Roulette Wheel Probability: Everything You Need to Know

roulette wheel probability

Learn how to calculate the roulette wheel probabilities to develop a strategy based on logic.

  • Roulette is one of the simplest casino table games as it simply requires you to guess the number that will win.
  • However, by understanding how to calculate the roulette wheel probability and the results, you'll be better equipped than most players to develop a logical strategy.

The Mathematics of Roulette Probability

While certainly not as fun as playing roulette, understanding the probability of a particular set of sequences requires some mathematics.

A typical European Roulette wheel has 37 numbers while an American Roulette wheel has 38. On every spin, there is a 1 in 37 or, in the case of an American wheel, a 1 in 38 chance of any number winning. Since odds are commonly shown in decimal, the odds of any number winning can be shown as 0.027 or 0.026 respectively.

However, if you're planning to play multiple rounds of roulette, as most players do, you need to concern yourself with more than just a single round. This is where mathematics may get somewhat more complex.

Understanding the Use of the Factorial Function in Roulette

The factorial function is a mathematical tool used to calculate the total number of possible combinations in a sequence of natural numbers. In roulette, this tool can help players calculate all the different sequences of results where each number on the wheel wins once.

Considering a European Roulette wheel, calculating the factorial function would require a multiplication of all natural numbers from 1 to 37. The '!' symbol after a number is used to outline the factorial function of the number, which, in this case, would be '37!'. Needless to say, the resulting value is huge and surpasses the trillions of trillions. It gets significantly bigger if you include the reasonable probability that a number will win twice or more before another number might win once.

Using an Equation to Calculate Roulette Probability

There's a mathematical equation that's used to calculate the probability of any result or sequence of results in roulette. This equation is P(e) = [n!/(x!(n-x)!)]x[P(b)^x]x[((1-P(b))^(n-x)].

This is a fairly complicated equation, but fortunately, you can find free online calculators that work out the calculation once you enter the required parameters. These parameters are:

  • n: the number of roulette rounds played
  • x: the number of times a bet wins
  • P(b): the probability that a selected bet wins in one round

How to Use the Roulette Wheel Probability Equation

roulette wheel equation.png

As with any mathematical formula, the best way to understand it is by using an example.

Let's check the probability of a single number winning just once across 37 rounds. We've already seen that all numbers in European Roulette have a 1 in 37 chance of winning, but now we want to find out what are the odds that no number wins twice in 37 consecutive spins.

In this example, n is 37, x is 1 and P(b) is 1/37 or 0.027. The result is 0.373 or 37.3%. If we amended the parameters accordingly, we could also calculate the probability that a number would not win at all in 37 spins or that it would win twice. The results are 36.2% and 18.6% respectively.

Remarkably, the odds that a number wins once or twice in 37 spins are almost equal. While such a result does not provide any guarantees about what will happen when you play a few rounds of roulette, it does explain the gambler's fallacy which many players believe. This cognitive bias tries to link random events in the past with random ones in the future. So, for example, players might believe that if a number wins consecutively or within a few rounds, it's likely to win again very soon.

Calculating the Odds of Streaks in Roulette

For the purposes of this calculation, we'll consider streaks to be a series of winning results from even-money bets. Roulette offers six such bets, namely Black/Red, Even/Odd, and Low/High.

The odds of winning a single even-money bet, such as Red, in European Roulette is 18/37. We can use the probability equation to calculate the likelihood that Red wins multiple consecutive times. For example, there is a chance of 0.115 that this bet will win three times in a row. To make this calculation, both n and x have a value of 3 while P(b) has a value of 18/37.

If you'd like to calculate the mathematical likelihood of a losing streak, you need to replace x with zero and P(b) with 19/37, since due to the zero, 19 numbers can win for this bet to lose. Entering these parameters into the equation results in a 1 in 7.38 chance of having a losing streak over three consecutive spins.

The Limitations of Roulette Wheel Probability

Similarly to the Return to Player (RTP) percentage of online slots, the roulette wheel probability is based on sound mathematical principles. However, these results are based on an infinite number of spins, and you would only get them consistently throughout millions of spins. Only then can these mathematical principles be proven.

Since you're unlikely to play so many rounds of online roulette, you should not expect any guarantees from these formulas. No matter whether you play virtual or live roulette, the outcome of each spin is random, and while it is possible to note patterns in historical results, you should not use these to try and predict future results.

Key Takeaways

  • It is possible to calculate the mathematical probability of winning and losing streaks in roulette.
  • Understanding how roulette wheel probability is calculated can help dispel the gambler's fallacy.
  • Probability results are only proven when a roulette wheel is spun millions of times.
  • Roulette is a game of chance that should be played responsibly and for fun, not as a source of financial income.

Prices quoted in copy are correct at time of publication but liable to change.