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Roulette Probabilities and the Maths behind them

Although the rules for roulette tables are designed in such a way that the bank makes money in the long run, it must be said that roulette is a pretty fair game. If you play with only small amounts (relative to your chips), then you can gamble for a long time at the roulette table whilst having a lot of fun.

This article shows you how to calculate probabilities in roulette, including the odds for each type of bet, allowing you to analyse all the possible variations.

Probabilities – A Brief Introduction

The calculation of roulette wheel probabilities is very simple.

Let us take the classic example of a dice with the numbers one to six. How large is the probability that the next throw is the number five?

Of course, it is 1/6 or 16.67%: only one side of the dice has the number five, one of six numbers in total.

And how big is the probability of getting a number that is at least four?

The answer is 50%: there are three possible throws (numbers four, five and six), meaning a three in six chance, or 1/2.

If you have understood these simple examples then you will also be able to understand the probabilities expected in roulette.

Roulette Probabilities

Roulette has a total of 36 numbers, one to 36. The 36 numbers are divided into the following equal sized ‘sets’:

• 18 red numbers (Rouge) and 18 black (Noir)
• 18 even numbers (Pair) and 18 odd (Impair)
• 18 numbers in the lower half (Manque) and 18 in the higher half (Passe)

Lastly, there is an additional stand-alone number, which is the green “zero”, making 37 numbered zones in total. The zero is not included in any of the three sets above.

So, if you are gambling on any of the three sets, in other words;

• red or black
• odd or even
• low or high

then your likelihood to win is 18/37 = 48.65%. If your bet wins, your stake is returned plus 100%. In effect you double your money as each of these sets is an ‘evens’ bet.

Assuming the frequency of results is uniform (i.e. each number appears once during every 37 rotations), the bank will win all bets placed on the three sets every 37th round, when the green zero arrives. Thus, players lose an average of 1/37 of their stakes (assuming that they always play with the same stake per round).

Putting this another way, imagine you are betting on both the “red” and “black” sets at the same time. On average, 36 out of 37 times, the roulette ball will land on a number from one to 36. You are backing all 36 numbers and will lose one bet each time but receive full compensation on the other. (A zero sum game). But you will make a total loss (lose both bets) when the ball lands on zero, and this is precisely the case in 1/37 of the games.

So you can say that with simple chance you should expect returns of 36/37 of your stake money, which is the expected value in roulette (97.3%). In other words, for every 100 units staked you will receive back an average of only 97.3 units.

There are some casinos which are a little fairer to players who rely on simple chance. In such a casino when betting on, for example, red, if a zero appears your stake is not lost immediately but is left to ride on the next spin of the wheel.

In this case, the stake is returned to you (without winnings) if the next spin is “red”. The expectancy value for this type of house increases a gambler’s chances to 98.65%, or 98.65 units returned per 100 unit stake. Wagering with 100 units in a session will cost an average of 1.35 units (the house “edge”). You’ll find further maths and explanations in Wikipedia here.

Probability Table for Roulette Table Bets:

Expected Sequences for Simple Probabilities

If you witness “red” five times in a row, what do you think should come next? Red or black?

Many will believe that the chances for “black” are now higher. Black has not appeared for a while so it must surely arrive again soon to ‘balance out the frequency’.

Unfortunately, this thinking is incorrect. Following a series of “red” five times in a row, the probability for black always remains the same: 18/37 = 48.65%. Just remember the roulette wheel has no memory – it simply does not know what the last number was.

This is why it makes no sense to write down the sequence of red and black. (Surely you must have observed gamblers in casinos engaging in this habit?). It is impossible to detect even the smallest patterns in the roulette wheel because the ball has no memory and will always churn out numbers completely at random.

However, it is unlikely that “red” will appear six time in a row. The chance of six reds in a row is
(18/37) ^ 6 = 1.3% (i.e. 48.65% x 48.65% x 48.65% x 48.65% x 48.65% x 48.65%). (For more explanations see also our combinatorics article). By the way, this is approximately half the chance that any one single number appears (1/37 = 2.7%) and, of course, it is just as unlikely that there will be red numbers five times in a row followed by a black number; the probability for this sequence is also 1.3%.

Examples of Probabilities

To give you a better feel for roulette probabilities, here are a few examples. Here we use “r” for red and “b” for black:

• r or b: 18/37 = 48.65%
• rr or bb: 18/37 x 18/37 = 24% (two reds in a row or two blacks in a row)
• rb or br: 36/37 x 18/37 = 47.3% (one colour followed by the other)
• rrr or bbb: 18/37 x 18/37 x 18/37 = 11.5%
• rbr or brb: 18/37 x 18/37 x 18/37 = 11.5%
• rrrr or bbbb: 18/37 x 18/37 x 18/37 x 18/37 = 5.6%

Never confuse these probabilities with the “conditional probabilities”. The chance of red four times in a row is 5.6% but if you come to the table immediately after a red number has appeared, the probability that you will witness a further three reds (making a total of four times in a row) is 11.5% – purely because you will be observing only a series of three rounds.

The Law of Small Numbers

The law of small numbers was first conceived in 1898 by Ladislaus Bortkiewicz (1868-1931). His book is still available in German, free of copyright, and can be downloaded as a PDF or Kindle version. An English language introductory text is also available.

The Law of Small Numbers is not exactly bedtime reading material, but here is a brief summary. As roulette is a game of repetitions the “law of a third” and the “law of two-thirds” should become evident as play continues.

If there are exactly 37 rotations performed at roulette (as many as there are different numbers), according to the book:

• approximately one third of the numbers (around 12-15 numbers), will not be hit at all
• approximately one third of the numbers will appear exactly once
• approximately one third of the numbers will appear twice or more
• approximately two thirds of the numbers will appear at least once

The Law of Large Numbers

The law of large numbers states that with an increase of repetitions the actual observed frequency comes closer to the theoretically calculated probabilities.

Although it is possible to observe on one particular evening, for example, red numbers coming up 60 times in a row, and another evening 40 times black, ultimately, when you observe the distribution of red and black over a longer period, the frequency will be closer to 50:50.

Therefore, there is no roulette system with which you can beat the probabilities. It may seem for a short time that a certain system functions but, in the long run, you will lose money with every system because roulette rules are designed in a way that the casino earns money. Period. Bear this in mind when you play roulette, but it is still fun!

We hope that this article about roulette and its probabilities has helped you. Now it’s time to enjoy. All you need is a serious and decent online casino where you can test your newly acquired knowledge. Choosing a suitable casino is best done via a comparison site. Here is one: Australia Online Casinos – Check it out, choose your casino and then put your roulette theory into practice!

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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.

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.

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.

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.

Roulette Odds Calculator

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.

Recommended Betting Units

Roulette Winnings Calculator

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. 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.