## Roulette probabilities and math

Because the rules are designed so that the bank / casino can earn money in the long run, there is no system to beat roulette. In a very long run you will make a loss.

Nevertheless, roulette is quite a fair game. If you only play with small amounts (in relation to your chips), then you will be able to gamble for a long time and have a lot of fun.

We will show you how to calculate roulette probabilities, chances of winning for the individual bets and then analyse the possible fluctuations (the so-called winning and losing streaks) in roulette.

## Probabilities – an introduction

Calculating simple probabilities is very simple. Let’s take the classic example of a cube with the numbers 1 to 6. How big is the probability that the toss shows the number five? It is 1/6 or 16.7%: one side of the cube has the number 5 and in total there are six sides.

And how big is the chance to get a number that has at least four eyes? It is 50%: three possible sides (the 4, 5 and 6) are helping us and in total there are 6 sides.

And the same is applied for roulette.

## Roulette probabilities

Roulette has a total of 37 numbers: the digits from 1 to 36 plus the green zero (the null). There are 18 red and black (rouge and noir), 18 even and odd (pair and impair) and 18 numbers in the low and high (manque and passe). If you now put on a simple chance, like

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

then your chance to win is 18/37 = 48.6%. If you win, your bet doubles. One can thus also say that the bank wins in one of 37 cases, so when the zero comes. Therefore players lose an average 1/37 of their bet. You can also visualise this by setting “red” and “black” at the same time. In 36 of 37 cases, the ball will land on a number between 1 and 36. So you lose one bet but double the other. If you have bad luck and the ball falls on the zero, then your entire bet is gone. And this happens in 1/37 of the turns. You can also say that in the game with even bets you get back 36/37 per bet. And this is nothing else than the expectation in roulette. This is 97.3%. So you get an average of €97.3 per €100.

Depending on the provider, some casinos are even a bit more fair with the players which put an even bet. In such casino, if you place an even bet like red and get the zero, you can leave your bet and get it back when the next round is “red”. The expected value of such even bets increases to 0.9865 or €98.65 per €100. With this rule, a game with €100 bet “costs” you so on average €1.35 (see also here for further explanations).

## Table probabilities for individual chances

 Bet Number of fields Profitability Even chances 18 48.6% Dozens, columns 12 32.4% Transversale simple 6 16.2% Carré 4 10.8% Transversale pleine 3 8.1% Cheval 2 5.4% Plein 1 2.7%

## Probabilities of evens bet consequences

When five times in a row comes “red”. What do you think comes next? Red or black? Many think that now the probability of “black” is higher. After all, it has been a long time since black, so “finally” has to be black again in order to “balance the probabilities”. But this is nonsense. Even if red came 5 times in series, the probability that now comes black is still 18/37 = 48.6%. Remember: “The roulette ball has no memory!”. It does not know where she landed previously.

That is why it makes no sense at all to record the sequence of red and black. Sure, you’ve seen gamblers who record the results in the casino. Although, it is not possible to detect the smallest irregularities in the roulette wheel in one evening (and not even in a year), the ball has no memory and will always land by chance on a certain field.

Although it is very unlikely to get 6 times in a row red. The chance for that is only (18/37)^6 = 1.3%, which is about half as much as a single number with a stake (1/37 = 2.7%). But, it is just as improbable that first comes 5 times red and then black. Again, the probability is 1.3%.

### Examples of probabilities

To give you a better understanding about roulette probabilities, we have put a few examples here. “R” stands for red and “b” stands for black.

• r oder b: 18/37=48.6%
• rr oder bb: 18/37 x 18/37 = 24% (red two times in a row)
• rb oder br: 36/37 x 18/37 = 47.3% (once red and once black)
• rrr oder bbb: 11.5%
• rrrr oder bbbb: 5.6%

Not to be confused are probabilities with the “conditional probabilities”. Thus the chance of 4 times red in series is only 5.6%. However, if you observe at the table as the ball rolls on a red number, then the probability of 3 times red in series (ie a total of 4 times) is 11.5%. Finally, you only consider the 3 series.

## Law of small numbers

The term “law of small numbers” refers back to the book of the same name by Ladislaus von Bortkiewitz. The Russian-German private lecturer lived from 1868 to 1931 and wrote his work in 1898. The book is free of copyright and can be downloaded here as a PDF or Kindle version. His law of small numbers is presented briefly in the following:

If the roulette 37 rotations are carried out, so exactly as many different numbers exist. Now, according to this “law of the third” or “two-thirds law”:

• about 1/3 of the numbers, i.e. about 12-15 numbers did not showed up at all
• about 1/3 of the numbers are hit exactly once
• About 1/3 of the numbers are hit twice or more

The name “two-thirds law” comes from the fact that in a rotation, i.e. in 37 games, about 2/3 of the numbers will be hit at least once.

## Law of large numbers

The law of large numbers states that the measured frequency with increasing number of passages approximates the theoretically calculated probability. Therefore, it is possible to see in one evening for example, 60x red and only 40x black. If you observe the distribution of red and black for a longer period of time, then this ratio becomes more and more the value 50:50. There is no roulette system which can increase the probabilities even a bit. For a short time it may seem as if the system works. In the long run, however, you will lose money with every system because the roulette rules are designed so that the casino earns money.

We hope the article about roulette probabilities has helped you. Now it’s time to practice. For this you need a serious and decent online casino. And since you will practise a lot at the beginning, we recommend you to choose a casino with a free bonus. At 888 Casino you get €88 for free after registration. You can withdraw that money later if you win with it. So just sign up here and put the roulette theory into practice!

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