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Roulette by the Numbers

It’s a wheel game with easy-to-understand mathematics, so why is it so misunderstood?

Roulette by the Numbers

To stress the importance of knowing numbers, I usually begin my game protection sessions in table games with a simple gambling question. “What is the mathematical house advantage on your double-zero roulette game?” I start out explaining that anyone who has been in the casino gaming field for more than five years can easily be considered a professional and should be able to answer a basic question such as the exact mathematical house edge in roulette.

I’m not surprised when no one can come up with the correct answer, which is 5.26 percent, because people in our profession rarely take time to learn the different percentage numbers of the very precious edge casinos hold over the player. I find it amazing that in a field ruled so much by numbers, most people, executives included, have only a general idea, if that. The standard reply: “I don’t know exactly, but I’ll bet it’s pretty high.”

How do I come up with the percentage 5.26 percent? Roulette is easy to calculate because it’s straight mathematics. You can figure it out using a hand calculator, but I personally prefer using an Excel spreadsheet.

First, one needs to develop a clear picture of the logic behind the math. On a double-zero roulette wheel there are 38 possible pockets for the ball to land (36 numbers plus the zero and double-zero). If a person were to wager on one number straight up to hit, that person has one opportunity for success and 37 chances to fail. If the ball were to fall into the losing pockets, the person loses one chip, while a ball dropping into the wagered number’s pocket provides the player with a 35-chip windfall.

Chart 1 illustrates how the 5.26 percent mathematical house advantage is calculated. The “probability” numbers are the decimal point equivalent to 1/38 and 37/38. It’s simple to see that by multiplying the payout result by the probability, one can calculate the return. By adding the two returns together, the positive and negative return numbers will produce a total return when one unit is wagered. This decimal result is easily converted to a percentage by moving the decimal point two positions to the right and adding the percentage sign.

Chart 1-Advantage of 1 Number Wagered

What about the other wagers on the double-zero wheel which are subject to a host of different payouts? Believe it or not, all but one of the possible wagers experience the same mathematical edge.

Chart 2 recaps several of the other wagers including the five-number wager when covering the 0, 00,1, 2 and 3. The five-number wager is an “odd fellow” compared to all other roulette wagers, in which payouts are symmetrical to the probabilities. When examining the properties of a single-zero wheel, this same unconforming wager situation does not occur.

How then does this common mathematical advantage apply to the actual situation on the game itself? Basically, no matter where you wager your chips (bar the five-number wager), the house advantage remains the same. There is no difference between the player who sticks to precise bet placements and the player who tosses a handful of chips onto the table and has the dealer place them as close as possible to the wager in which they landed.

Question: Since all wagers (bar the five-number bet again) are mathematically the same, what constitutes a “tough” roulette player? If all bets are subject to the same house advantage, there is no such thing as a tough roulette player. Usually, the casino executive making this comment is referring to a person who used some type of money management scheme and is rated “tough” since it takes a long time to grind him or her out of their money.

Chart 2 -Advantage of Other Numbers Wagered

Across the Pond…

In Europe, the more popular form of roulette is the single-zero wheel. On both the single-zero crown and the corresponding layout, the double-zero pocket and number have been eliminated. Note: There is a noticeable difference between the Western double-zero wheel crown and the European single-zero wheel configuration.

Based on this premise, how would that change the mathematics on the European wheel? Can you figure it out?

The difference between single-zero and its double-zero cousin lie in the probabilities. Where the double-zero offers 38 possible pockets, the single-zero has one less with 37. If you analyze the math on the other remaining wagers, you will see that they all are subjected to the same 2.7 percent house edge as the straight-up wager.

Chart 3 – Single-Zero Wheel

To confuse the European situation even more, most roulette games in France are subject to an additional rule known as “en prison” or “la partage.” This rule situation occurs when the player wagers on any even-money bet (black/red, even/odd, or first/last 18) and the ball lands in the zero pocket. In some casinos, the player receives half the original wager back, and in other casinos, the bet is placed in “prison” and is subject to a second roll. On the second roll, if successful, the player is returned the original bet, and if not, it is taken. This rule reduces the mathematical advantage of all even-money wagers on the single-zero wheel to approximately 1.38 percent.

Don’t forget “quadrant” wagering! Many single-zero layouts include the ability to wager on sections of numbers located on the wheel head. These quadrants, known as (in English) “Neighbors of Zero,” “Zero Game,” “Thirds of the Wheel” and “Orphans,” pay in respective multiples per unit wagered.

In addition, there are several variations of these sections wagers which depend primarily on the casino and the clientele. The numbers covered and the different payoff odds are an entirely different article; however, the mathematical house advantage is the same as the rest of the single-zero wagers at 2.7 percent. The area for wagering these quad options appears on the layout close to the dealer and looks like an elongated racetrack.

Side Note: For those of you in North America concerned with being attacked using a computer to “clock” the properties of the wheel head, i.e., the spin of the crown and ball to determine the approximate landing area of the ball, Europe is where it is happening. There are three reasons for that: slower crown head speeds, lower house advantage to overcome, and the ability to wager quadrants. I’m not saying it cannot happen in North America, but why travel that great a distance to the states to take on a much more difficult game?

Estimating the Unknown ‘Oops’

TCSJohnHuxley’s Blaze Roulette at Okada Manila

Ever wonder what mistakes in payouts do to your roulette game? I don’t mean the effect of mistakes we are apt to catch; I’m referring to the mistakes made when the dealer calls out a payout, and the floor supervisor says “go ahead” while inputting rates into a computer screen several tables away.

Several years ago, I investigated an alleged roulette incident for a regulatory agency of one of the European countries. My mission was to determine if roulette dealers at a certain higher-limit casino were purposely overpaying certain customers on winning outcomes.

After reviewing several hours of video footage, I determined that no misappropriation of funds was evident. However, I did observe payout mistakes made by different dealers. In my opinion, the mistakes were not as numerous as they could have been based on the level of play and chip action, but it begs the question, what do these occasional mistakes cost the casino? How much does the occasional mistake affect the overall mathematics of the roulette game?

A majority of the mistakes happen when either an under-experienced dealer or a dealer buried in a high chip action game makes calculation errors. Errors that are common are overpayment that results from addition errors or from using the wrong “key” to estimate the total payout. Other than incorrectly calculating the total payout of the same color, mistakes are made by overestimating the correct number of 20-chip stacks needed for the payout. It was not uncommon to witness payouts made where the dealer inadvertently overpaid with one or two additional 20-chip stacks.

The costliest mistakes occur when stacks are converted to larger casino value chips. Dealers are usually accurate when converting color stacks to casino value chips with wagering levels they are familiar with, but they seem to stumble with unfamiliar conversions usually involving higher-value color chips and higher-denomination casino value cheques. On a busy game, the chances of pushing out too much money are much greater than doing so when the business level is more manageable. This problem happens to even the most experienced dealers when they get overloaded with bets and subsequent payouts.

What about the mistakes accidentally made in the casino’s favor?

Wouldn’t those mistakes help balance out the losses to some extent? The answer is that they will to a point, but the players are more likely to spot and alert the dealer or floor supervisor to mistakes made against them, and are somewhat reluctant to mention payoff errors made in their favor.

So what is the cost? If all mistakes made by dealers resulted in a 1 percent overpay of total payoffs, roulette’s mathematical advantage would drop from 5.26 percent to 4.34 percent. Using this error estimation method, a 5 percent mistake rate would render the game almost even money. Remember, it will be the bigger payouts, not the smaller ones, that will be prone to errors.

This begs the question, why don’t casino managers want calculators on the tables? I posed the same question years ago when I was still working as a floor supervisor. Why not have a calculator available so the floor can accurately estimate and confirm a multi-wager total payoff or a casino value chip conversion?

It was explained at that time that using a calculator would reduce the skills level of the dealer and floor person for totaling and converting payouts, and would be more detrimental to the game than helpful. At that time, I could see the point, but today I must rethink that logic. By reducing mistakes, the industry would increase win, which improves gaming revenue. If reducing payout errors in roulette were to increase revenue annually per casino by 1 percent of total wagers handled, wouldn’t it make sense to develop an “on table” calculator that accurately totals and converts?

Is Your Wheel on the Square?

Confiscated roulette computers that purport to be able to track where the ball is going to land

A casino-quality roulette wheel head is considered a tool of precision that is used to determine a random outcome. But what happens when your roulette wheel is no longer a precise instrument? Would you know it was no longer accurate? Back in the late 1980s, professional gambler Billy Walters attacked a number of roulette tables in Nevada and Atlantic City based on wheel head shortcomings known as wheel bias. Between 1986 and 1988, Walters played over a dozen different roulette wheels in Lake Tahoe, Las Vegas and Atlantic City, and beat those games for over $6 million.

A team of people employed by Walters tracked the numbers spun on several wheels and his statisticians analyzed the results in search of numbers that “hit” more often than probability dictated. Once these bias roulette wheels were located, Walters approached each casino, placed money on deposit in the cage, and convinced management to raise the straight-up number limit. From what I understand, Walters never had a losing session, and only quit play when the casino had enough and closed the game.

How can a casino determine if their roulette wheel head is still the precise piece of equipment they originally put on the casino floor? First, the casino needs to record the results of anywhere from 4,000 to 10,000 spins. Next, management needs to determine the average number of hits per pocket and measure the plus or minus difference from that average for each pocket.

The next step is to determine the average or standard deviation statistically allowed for the total number of spins observed. Once this average is determined, the standard deviation for each pocket from the norm, both plus and minus can be determined, and any deviation greater than -2 or +2 from the average norm should be called into question.

Determining if the entire wheel is “bad” is another matter. When totaling all the different plus and minus deviation numbers from the norm, the total will not tell you anything is wrong or normal because the sum will always be zero. It doesn’t matter the range of the different number deviations; for every seriously plus deviation there are equal minus deviations.

Enter the Chi Square Goodness of Fit test. Chi Square is the process of “squaring” or multiplying each pocket’s standard deviation by its plus or minus number. Squaring the number will then remove the -/+ sign and produce integers that can be added together. The sum of these integers will produce a number that will indicate whether the roulette wheel head is random or contains problems on the wheel head crown that could result in wheel bias attacks. For instance, one should investigate any wheel head that exceeds a Chi Square sum of 90 as being a problem. The roulette wheel head Walters attacked in Lake Tahoe in 1986 had a calculated Chi Square sum of 142.

The Chi Square Goodness of Fit test is considered the “gold standard” of testing for all kinds of problems surrounding the randomness of the wheel head. Not only will it determine if standard wear and tear has created a problem, Chi Square will also determine if the wheel head has been altered by a customer or a dishonest employee. Chi Square tests should be used to check all wheel heads at least once every year.

Many of the roulette score board or totem manufactures provide a Chi Square test within their equipment’s software package. Management should contact the manufacturer and either install or activate the software. You may be able to view real-time results on a continuing basis with a simple command stroke of your computer. If there isn’t a Chi Square test software available, please contact me and I will provide you with the Excel spreadsheet I use for that specific purpose (

Note: Don’t forget to conduct the same test on your electronic table games (ETGs) as well. Some of the mechanical wheel heads used in the electronic platforms are not constructed with the same precision as the tabletop devices. You may find problems that could be exploited where management would not expect to look.

Back to the Beginning

What’s up with the roulette variation known as triple-zero roulette? Where the Europeans are offering roulette games with a lower mathematical advantage, several North American casinos have decided to take a different direction. By adding a third zero to both the wheel head and the layout, the mathematics change once again, and this time the advantage increases.

Chart 4 – Triple-zero Wheel

Well, well. The casino industry has taken a game with a monstrous mathematical advantage and found a way to make it even stronger with an edge of 7.69 percent. Approximately 2.4 percent higher than the standard double-zero wheel, and 5 percent higher than the single-zero variation (without the “en prison” rule). This game has been out on the casino floor in Las Vegas for a dozen years, and the initial feedback I received is that play definitely caters to the lower-limit clientele.

One Las Vegas Strip casino’s philosophy for the triple-zero wheel is to keep offering it until the high house advantage hits a “pain point” with the off-the-street roulette players. In other words, they are going to keep offering it until the novice roulette gamblers wise up. This might work well in a destination-resort location like Las Vegas with a consistent turnover of novice roulette players, but what about a casino where most customers are local players?

Please remember an important fact about casino gambling—it’s another form of entertainment for the adult population. Not only is your casino competing with other casinos in your immediate location, your casino is also competing with other leisure venues for the customer’s entertainment dollar such as nightclubs, theaters, sporting events and concerts. Most gamblers enter a casino believing that they will not walk away a winner. They are there to enjoy the thrill received by placing something of value in jeopardy on an unpredictable outcome. If the casino raises their edge so high the customers wagering on the games do not get their anticipated “bang for their bucks,” next time they may opt to spend their entertainment dollars seeing a good movie or cheering for their favorite sports team.

As former Nevada Gaming Control agent, casino operator, professional card counter and casino consultant, Bill Zender has been involved in various areas of gaming and hospitality since 1976. He has instructed courses on game protection, card counting, advantage play and gaming operations at various colleges and institutions throughout the country. As a member of JMJ, Inc., Zender was an owner and operator of the Aladdin Hotel and Casino. He has additional operational experience in card room casinos in California and is considered an expert in Asian gaming. For more information, visit

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