In roulette, calculating expected value is relatively simple for a few reasons. First, almost all of the bets on a roulette table have the same expected value, simplifying the process significantly. Secondly, roulette is straightforward from a gaming perspective: you place one bet, one event happens (the spin of the wheel), and then the bet either loses or is paid out.
The following section will teach you everything you need to know about expected value. Then, we’ll move on to showing you what the expected value is on each bet in both European and American roulette.
For most bets, calculating your expected value (or EV) is simple. It can become harder in games where you don’t have all the information you need to make the calculations; in games like that (such as poker), expected value often requires a little estimation along with the hard math.
In order to make an expected value calculation, you’ll need to know four things: the probability of winning the bet, the amount you’ll win if you the bet is won, the probability of losing the bet, and the amount you’ll lose if the best is lost. You can plug actual dollar amounts into the calculations if you want to know the expected win or loss on a particular bet (as we will do) you can just use “units” for more general calculations that cover bets of any size.
Here’s an example. Imagine you are flipping a coin with a friend. However, the payouts aren’t quite fair: you only have to bet $10 on each flip, while your friend has to bet $11. What is the expected value for you?
Well, we know that you have a 50% chance of winning, and when you win, you’ll win $11. We can multiply those numbers together to come up with a total of $5.50. We also know that you’ll lose half the time, and will lose $10 each time you lose. $10 multiplied by .5 is $5. We can then take those two figures, subtract the average loss from the average win, and find that you expect to make a profit of $0.50 on each flip. The math looks like this:
($11 * .5) – ($10 * .5) = $0.50
This is the same math used in any EV calculation, though things can get more complicated if you have to include the possibility of winning different amounts or that the bet could push.
Let’s start with betting on a single number. If a single number bet wins, it pays out at odds of 35-1. However, there’s only one winning result out of a possible 37 (the 36 numbers along with the single zero). That means the odds of winning on a single spin are 1/37, or 2.70%. Meanwhile, the odds of losing on any given spin are 36/37, or 97.30%. So the math looks like this:
(35 * .027) – (1 * .973)
.946 – .973 = -0.027
Or a 2.7% advantage for the casino. We’ve actually engaged in some rounding in the numbers above, but you may be able to figure out that 0.027 is also 1/37 — or the number of zeroes on the wheel divided by the total number of pockets. This is why it is often said that the casino earns its advantage from the zero.
It’s simple to show that this expected value is the same on every bet. Take, for instance, a corner bet, which pays out 8-1 if any of four numbers covered by the bet win. The math:
(8 * .1081) – (1 * .8919)
.865 – .892 = -0.027
Or, once again, a 2.7% edge for the house.
The expected value is somewhat higher for the casino (or worse for the player) in American roulette. This is due to the fact that while the payouts are the same, there is an extra number on the wheel in the form of a double zero (00). This makes the game inferior for players. You won’t see this version of the game in Australian casinos, but it is available in online casinos. We don’t recommend you play it when European roulette is available, and here’s the math that explains why.
Let’s take that same single number bet we looked at above. The payout is the same (35-1), but the odds of winning are lower, since there’s now only one winner out of 38 numbers. Meanwhile, you’ll now lose 37 times out of 38. That’s seems like a small change, but it makes a significant difference when it comes to the expected value:
(35 * .0263) – (1 * .9737)
.921 – .974 = -0.053
Or a 5.3% edge for the casino. The expected value is actually slightly better than this; we rounded a bit there, and the house edge for the casino in American roulette is actually closer to 5.26%. Once again, this is exactly equivalent to 2/38 — the number of zeros divided by the total number of pockets on the wheel. This is almost double the house edge as European roulette. From the player’s perspective, this means that while both games have a negative expected value, the EV in European roulette is still much better than the American version.