From thought experiments to movies, the roulette wheel represents pure chance at its finest. There are 38 slots into which the ball could land, and all are equally likely. Thus, a roulette wheel can be treated as a random number generator. From the tense moments in Las Vegas when life savings are on the line, to the cool and calm James Bond who collects his roulette winnings, it seems the outcome is merely determined by luck.
However, roulette is a game governed by classical mechanics, and classical mechanics is at its heart deterministic. If one knows all the relevant information of initial states precisely enough, then one could - in theory - compute the final destination of the ball. Of course, such precise measurements and detailed calculations are impossible even in a laboratory setting, let alone in a casino. The chaotic motion of the ball as it hits the bumpers and scatters across the wheel interfere with any prediction. In spite of this, it is possible to create a prediction that is 'good enough.' By predicting the most likely region of the wheel in which the ball will land, though any single victory is not guaranteed, the odds can be shifted in favor of the player.
There are several methods of arriving at such a prediction. From visual cues to the use of lasers, there are many people online willing to sell such (illegal) systems for a chunk of change. The theory behind these tools is of far more academic value than the tools themselves. The method of prediction we will focus on is using tilt to find a bias in the wheel itself. If the ball is inherently more likely to land in one section of the wheel than another, then the 'pure chance' of the wheel has broken down. As we will see, if the wheel is not perfectly level, such a bias appears. Even with minor tilt, the wheel can create a forbidden zone where the ball is very unlikely to land.
We will begin by analyzing the physics of the roulette wheel, using the work of Thorp  and Eichberger . Though the essence of roulette is simple, in order to make meaningful predictions, friction and air resistance need to be taken into account. These considerations will lead to a forbidden zone: regions in which the ball will not leave the rim of the wheel. After looking at the theory behind the forbidden zone, we will look at an experiment outlined by Dixon  that shows how this forbidden zone translates into biased results.
At its heart, the physics of an idealized roulette wheel is fairly simple. The basic geometry of the roulette wheel is illustrated in Figure 1. The wheel contains a rim around the outside, along which the ball initially rolls. The wheel itself is tilted inward, as shown in Figure 2, and thus at some point the ball will leave the rim and head towards the center of the wheel. At this point, the ball encounters a set of bumpers, whose purpose is to chaotically scatter the ball. Finally, the ball reaches the innermost part of the wheel, with 38 equally sized slots into which the ball can land.
|Fig. 1: Diagram of Roulette Wheel|
|Fig. 2: Cross-section of Wheel|
|Fig. 3: Finding the Departure Angle of the Ball|
For this experiment, a regulation roulette wheel was given a certain tilt (0°, .1°, .25°, .5°, and 1.35°). The wheel was then spun 176 times, and the results were recorded of where the ball landed. For the purposes of recording the results, the wheel was split into eight equal sections - in the stationary reference frame of the table, and the ball was always started in section 5. It is much easier to detect a tilt-based bias by looking at these stationary sections, ignoring the details of predicting how the wheel itself spins and which numbered slots will be where in the final state. The results of a large number of spins were binned and analyzed for statistical bias. The results of this experiment for several amounts of tilt are shown in Table 1 and also plotted in Fig. 4.
|Fig. 4: Plot of Results with Tilt and no Tilt|