Roulette probability formula

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Roulette is one of oldest casino games and its origins could easily be traced to more than a millennium ago when various simplified versions of it existed as a form of entertainment. Today, gamblers have access to dozens of betting systems, strategies, and even cheating devices that promise long-term success to those who dare use them. Still, there are only two simple concepts they need to grasp in order to improve their play and their chances of winning and that is the odds and the casino advantage that is built-in in every roulette game.

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Knowing the odds is essential in playing roulette or any other game of chance where players’ decisions have no impact on the outcome of the round. Games of skill, in contrast, typically require at least basic knowledge of the rules and give skilled players the opportunity to improve their chances of winning. Unlike classic casino games such as blackjack or poker where individual actions can significantly change the outcome, roulette is a game of pure chance where the outcome is completely random.

Once players recognize this fact, they can truly understand the nature of roulette and learn how to optimize their play to avoid significant losses on the roulette table. Those who want to be successful in the long term need to learn the basic probabilities in this game and determine how likely to win or lose any given bet is. Moreover, all roulette fans should bear in mind that every bet they place is more likely to lose than to win – this is how casinos manage to always make a profit even with the occasional big wins they have to pay out.

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So, what is probability and is there a difference between the true odds in roulette and the casino odds offered to players? These are fundamental questions that need to be answered honestly before gamblers sit around the roulette table.

Probability and True Odds

Often, the terms “probability” and “odds” can be confused by even professional casino players. Although they are, indeed, very similar in terms of their usefulness and relevance to evaluating different roulette bets, the two are quite different – at least theoretically.

First of all, players should know that if fair and unbiased, roulette wheels produce random results with every spin. Each of the numbers is equally likely to win or, rather, equally likely to lose. At the same time, for each number from 0 to 36, there are only two possible outcomes – a win or a loss.


The probability is the likelihood of any given outcome compared to all outcomes that are possible. Expressed as a number, the probability always exists between 0 and 1 – 0 would indicate that something is impossible to occur, while 1 would mean absolute certainty for the outcome. Even chances, for example, would be expressed as a 0.50 probability. In roulette, we have either 37 or 38 numbers in total, so these are all the possible outcomes.

Let us take the single-zero roulette wheel where for each number, there is only 1 way to win and 36 ways to lose. So what is the probability that the ball will fall on 17 in the next spin, for example? We can calculate it by dividing the number of ways to win by all possible ways – 1/37, which is 0.027. For most people, the probability is easier to understand as a percentage, so we simply multiply the decimal by 100 and we get 2.70%.

True Odds in Roulette

Now that we have established what probability is and how it is determined in roulette, we can compare it to the notion of “odds”. Odds are also used to describe the chance of an event occurring but they compare the number of ways it can occur to the number of ways it cannot occur. The odds of any particular number winning in roulette could be simply displayed as 1:36 or 1/36 where 36 is, once again, the number of ways to lose.

Sometimes, when it comes to expressing the odds of a particular bet in roulette, they would be in reverse, indicating the odds against winning. Using the same example, the odds of the number 17 against winning would be 36:1, or 36 to 1. These are known as “true odds” as compared to what some players refer to as “casino odds”. There is a huge difference between the two terms, but it will be explained in detail in the following section.

To make this even clearer, let us see the odds for probably the most commonly placed bet in roulette – red/black. If we place our chips on red, the probability would be 18 out of 37, while the odds would be 19:18 against us because there would be 19 ways to lose against 18 ways to win. In percentages, the probability would be 18/37 = 0.48648648648, 48.65%.

Casino Odds

When determining whether a roulette bet has good or bad odds, experienced players take into account not only their probability of winning and the true odds but also the potential reward they could bring. Each bet pays out differently, depending on its likelihood of winning – the less likely a given outcome is to occur, the more its potential payout would be. There is a sound logic behind this and in a perfect world where casinos would not have to make a profit, the payout of every bet would be equal to the odds against winning.

Let us take a look at the straight bet in European roulette – the odds against the player here are 36 to 1 (36:1) because as we have already shown above, there are 36 ways for this bet to lose and only 1 way to win. In order to offer a reward that would match the risk, the casino would be expected to pay players 36 to 1. In other words, it would be expected to return the original stake and to pay out winnings that are worth 36 times the amount of the bet.

Casino Odds Additional TipsIn reality, this does not happen, however, and the house gives a slightly lower payout of 35:1 on winning straight-up bets. The idea is that on every winning wager, players pay a small fee to the casino and in this case, it is one unit. The difference does not seem significant but it is how casinos gain their advantage over players – by paying every winner just slightly less than they should have. Over time, this tiny difference makes a large profit for the house and provides a guaranteed income in any possible scenario.

These payouts offered by casinos are sometimes referred to as “casino odds” due to their visible similarity. And while both are expressed as ratios of two numbers, they are never equal – the odds against winning are always slightly higher than the casino odds. The closer the casino odds get to the true odds of any given bet, the lower the advantage of the casino.

Roulette House Edge Explained

Clearly, the payout does not correspond to the true odds of roulette bets. This advantage of the casino over its patrons is called house advantage and can be easily demonstrated with the following example – we bet $1 on the number 17 and win. If given the true odds for this bet (36:1), we would receive our initial $1 stake back plus $36 as a payout. Instead, casinos pay us $35 and the original $1 bet but they keep the $1 difference.

The same concept applies to all bets in the game, which results in an average house edge of 2.70% for single-zero roulette and 5.26% for double-zero roulette. There are several formulas for calculating the house edge but probably the simplest one is the following – we need to subtract the casino odds from the true odds against success and then, to multiply that by the probability of success. So, the formula will look like this:

House Edge = (True Odds – Casino Odds) x Probability

When we substitute with the numbers we have already used for the straight bet, we get – (36/1 – 35/1) x 1/37 = 1×1/37 = 0.02702702702, or 0.027. To express it as a percentage, we multiply by 100 and get 2.70%. Several other formulas exist, but they all eventually come to the same conclusion and percentage for the house edge.

Roulette House Edge Additional TipsWhat this means for the players is that they can expect to lose 2.70% of their wager when playing a single-zero roulette. For every $100 they bet, they would lose on $2.70 on average. Of course, this is a theoretical ratio between the stake and the expected loss but things could be very different in real life. If we place a $100 chip on red, we can either win or lose its entire value. But we cannot win the bet and lose $2.70 at the same time.

So, the house edge can only get closer to reality with hundreds or even hundreds of thousands spins of the roulette wheel. It is a theoretical concept and while this may sound too vague to players, it is a good indicator of how much money they can expect to lose over time. Of course, they might win hundreds of dollars by the end of one gaming session or lose hundreds more in the next one. Still, they are less likely to lose their bankroll if they stick to specific games and bets where the casino has the lowest advantage.

Beating the Odds in Roulette

The idea that by using a complex strategy one could beat the odds in roulette is very popular but equally misleading. There are countless guides, books, and websites dedicated to convincing people that there is a guaranteed method of winning in this game. In fact, various betting systems, often inaccurately called “strategies”, have been developed over the years, as well as ways that would supposedly help players exploit the imperfections of physical roulette wheels and overcome the house edge.

Unfortunately, these methodologies have been repeatedly proven to be inefficient in securing winnings over the long term. Moreover, their short-term usefulness is doubtful and the reason is obvious to most expert-level players – roulette is a game with fixed odds that cannot be changed even with the best strategy. As explained above, the outcome of every spin of the roulette is random and based on probability. The house edge, therefore, also remains the same almost as a proof for the popular saying that the house always wins.

Roulette Strategies

The so-called roulette strategies are, in their essence, betting systems based on a progression where the amount of the stake changes after a certain outcome. Famous methods such as the Martingale or the D’Alembert suggest that you increase your bet after every loss, hoping that, in the end, one winning bet will compensate for all the losses. There are also systems where the bet is decreased, while in others, it remains the same throughout the entire game session. Overall, the idea is to help players minimize their losses or even generate some decent winnings over the long term.

While none of these betting progressions provides a fool-proof way to win, another type of strategies comes with exactly this claim. These strategies are based on the notion that you can increase your chances of winning by covering a large part of the table. Indeed, this sounds logical at first and certain methods even suggest that you should cover much more than 50% of the table. This strategy, however, will be too costly for most players, especially after a few losing spins.

If not used carefully, both betting systems and roulette strategies may “eat up” players’ entire bankroll within a short period of time. Placing neighbor bets or any other type of announced bets could be detrimental to those who are not well prepared to suffer serious losses. In conclusion, even the best methods and combinations of bets cannot aid you in beating the odds and overcoming the built-in casino advantage.

Advantage Play

Some roulette players rely on quite different methods for securing winnings. Known as advantage play methods, these include any strategies that give players either a statistical or a mathematical advantage over the casino. If employed successfully, they can beat the standard roulette odds and even if it is just by a little, it should be enough to provide players with long-term winnings. Unlike the betting strategies and systems described above, advantage play does not revolve around the betting layout but rather, around the wheel.

It can be used in both online and land-based casinos and online, advantage players would try to use special software that records and analyzes the results of hundreds of spins. The aim is to find a pattern in the winning numbers (such as repeating sequences of winning numbers) and then, to use it to make predictions. However, most online casinos nowadays offer roulette games based on RNG (random number generator) so finding patterns would be impossible as the outcome of every spin is random.

When it comes to playing a roulette game with a physical roulette wheel, however, advantage play is much more different. Players who try to use this technique would typically stand by the roulette table for at least 40-50 spins and write down all the winning numbers in the hope that they would be able to spot numbers that come out more frequently than others. In fact, sometimes they observe the wheel for hundreds of spins before they can notice repeating numbers, patterns or some irregularities.

Advantage Play Additional TipsThis method was mostly used in the past when casinos did not have such strict maintenance rules and protocols, while the wheels and other gaming equipment were manufactured in a manner that today would be deemed unacceptable. Players who wish to turn the odds in their favor these days need to be extremely discreet if they plan to observe the wheels before the casino security become aware of them. Exploiting roulette wheels’ bias and imperfections for one’s profit is not usually met with understanding from casinos.

How to Increase Player’s Chances of Winning

There are no reliable ways to actually beat the odds when playing roulette and even if such methods exist, they are neither simple nor acceptable by casino standards. Yet, players can increase their chances of winning by following several basic principles that can be applied to both online and brick-and-mortar casinos.

First of all, choosing a good roulette table is essential and obviously, single-zero roulette games are a much better option than double-zero games, which are mostly offered in casinos across the United States. The house edge in the American-style roulette is twice as high due to the additional sector on the wheel, the green 00. But picking French or European-style roulette variations is just the first step in learning how to maximize players’ expected value.

Second, players should always base their play around wagers with the lowest possible house edge. Of course, the best bets in roulette are the outside bets, which cover large portions of the wheel and require a single chip to be wagered. These include black/red, even/odd, and low/high, where the advantage of the casino is 2.70% while the player’s odds of winning are the highest. It is true that the payouts are not particularly attractive but these bets are less risky options in the game.

When choosing a bet, players need to compare the casino odds to the true odds and find the type of bet where these two as close to each other as possible. Often, the most attractive payouts are offered for bets which the casino pays much less than their actual worth. As a general rule of thumb, players should remember that the less risk they take, the less they can expect to win. All the good payouts, however, come with exceptionally bad odds and require a great tolerance for risk.

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.

Press a button below where the ball lands

CalculationsCalculate the roulette probability of RedBlackOddsEvens(1-12)(13-24)(25-36)StraightSplitStreetSquareSix Line Bet along with the expected return on your bet
Outcome Probability = 181212346 Numbers
37 Total Numbers
Success Percentage = Outcome Probability x 100%
Success Percentage = 181212346 x 100 = 48.64%32.43%2.70%5.41%8.11%10.81%16.2%
Expected Return on your Bet = Bet * (Success % - (1 - Success %))
Expected ReturnSuccess Percentage = %
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.

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.

Roulette Wheel Probability Statistics

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.

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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.

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. Casino offers no deposit. 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

Roulette Probability Statistics Problem

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.

Roulette Probability Calculator

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.

Roulette Patterns And Probability Calculations Chart

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.