How to Gamble (Roulette)

This is a post from a series in my project.


Historically, many people have been drawn to the allure of roulette (the gambling game) due to its irresistible deterministic nature (a system in which no randomness is involved; it always produces the same output from a given starting condition). This research paper by Small and Tse explores the mathematics (specifically the chaotic dynamics) in a roulette wheel. They outline a short review of the history of the game and provide their own model to maximize profits. The authors believe that the “relatively simple laws of motion allow one, in principle, to forecast the path of the ball on the roulette wheel and to its final destination.” Small and Tse show that if a player knew the position, velocity, and acceleration of the ball, the player would be able to predict the outcome of the game with reasonable accuracy. They also explore how a very slightly tilted surface has an enormous effect on the results of the game. They claim their model can achieve an expected return of 18%, which is significantly higher than the commonly accepted 2.7% return from a random bet. This paper draws from chaos theory and probability theory to make its conclusions, and is a good example of how mathematics can be used in practical applications.

I found a neat YouTube video about the expected return of roulette. There’s some cheesy humor in the video, but overall, this woman clearly walks you through how you would calculate your expected winnings (or losses) in 1 game of roulette with specific parameters.

It is generally accepted that the house will pay 35 to 1 when you win at roulette. Therefore, on a wheel with 37 pockets, the expected return from one bet is:


A brief history

Roulette is the oldest casino game. Like other casino games, it is carefully designed to take you for all you’re worth. Basically what happens is: there’s a heavy wheel, machined and balanced to minimize friction, with colored notches and a plastic ball that whirls around the wheel as you spin it. The wheel is designed to spin for a really long time with a slowly decaying angular velocity. When the wheel stops spinning, the ball will too. The aim of the game is to bet on where (which numbered notch/pocket) you think the ball will land. The wheel is designed in such a way that the pockets are numbered consecutively and red and black colors alternate. I can’t summarize the game’s history better than the original authors:

The origin of the game has been attributed, perhaps
erroneously, to the mathematician Blaise Pascal. Despite
the roulette wheel becoming a staple of probability theory,
the alleged motivation for Pascal’s interest in the device was
not solely to torment undergraduate students, but rather as
part of a vain search for perpetual motion. Alternative stories
have attributed the origin of the game to the ancient Chinese,
a French monk or an Italian mathematician. In any case,
the device was introduced to Parisian gamblers in the mid-eighteenth
century to provide a fairer game than those currently in circulation.

I’ll be honest. Before reading this paper, I had no interest in playing roulette. Even worse, I had no idea even how to play. So for those of us who either live under a rock or are generally unaware of the goings-on at casinos, let’s start at the beginning: How do you play roulette? (I started at this site.) Wikipedia has a neat how-to-play article, too.

How to play roulette

  1. Figure out whether you’re playing on a European (more fair) or American (less fair) wheel. (American wheels have slots that go from 00, 0, 1, 2, … 35, 36, whereas European wheels do not feature the 00. Their numbers go from 0, 1, 2, … 35, 36)
  2. Get some $$dollarz$$ to play.
  3. Decide on a bet to place. You can bet on as little as just 1 number or as many as 18 numbers. You increase your chances of winning if you bet on more numbers.
  4. Place your bet. Where to place your bet depends on what kind of bet you decided to pick in the first place. There are different places for different types of bets.
  5. When the dealer says, “Place your bets!”, you should place your bets.
  6. The dealer will then spin the wheel.
  7. It should go without saying that when the dealer says, “No more bets.”, you can’t place any more bets.

This video shows you how the wheel and ball are spun:

Winning strategy

There are two ways to win at roulette:

  1. Hope you get an unbalanced wheel and be prepared to exploit this handy fact.
  2. Base your strategy off of the deterministic nature of the spin of the ball and the wheel.

Casinos will do their best to prevent (1) from happening, but they can’t do anything about (2). The reason why they can’t prevent (2) from happening is because bets are allowed to be placed after the wheel has started spinning. This means you can observe the motion of the ball and the wheel before placing a wager.

People who have (successfully) tried to cheat

History is littered with people trying to squeeze money out of casinos. Here is a brief overview of a few that stand out.

  • In 1873, an English mechanic and amateur mathematician who went by the name of Jagger carefully observed six roulette wheels at the Monte Carlo casino. He and his assistants logged the outcome of the each spin of the wheel for five weeks. Analysis of their data showed that each wheel had a unique bias due to factory imperfections which Jagger was quick to exploit. According to some sources, Jagger walked away with £65,000.
  • In his daydreams about the nature of chance, Henri Poincaré used a slightly different version of a roulette wheel to show how sensitivity to initial conditions can be used to figure out the ultimate resting state of the wheel. Aside: sensitivity to initial conditions forms a cornerstone of modern chaos theory.
  • This report from the BBC in 2004 describes three gamblers who were arrested for hiding a laser scanner in their smartphones to predict where the roulette ball would most likely stop (based on the laser’s estimation of the ball’s velocity and position). Luckily for the gamblers (and not for the casino), they were released without any charges and were allowed to keep their £1.3 million winnings. The laser from the smart phone was linked to a computer that was able to make calculations fast enough for the three to make their bets before the dealer said, “No more bets.” The work of Small and Tse are extremely similar to this case.

The model

Let’s use polar coordinates to describe the position of the ball, since we are on a wheel. We normally use Cartesian coordinates – the x-y plane – but due to the symmetry and naturally circular state of our system, polar coordinates are more intuitive. Polar coordinates describe location in space in terms of the distance from the origin r and angle measurement \theta. The picture below describes the transformation from Cartesian to polar coordinates.

The authors model the ball as a single point, so its position in r can vary between [0, r_{rim}], where r_{rim} represents the farthest point from the middle of the wheel that the ball can be – the very edge of the wheel. Similarly, r_{defl} is the radial distance to the location of the metal deflectors on the fixed part of the wheel (stator). \phi is the angle of rotation of the inner wheel.


The picture above also shows a free-body diagram of the forces acting on the ball as it travels on the roulette wheel. m is the mass of the ball. The angle \alpha is the angle of incline of the stator, and a_c is the radial acceleration of the ball. Obviously, g is gravity. The authors assume that the deflectors are evenly distributed around the stator at constant radius r_{defl} < r. In terms of a derivative, this means \frac{d r_{defl}}{d \theta} = 0. There are going to be two models here: one model with a completely horizontal table and one model with a tilted surface. Comparing the two results shows how critical it is for the roulette wheel to be on a totally flat surface.

The completely flat table

The point in time t_{defl} we are interested in is when r = r_{defl}. In other words, we are looking for the time when the ball settles down in a pocket. Before the ball settles down, it will pass through four distinct states:

  1. Sufficient momentum to remain on the rim
    • While on the rim, r is constant, and the ball has a varying angular velocity \dot{\theta}. \theta decays only due to a constant rolling friction.
  2. Insufficient momentum; leaves the rim
    • The ball leaves the rim at the point where its velocity squared is: \dot{\theta}^2 = \frac{g}{r}\tan \alpha.
    • The time this happens is: t_{rim} = -\frac{1}{\ddot{\theta}(0)}(\dot{\theta}(0)-\sqrt{\frac{g}{r}\tan \alpha})
  3. Moving freely on the stator
    • Due to the force of gravity and the centripetal force, the ball’s radial position decreases. Integrating this differential equation reveals the ball’s position: \ddot{r} = r\dot{\theta}^2\cos \alpha - g\sin \alpha
    • As it turns out, for a level table, the time spent from leaving the rim of the wheel (1) until the ball is about to hit the deflectors (4) is fairly consistent because the ball leaves the stator with exactly the same velocity \dot{\theta} each time. Thus, it is possible to estimate the position of the ball without having to integrate the equation above by using tabulated numbers.
  4. Settling into a pocket
    • The instantaneous angular deflection of the ball: \theta(t_{defl})=\theta(0)+\dot{\theta}(0)t_{defl}+\frac{1}{2}\ddot{\theta}(0)t_{defl}^2
    • The instantaneous angular deflection of the wheel: \phi(t_{defl})=\phi(0)+\dot{\phi}(0)t_{defl}+\frac{1}{2}\ddot{\phi}(0)t_{defl}^2
    • The angular location on the wheel where the ball strikes a deflector: \gamma = |\theta(t_{defl})-\phi(t_{defl})|_{2\pi}. Here, |\cdot|_{2\pi} represents modulo 2\pi. A modulo operation returns the remainder after division of two numbers (e.g., 5 mod 4 = 1, 9 mod 10 = 9, -2 mod 3 = 1, …)

The tilted table

  • Due to the change in the position of the wheel, it now has new equilibrium points. As a result, its angular velocity changes. Now, \dot{\theta} = \sqrt{\frac{g}{r}\tan(\alpha+\delta \cos \theta)}
  • The intersection of the above equation and this one will be the point the ball leaves the rim: \dot{\theta} = \dot{\theta}(t_1) - \frac{1}{2\pi}(\dot{\theta}(t_2) - \dot{\theta}(t_1))\theta
  • For a tilted roulette wheel, the ball will favor one half of a wheel more than the other.
  • A tilt of about 0.2 degrees is more than sufficient to bias the wheel.

After presenting these mathematical models for the motion of the wheel and the ball, the authors delve into some pretty lengthy and intense numerical simulations that support their model. They examine the sensitivity of their model to parameter uncertainty to see what sort of changes casinos may have to make to their roulette tables in order for the house to have a natural advantage (good news for the casinos; it’s only minor adjustments). The authors also realize that the game is truly predictable with some degree of certainty (good news for gamblers; this is a relatively honest game).

And finally for a fun bit of trivia: According to, the most frequently chosen number in roulette is 17 because that’s the number James Bond played in the movies.