Probability and Odds in Roulette

Having explained the basic terms and the variations of the game, we move forward to discuss the mathematical side of roulette. Although some of you may ignore this topic, we should underscore that knowledge of such concepts as probability and odds is crucial before one digs deeper into the various strategic approaches to the game.

What do we mean by the term ”probability”?

If we are to offer a formal definition of the term ”probability”, it may sound as follows: ”The probability of the occurrence of an event is defined as the number of cases favorable to the event, divided by the number of equally-likely possible cases.”

In case we presume that all possible outcomes have an equal chance to occur, as they are a result of a random device (the roulette wheel), and for every single outcome there are only two possible scenarios (a success or a loss), the definition pinpointed above may be transformed into the following: ”The probability of winning equals the number of ways to win, divided by the number of possible outcomes.”

The number of possible outcomes includes the possible ways to win and the possible ways to lose. Having said this, the probability of success may be presented with a simple formula:

PROBABILITY OF SUCCESS = Ways to Win / Ways to Win + Ways to Lose

Probability always belongs within the range between 0 and 1 and can be expressed as a fraction or a decimal. If one is to present probability as a percentage, then the decimal needs to be multiplied by 100.

Let us take the simple example, when tossing a coin. As the coin is two-sided, it has one way to win and one way to lose. Therefore, in this case the probability of success will be as follows:

PROBABILITY OF SUCCESS = 1 / 1 + 1 = 1 / 2 = 0.5, or 50%

An expert in statistics would interpret the probability of success when tossing a well-balanced coin as one half. An expert in gambling, however, would interpret it as 1 chance out of 2. We would interpret it as 50 percent.

Let us have another example – the tossing of an ordinary six-sided die. As there is one way to win and five ways to lose, the probability of success (rolling a specific number) will be as follows:

PROBABILITY OF SUCCESS = 1 / 1 + 5 = 1 / 6 = 0.1667, or 16.67%

The expert in statistics would now interpret the probability of rolling the particular number as one sixth, while the expert in gambling would say it is 1 chance out of 6. We would interpret it as 16.67 percent.

If we take the French roulette wheel, having 37 numbers, and we place a straight-up bet, then we have 1 way to win and 36 ways to lose. Therefore, the probability of success (the number we selected to win) will be as follows:

PROBABILITY OF SUCCESS = 1 / 1 + 36 = 1 / 37 = 0.0270, or 2.70%

If we take the French wheel again and place a street bet on a row of three numbers, then we have 3 ways to win and 34 ways to lose. Thus, the probability of success (one of the three numbers to win) will be as follows:

PROBABILITY OF SUCCESS = 3 / 3 + 34 = 3 / 37 = 0.0810, or 8.10%

And lastly, if we place a dozens bet on the French wheel, then we have 12 ways to win and 25 ways to lose.

PROBABILITY OF SUCCESS = 12 / 12 + 25 = 12 / 37 = 0.3243, or 32.43%

Bet Payout Probability(American) Probability(European)
Straight Up 35 to 1 2.63% 2.7%
Split 17 to 1 5.26% 5.41%
Street 11 to 1 7.89% 8.1%
Corner 8 to 1 10.53% 10.81%
Line 5 to 1 15.79% 16.22%
Dozens 2 to 1 31.58% 32.43%
Columns 2 to 1 31.58% 32.43%
Red/Black 1 to 1 47.37% 48.65%
Even/Odd 1 to 1 47.37% 48.65%
Low/High 1 to 1 47.37% 48.65%

What do we mean by the term ”odds”?

The majority of players at a casino will use the term ”odds” instead of ”probability”. There is a certain difference between odds and probability and it lies in the way the two terms are mathematically expressed. Once we have calculated the probability of success, we can easily estimate the odds for success by using the following formula:

ODDS FOR SUCCESS = Ps / 1 – Ps, where

Ps equals the probability of success.

If we go back to the coin tossing example, then we will get the following result:

ODDS FOR SUCCESS = 0.5 / 1 – 0.5 = 0.5 / 0.5 = 1 / 1

Odds are usually not transformed into a decimal, but are rather presented as a ratio of whole numbers. In the case above we simply multiplied the numerator and denominator by 2, so that we come up with the result we wanted. Odds are presented as 1:1 and pronounced as ”one-to-one”, or ”even odds”.

We should note that there is another way to calculate the odds for success. There is no need to estimate the probability first and then the odds. We can get straight to the desired result by using the following simple formula:

ODDS FOR SUCCESS = Ways to Win / Ways to Lose

If we go back to the die tossing example and use the above mentioned formula, we will get:

ODDS FOR SUCCESS = 1 / 5, or

the odds are 1:5 (one chance to be successful and five chances to lose). This is different from the probability of success, which, as we estimated, is 1 chance out of 6. And it is what causes confusion.

Bets Payout Probability
Even 1:1 48.6%
Odd 1:1 48.6%
Black 1:1 48.6%
Red 1:1 48.6%
1-18 1:1 48.6%
19-36 1:1 48.6%
1-12 2:1 32.4%
13-24 2:1 32.4%
25-36 2:1 32.4%
Single Number 35:1 2.7%
Combination of two numbers 17:1 5.4%
Combo of three numbers 11:1 8.1%
Combo of four numbers 8:1 10.8%
Combo of six numbers 5:1 16.2%

People may end up confused, also when the odds are presented in reverse. If the odds for success are 1:5, then the odds against success are 5:1. It is so, because we have the following formula:

ODDS AGAINST SUCCESS = Ways to Lose / Ways to Win

If the two above mentioned odds are closer to even, it becomes a more delicate situation. If we have odds for success of 5:6, then the odds against success will be 6:5. An experienced player will surely see the difference, while the beginner may easily be misled.

House odds (odds paid) represent the opposite of odds for success. This explains why the lower the chance of success is, the higher the payout is. As the house is playing against the player, the odds paid equal the odds against success, if, of course, we presume that the casino does not take a share. The reality is that house odds are always set a bit lower than the odds against success, which provides casinos with a profit margin.

Let us get back to the examples concerning the French roulette wheel. If we place a straight-up bet, we have one way to win and 36 ways to lose.

ODDS FOR SUCCESS = Ways to Win / Ways to Lose = 1 / 36

Or, the odds for success are 1:36 and the odds against success are 36:1. If we indeed emerge successful with our bet, the casino will pay us 35:1, not 36:1. The obvious difference represents the house advantage (a 1-unit profit).

If we are to place an even-money outside bet (black), we have 18 ways to win and 19 ways to lose (18 red numbers and one green single zero).

ODDS FOR SUCCESS = Ways to Win / Ways to Lose = 18 / 19

Or, the odds for success are 18 to 19 and the odds against success are 19 to 18. If we indeed win, the casino will pay us 18:18 and not 18:19. Again the casino will have one unit of profit.

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