Chapter 3
Roulette
Roulette is one of the simpler games available in modern casinos and has captured the popular imagination like no other. In fact, the game has been featured in countless movies such as Humphrey Bogart's 1942 Casablanca, Robert Redford's 1993 Indecent Proposal, and the 1994 German movie Run, Lola, Run.
3.1 Rules and Bets
Roulette is played using a revolving wheel that has been divided into numbered and color-coded pockets. There are 38 pockets in the American roulette (popular in the United States), or 37 pockets in the European roulette (common in Monte Carlo and other European locations); see Figure 3.1. The croupier (as the casino employee in charge of the table is known) spins the wheel and a small ball in opposite directions. The outcome of the game depends on the pocket where the ball falls.
Figure 3.1 The wheel in the French/European (left) and American (right) roulette and respective areas of the roulette table where bets are placed.
Bets in roulette are placed by moving chips into appropriate locations in the table. Roulette bets are typically divided into inside and outside bets. Outside bets derive their name from the fact that the boxes where the bets are placed surround the numbered boxes.
The simplest inside bet is called a straight-up, which corresponds to a bet made to a specific number. To place this bet, you simply move your chips to the center of the square marked with the corresponding number. The payoff odds from a straight-up bet are 35 to 1, which means that if your number comes up in the wheel, you get your original bet back and get a profit of $35 for each dollar you bet. Other inside bets, such as the split or the street, are described in Table 3.1.
The simplest of the outside bets are the color (or red/black) bets, and the even/odd bets. As its name indicates, you win a color bet if the ball falls on a pocket that has the same color as the one you picked. Similarly, you win an even bet if the number that comes up in the roulette is a nonzero even number. In both cases, the payoff odds are 1 to 1, so they are often called even bets. However, as we will see below, these even bets are not fair bets because the winning odds are not 1 to 1. A list of outside bets is presented in Table 3.2. This list corresponds to the bets and payoffs most commonly used in the United States; some casinos allow for additional bets, or might slightly change the payouts associated with them.
Table 3.1 Inside bets for the American wheel
| Bet name | You are betting on… | Placement of chips | Payout |
| Straight-up | A single number between 1 and 36 | In the middle of number square | 35 to 1 |
| Zero | 0 | In the middle of the 0 square | 35 to 1 |
| Double zero | 00 | In the middle of the 00 square | 35 to 1 |
| Split | Two adjoining numbers (horizontally or vertically) | On the edge shared by both numbers | 17 to 1 |
| Street | Three numbers on the same horizontal line | Right edge of the line | 11 to 1 |
| Square | Four numbers in a square layout (e.g., 19, 20, 22, and 23) | Corner shared by all four numbers | 8 to 1 |
| Double street | Two adjoining streets (see Street row) | Rightmost on the line separating the two streets | 5 to 1 |
| Basket | One of three possibilities: 0, 1, 2 or 0, 00, 2 or 00, 2, 3 | Intersection of the three numbers | 11 to 1 |
| Top line | 0, 00, 1, 2, 3 | Either at the corner of 0 and 1 or the corner of 00 and 3 | 6 to 1 |
From a mathematical perspective, the game of roulette is one of the simplest to analyze. For example, in American roulette, there are 38 possible outcomes (the numbers 1–36 plus 0 and 00), which are assumed to be equiprobable. Hence, the probability of any number coming up is 1/38. This means that the expected profit from betting $1 on a straight-up wager is
Note that the number is negative. Therefore, in the long term, you lose about 5 cents on each dollar you bet. This number is called the house advantage, and it ensures that casinos remain a predictably profitable business (remember the law of large numbers for expectation from Chapter 2).
Take now an even bet. There are 18 nonzero even numbers; therefore, the expected profit from this bet is
Table 3.2 Outside bets for the American wheel
| Bet name | You are betting on… | Placement of chips | Payout |
| Red/black | Which color the roulette will show | Box labeled Red | 1 to 1 |
| Even/odd | Whether the roulette shows a nonzero even or odd number | Boxes labeled Even or Odd | 1 to 1 |
| 1–18 | Low 18 numbers | Box labeled 1–18 | 1 to 1 |
| 19–36 | High 18 numbers | Box labeled 19–36 | 1 to 1 |
| Dozen | Either the numbers 1–12 (first dozen), 13–24 (second dozen), or 25–36 (third dozen) | First 12 boxes (first dozen), second 12 boxes (second dozen), third 12 boxes (third dozen). | 2 to 1 |
| Column | Either on 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34 (left column) or 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35 (middle column) or 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36 (right column) | Marked box below the corresponding column | 2 to 1 |
This same calculation applies to odd, red, black, 1–18 and 19–36 bets. On the other hand, for a split bet, we have
and for a street bet
As a matter of fact, the house advantage for almost every bet in American roulette is the same (
). Among the bets discussed in Tables 3.1 and 3.2, the only exception is the top line bet, which is more disadvantageous than the other common bets:
Indeed, it is more disadvantageous to play a top line bet than to simultaneously bet in each of the numbers included in it using straight-up bets! To see this, consider betting $5 on a top line bet versus betting $1 simultaneously on each of the numbers in the top line bet (0, 00, 1, 2, 3). Due to the properties of expectations (recall Chapter 2), the expected profit from the first wager is
while, the expected profit from the second bet is
This means that with the top line bet you lose on an average 50% more even though you are betting the same amount of money to exactly the same numbers!
Although most bets in roulette are equivalent in terms of their expected value, the risk associated with them differs greatly. Take, for example, the straight-up bet:
while the variance of a color bet is
These calculations highlight that the risk associated with the color bet is much smaller than the risk associated with the straight-up bet. To verify this intuition, let's simulate 10,000 spins of an American roulette and plot the running profit associated with betting $1 every time on each of the two bets:[
Figure 3.2 Running profits from a color (solid line) and straight-up (dashed line) bet.
Figure 3.2, which shows the results from these simulations, is consistent with the discussion we had before: although the average profits from both bets eventually tends to converge toward the expected value of 
, the straight-up bet (which has the highest variance) has much more volatile returns.
What should you do next time you visit a casino? It depends on what your goal is, and how much money you have in your bankroll. If you want to play for as long as possible before exhausting your money, you should only play color bets (or bets with similar payouts such as the even/odd bet or the 1–18 or 19–38 bets). On the other hand, if you want to maximize the amount of money you can potentially make from a single bet, then you should play only straight-up bets (just like the main character in Run Lola, Run). However, in that case, you are also maximizing the probability that you will go bankrupt very quickly. There are no free lunches…
European roulette can be analyzed in a similar way. Payout odds are the same as in American roulette, but now the probability of a single number is 
. Hence, the expected value of a straight-up bet is in this case
This calculation shows that the house advantage for straight-up bets in American roulette is almost twice as large as for European roulette. If you have a choice, you should always prefer roulette in Europe rather than in the United States.
3.2 Combining Bets
Sometimes players like to place multiple bets simultaneously on the same spin of the roulette wheel. As an example, consider placing $2 on a bet of red and $1 on the second dozen. The payout of this simultaneous bet is going to be different depending on the number that comes up. If a red number among the second dozen comes up (such as 16), you win both bets and you get back the $3 you originally bet plus a profit of $4 more (recall that the payoff odds for reds is 1 to 1, which means that you profit $1 for each dollar you bet, while the payoff of a dozen bet is 1 to 2, meaning that you profit $2 for each dollar you bet). On the other hand, if an odd number in the second dozen comes up you lose the even bet, but you win the dozen bet. Table 3.3 describes the possible outcomes along with their probabilities, payouts, and profits (remember that the entry cost of the bet is always $3).
Table 3.3 Outcomes of a combined bet of $2 on red and $1 on the second dozen
| Outcome | Prob | Payout ($) | Profit ($) |
| Red in second dozen (14, 16, 18, 19, 21, 23) | 6/38 | 2 + 2 + 1 + 2 = 7 | 7 3 = 4 |
| Black in second dozen (13, 15, 17, 20, 22, 24) | 6/38 | 1 + 2 = 3 | 3 3 = 0 |
| Red in first or third dozen (1, 3, 4, 5, 9, 12, 25, 27, 30, 32, 34, 36) | 12/38 | 2 + 2 = 4 | 4 3 = 1 |
| All other numbers (0, 00, 2, 6, 7, 8, 10, 11, 26, 28, 29, 31, 33, 35) | 14/38 | 0 | 0 3 = 3 |
From Table 3.3, it is easy to see that the expected profit of this combined bet is
This is the same expected profit that you would get from betting $3 on pretty much any simple bet! This result suggests that you cannot decrease the house advantage by combining bets. On the other hand, the variance of this bet is
Note that this value is smaller than the variance of the dozen bet, but larger than the variance of the color bet. Therefore, even though it will not increase your probability of winning, mixing bets allows you to tailor the risk that you assume.
3.3 Biased Wheels
So far, our analysis of the game of roulette has assumed that all numbers have the same probability. However, in reality, the wheel is a mechanical device subject to wear and tear so, in time, even the best roulette tends to slightly favor some numbers (in other words, it becomes biased). For example, this might happen because of warped axels, because of chipped/battered pockets, or simply because the wheel is not level. Alternatively, the bias might not be due to the wheel itself, but to the way the croupier spins the wheel or the ball. Whatever be the reason, a biased roulette hurts the casino only if players are aware of it, in which case they can exploit the bias to reduce (or eliminate) the house advantage.
Let's consider the analysis of a biased wheel. For the sake of concreteness, we will work with a wheel that has a slightly positive bias toward three numbers and a negative bias toward the others. A player who knows which three numbers have a positive bias can exploit it by making simultaneous straight-up bets to these numbers. For example, consider a biased wheel in which the numbers 2, 4, and 21 each have a probability of 0.028 (which is slightly larger than the typical 
associated with an unbiased wheel), while the other 35 numbers have all the same probability of 0.02617143 of coming up (which is necessarily slightly lower than 
). If the player only makes straight-up bets of $1 to each of the three numbers favored by the wheel, the profit of the bet is
which reduces to
Since the expected value is positive, the player will actually make money in the long run by betting on the numbers favored by this biased roulette! The fact a player can potentially exploit any bias in the wheel means that casinos are unlikely to bias their wheels on purpose.
Now, let's turn the previous calculation around to answer the following question: how large does the combined bias of the three numbers needs to be in order to eliminate the house advantage (and make the game of roulette fair) if the player is able to discover it? Let 
be the combined bias and assume that we bet $1 on each number. That means that the probability of winning with the simultaneous straight up bet is 
(with a profit of $ 
), and the probability of losing is 
(with a expected profit of $ 
). The expected profit is therefore
and to make the game fair we need 
to satisfy 
, or 
. Therefore, it is enough to change the probability of three numbers from 
to 
to make the house edge disappear.
We can use R to simulate a biased wheel in which the numbers 2, 4, and 21 are favored:[
Figure 3.3 Empirical frequency of each pocket in 5000 spins of a biased wheel.
Figure 3.3 presents a bar plot of the observed empirical frequencies observed in 5000 spins of a biased wheel coming out of the simulation above. Note that even with 5000 spins it is not easy to identify which pockets (if any) are favored. Indeed, from this simulation, it would appear that the wheel is biased toward the number 28!
The previous simulation highlights that, even though exploiting a biased wheel is one of the very few ways in which a player can reduce the house edge, detecting a biased wheel with a good degree of certainty is not easy and might require that we observe the wheel for a really long time (particularly if the bias is small). The exact number of spins to be recorded depends both on the size of the bias and on how certain you want to be about the existence of the bias; a rough approximation of the necessary number can be obtained using Chebyshev's theorem.
Figure 3.4 Cumulative empirical frequency for a single pocket in an unbiased wheel.
Let 
be the observed frequency of event 
after 
identical repetitions of an experiment, and let 
be the probability associated with the event 
. Then, for any desired precision 
,
Chebyshev's theorem links the number of repetitions of an experiment (in our case, the number of spins) with the error committed when approximating 
by the empirical frequency 
. We already knew (from the law of large numbers) that this error becomes smaller and smaller as 
grows, but we had no idea about exactly how fast it decreased. Chebyshev's theorem fills that gap, and it can help us determine how many spins are needed to figure out if a wheel is biased or not.
Understanding Chebyshev's theorem can be hard because there are so many frequencies and probabilities involved in the definition. To gain some intuition, consider running a large number of simulations of an unbiased wheel, each consisting of 10,000 spins. Figure 3.4 shows curves for the cumulative empirical frequency of any given pocket for each of 100 such simulations. Because these are random experiments, each curve is slightly different from the others. In spite of this, some patterns are clear. For example, you can see that the graph looks a little bit like a horizontal funnel, wider on the left size and narrower on the right.
We can relate the features of the graph with the different terms that appear in Chebyshev's theorem. For example, you can think about the width of the funnel as the error that is committed when we approximate 
by the empirical frequency 
. Hence, the width of the funnel is, roughly speaking, equivalent to 
. As expected, smaller values of 
(more precision in the estimation) requires larger values of 
, and vice versa. Furthermore, it should be clear that, for any 
, the width of the funnel is itself random. Indeed, if we do a second set of 100 simulations, the width of the funnel will be slightly different. Hence, the best we can do is to select an 
that will give us the desired width with high probability (but we can never be absolutely sure that the error is not bigger than we want). The desired high probability is something we need to decide ourselves (usually 0.95, 0.99, or 0.999 are used). Note, however, that the larger the desired probability, the larger the value of 
needs to be.
To illustrate how Chebyshev's theorem works, consider spinning a wheel 100,000 times to determine whether the probability of 00 is 
. If the wheel is unbiased, how large is the probability that the difference between the estimate we get from this experiment and the true probability is greater than 0.001 (which is the maximum error that I am willing to admit)? A direct application of Chebyshev's theorem yields:
This probability is relatively high, so we actually need a much larger number of spins to be accurate enough. How many more? Let's say that I do not want the probability of a 0.001 error to be more than 5%. Then, again from Chebyshev's theorem,
which implies that
3.4 Exercises
-
1. What are even bets in roulette? Are they really even?
-
2. Albert Einstein once said that no one could possibly win at roulette “unless he steals money from the table while the croupier isn't looking.” Explain this statement in the context of the law of large numbers.
-
3. What is the expected value and variance for simultaneous $5 street-bet on 22–23–24 and a $1 odd-bet in American roulette?
-
4. What is the expected value and variance for a simultaneous $2 square-bet on 1–2–4–5 and a $3 bet on the first column in American roulette?
-
5. What is the house advantage in European roulette for the split, color, and dozen bets?
-
6. In American roulette, almost all bets have the same expected value. However, do they all have the same variance? If you think that they do not, you should show a counterexample. If you want to play for as long as possible, what bet would you prefer?
-
7. What would be the house advantage in roulette if there were three “losing” numbers in the wheel (call them 0, 00, and 000)?
-
8. In American-style roulette, which one of the following bets has a higher winning probability? Which one has a higher expected payoff? Which one has the highest variance? Which one would you prefer, and why?
- Bet $18 on red.
- Bet $2 on a split.
-
9. In American-style roulette, which one of the following bets has a higher winning probability? Which one has a higher expected payoff? Which one has the highest variance? Which one would you prefer, and why?
- Bet $10 on a double-street.
- Bet $2 on a dozen (assume that the dozen does not contain any number from the double-street).
-
10. In the first and third column strategy in roulette, one bets two pieces in the first column, two pieces in the third column and two pieces in black. What is the expected value of this system in American roulette?
-
11. Consider a roulette wheel that is positively biased toward two numbers, 9 and 34. How large does the bias need to be in order for bets on this roulette to be fair?
-
12. Why are casinos unlikely to bias their roulette wheels on purpose?
-
13. The payoffs in roulette are selected assuming that all numbers have the same probability (in European roulette, i.e.,
). Assume that, after collecting many spins from one given European roulette, you find that three numbers, 25, 17, and 34 have a slightly higher probability of coming out (say 0.04), while the other 34 numbers have about the same probability (
). Assume that you pick a strategy where you make $1 straight up bets to each one of the high probability numbers. What would be the expected payoff from this bet? -
14. Making the same assumptions as the previous question, but now assume that the three biased numbers have a probability of 0.05, and the remaining numbers have a probability of
. Assume that you pick a strategy where you make $1 straight up bets to each one of the high probability numbers. What would be the expected payoff from this bet? -
15. How many spins of the wheel should you observe in order to be 99% sure that there is a bias of 0.0029238 in the usual probability for a single number (0.02163157) in an American roulette wheel (this is a bias that not only would cancel the house advantage but also would actually turn it into your advantage).
-
16. [R] Simulate and visualize the expected value and spread of the even bet in roulette.
-
17. [R] Simulate a biased roulette, where three numbers of your choice have a probability of occurring 0.3. Produce a histogram of the simulations.