend at the roulette table, while at the same time giving them a rush of adrenaline on every spin of the ball.
Also, sticking to the recommended unit may help you minimise your losses in case you experience a longer losing streak since you will not be risking a substantial percentage of your bankroll on each spin. Roulette novices should remember that it is better to settle for smaller wins than parting with their entire bankroll in a few spins by putting heftier sums on stake.
High-Risk Betting Units
The roulette betting calculators available online also compute high-risk betting units, suitable for the type of player, who prefers to bet big and collect greater profits, respectively. The roulette betting calculator has computed a high-risk betting unit of £8 for the bankroll of £200, used in this example. The sum corresponds to 4% of the overall bankroll amount and is considered the highest, yet the safest bet limit in the game of roulette.
Players, who opt for using higher-risk bet units, are recommended to be careful when putting their funds on stake in order to prevent losing more money they are comfortable with. If you are anxious or get infuriated each time you lose on a spin, this might serve as an indicator the high-risk betting unit is not for you. If that is the case, you should consider reducing the size of your next bets to the recommended bet unit to prevent being tempted to bet too much when you suffer a losing streak.
European Roulette Probability Calculator Formula
Determining the Best Base Betting Unit
Online roulette betting calculators predominantly focus on giving players an idea of the best range of units they should put on stake at the roulette table. At the end of the day, it is up to players to decide on the amount they wish to bet per each spin. However, players are recommended to join the roulette table only with money they are comfortable playing with, and potentially losing.
If the thought of losing a given amount of money per spin makes you sweat with anxiety, you should avoid putting this sum on stake. It is of utmost importance to stick to the limits you have set for yourself and feel comfortable with. Remember the best betting unit for yourself is the one that allows you to maximise the time you spend at the roulette table without causing you to risk a substantial part of your bankroll.
Another thing to take into account is the fact that these calculators are meant to be used solely for reference purposes. In order to find a balance, players are recommended to experiment a little with their base betting units and adjust them until they find a unit they are most comfortable with.
Most importantly, when you have found the betting unit that works best for you, you need to stick to it when you join the roulette table, regardless of what happens. Sometimes, players are tempted to increase their betting units whenever they suffer a lengthy losing streak in an attempt to recoup their losses. However, there is no guarantee the next spin will result in winning, so increasing the bet units dramatically might end up costing you your entire bankroll.
A Few Words in Conclusion
While online roulette betting calculators are useful in helping players determine the size of their base betting units, the effect they have on your bets winning or losing is equal to nil. In addition, if you search online, you may be able to find a range of roulette calculators that compute the probability of winning with specific roulette bets. Some calculators go as far as “predicting” which the next number to be spun will be.
There is one very important thing to remember, namely the fact there is no relation between the spins. Each spin is independent and does not affect the outcome of the spins that are to follow. This indicates all numbers on the roulette wheel have equal chances of coming up on the next spin, so there is no possible way to predict the outcome with absolute certainty. This practically renders calculators that compute the outcomes on the basis of previous winning numbers useless.
However, calculators that compute betting units might prove to be useful when it comes to players’ money management. But it is up to players decide whether or not they are comfortable with using the betting units, suggested by the calculators.