Roulette is one of the simplest games in the casino. There is only one event to consider, which is the winning number. So calculating the expected value is relatively simple.
Calculations for Expected Value
To determine the expected value, you need the probability of the event being tracked. You also need the amount you win if you predicted the correct number, and the amount lost on any losing spins. And finally you need the probability of losing your bet.
A Simple Example
Let’s consider a 50-50 game such as a coin toss. Say you are playing against a friend. You could bet ten dollars on each flip of the coin, and your friend bets eleven dollars. Now let’s calculate the expected value for you:
Firstly we recognize we have a 50% chance of winning. When you win, you win eleven dollars. Also keep in mind you will lose half the time. Each loss will cost you ten dollars.
The math is:
($11 x 0.5) – ($10 x 0.5) = $0.50
So it’s the amount you win times the probability, minus the same for your friend.
European Roulette Expected Value
European Roulette only has a single Green 0. The expected value for players is significantly better than the American 00 wheel.
Let’s consider the simplest bet possible, which is one unit on a single number:
If you bet on a single number and win, the payout is 35 to 1. This means a win gives you back your original bet, and 35 additional units (36 total).
There are also thirty-seven pockets on the single zero European wheel. The odds of losing are 36/37 spins, because you are only betting one number, and not betting on thirty-six other numbers. Expressed as a percentage, this is 97.30% or 0.973. Here’s the math:
(35 x .027) – (1 x 0.973)
0.946 – 0.973 = -0.027
And the -0.027 is where we get the 2.7% house edge. A simple way of looking at it is 2.7% is the same as 1/37.
Now let’s consider a different type of bet. For example, a square bet which pays 8 to 1. This type of bet covers four different numbers. The probability of winning is 4/37, which expressed as a percentage is 0.1081. Since this is the probability of winning, the probability of losing is 1 – 0.1081 = 0.8919.
Now the full equation is:
(8 x 0.1081) – (1 x 0.8919) = 0.865 – 0.892 = -0.027
So we can see even with a different type of bet, the house edge is still 2.7%.
Expected Value For American Roulette (Double Zero)
The payouts for American 00 Roulette are exactly the same as European Roulette. The one significant difference is that American roulette wheels have an additional pocket. The 00 wheels are most common in the United States of America, although also see service in various countries including Korea. American roulette wheels are quite rare in European casinos. If you ever have the choice of playing American or European wheels, European with the single zero is the obvious choice.
Now let’s repeat some of the equations above, but with the additional pocket.
(35 x 0.0263) – (1 x 0.9737) = 0.921 – 0.974 = -0.053
So this is a 5.3% house edge, which is nearly double the house edge of European roulette. As with our European Roulette example, we can obtain the same value with the fraction 2/38.
What the Difference In Edge Means
As the house edge is nearly double for American roulette, you basically lose money twice as fast. Let’s take a more detailed look.
With American roulette, there are 38 numbers. Let’s say you bet on one number for 38 spins. Statistically, you will win 1 in 38 times. On your one win, you will be paid 35 units, plus the original unit you bet. So after 38 spins, if you had bet on every spin, you will end up with thirty-six units. The ratio is 36/38.
Now with European roulette, there are thirty-seven numbers. And if you bet one number for thirty-seven spins, you can expect one win. So after the thirty-seven spins, you will be left with thirty-six units. The ratio is 36/37.
To summarize, after around thirty-seven spins on the European roulette wheel, you will be down about one unit. After around thirty-eight spins on the American wheel, you will be down two units. Understand the house edge is typically small and has a long-term effect. But those small differences in numbers can make a big difference over time. Again it helps to see a casino demo so you have a practical sense of how it works, and what to expect.
Let’s consider an example on the European roulette wheel. Again there are thirty-seven numbers, and statistically we expect to win one in thirty-seven spins if we are betting a single number. But this doesn’t mean we will specifically win every thirty-seventh spin. Sometimes we may win on consecutive spins. Sometimes we may not win once in two hundred spins. But on average, we will win about one in thirty-seven spins.
The difference between the average, and the deviation from averages, is basically variance. Another way of looking at variance is that it’s basically good or bad luck.
You may assume that casinos always win. But this is not the case. Even casinos can and do have losing days. For example, a player may have a lucky large win of two hundred thousand dollars. The win may occur on an individual table. In fact it may take this particular table several days to earn back the money for the casino.
Likewise, a professional roulette player with an advantage can still lose. Anything can happen in the short term. Even if you have an extremely high edge, the ball may land in the correct sector, but not specifically on the number you bet.
But probably one of the biggest mistakes made by system players, is that they think short-term winnings indicate their system works. The fact is even random bets can and do win for short periods of time. Only long-term results should be used to determine the viability of a system or strategy.