^{1,2,a)}and Chi Kong Tse

^{2}

### Abstract

There have been several popular reports of various groups exploiting the deterministic nature of the game of roulette for profit. Moreover, through its history, the inherent determinism in the game of roulette has attracted the attention of many luminaries of chaostheory. In this paper, we provide a short review of that history and then set out to determine to what extent that determinism can really be exploited for profit. To do this, we provide a very simple model for the motion of a roulette wheel and ball and demonstrate that knowledge of initial position, velocity, and acceleration is sufficient to predict the outcome with adequate certainty to achieve a positive expected return. We describe two physically realizable systems to obtain this knowledge both incognito and *in situ*. The first system relies only on a mechanical count of rotation of the ball and the wheel to measure the relevant parameters. By applying these techniques to a standard casino-grade European roulette wheel, we demonstrate an expected return of at least 18%, well above the −2.7% expected of a random bet. With a more sophisticated, albeit more intrusive, system (mounting a digital camera above the wheel), we demonstrate a range of systematic and statistically significant biases which can be exploited to provide an improved guess of the outcome. Finally, our analysis demonstrates that even a very slight slant in the roulette table leads to a very pronounced bias which could be further exploited to substantially enhance returns.

Among the various gaming systems, both current and historical, roulette is uniquely deterministic. 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. Perhaps because of this appealing deterministic nature, many notable figures from the early development of chaostheory have leant their hand to exploiting this determinism and undermining the presumed randomness of the outcome. In this paper, we aim only to establish whether the determinism in this system really can be profitably exploited. We find that this is definitely possible and propose several systems which could be used to gain an edge over the house in a game of roulette. While none of these systems are optimal, they all demonstrate positive expected return.

The first author would like to thank Marius Gerber for introducing him to the dynamical systems aspects of the game of roulette. Funding for this project, including the roulette wheel, was provided by the Hong Kong Polytechnic University. The labors of final year project students, Yung Chun Ting and Chung Kin Shing, in performing many of the mechanical simulations describe herein are gratefully acknowledged. M.S. is supported by an Australian Research Council Future Fellowship (FT110100896).

I. A HISTORY OF ROULETTE

II. A MODEL FOR ROULETTE

A. Level table

1. Ball rotates in the rim

2. Ball leaves the rim

3. Ball rotates freely on the stator

4. Ball reaches the deflectors

B. The crooked table

III. EXPERIMENTAL RESULTS

A. A manual implementation

B. Automated digital image capture

C. Parameter uncertainity and measurement noise

IV. EXPLOITS AND COUNTER-MEASURES

### Key Topics

- Kinematics
- 11.0
- Cameras
- 5.0
- Chaos
- 5.0
- Velocity measurement
- 5.0
- Acceleration measurement
- 4.0

## Figures

The European roulette wheel. In the left panel, one can see a portion of the rotating roulette wheel and surrounding fixed track. The ball has come to rest in the (green) 0 pocket. Although the motion of the wheel and the ball (in the outer track) are simple and linear, one can see the addition of several metal deflectors on the *stator* (that is, the fixed frame on which the rotating wheel sits). The sharp *frets* between pockets also introduce strong nonlinearity as the ball slows and bounces between pockets. The panel on the right depicts the arrangement of the number 0 to 36 and the coloring red (lighter) and black (darker).

The European roulette wheel. In the left panel, one can see a portion of the rotating roulette wheel and surrounding fixed track. The ball has come to rest in the (green) 0 pocket. Although the motion of the wheel and the ball (in the outer track) are simple and linear, one can see the addition of several metal deflectors on the *stator* (that is, the fixed frame on which the rotating wheel sits). The sharp *frets* between pockets also introduce strong nonlinearity as the ball slows and bounces between pockets. The panel on the right depicts the arrangement of the number 0 to 36 and the coloring red (lighter) and black (darker).

The dynamic model of ball and wheel. On the left, we show a top view of the roulette wheel (shaded region) and the stator (outer circles). The ball is moving on the stator with instantaneous position while the wheel is rotating with angular velocity (note that the direction of the arrows here are for illustration only, the analysis in the text assume the same convention, clockwise positive, for both ball and wheel). The deflectors on the stator are modelled as a circle, concentric with the wheel, of radius . On the right, we show a cross section and examination of the forces acting on the ball in the incline plane of the stator. The angle is the incline of the stator, *m* is themass of the ball, is the radial acceleration of the ball, and *g* is gravity.

The dynamic model of ball and wheel. On the left, we show a top view of the roulette wheel (shaded region) and the stator (outer circles). The ball is moving on the stator with instantaneous position while the wheel is rotating with angular velocity (note that the direction of the arrows here are for illustration only, the analysis in the text assume the same convention, clockwise positive, for both ball and wheel). The deflectors on the stator are modelled as a circle, concentric with the wheel, of radius . On the right, we show a cross section and examination of the forces acting on the ball in the incline plane of the stator. The angle is the incline of the stator, *m* is themass of the ball, is the radial acceleration of the ball, and *g* is gravity.

The case of the crooked table. The blue curve denotes the stability criterion (6), while the red solid line is the (approximate) trajectory of the ball with indicating two successive times of complete revolutions. The point at which the ball leaves the rim will therefore be the first intersection of this stability criterion and the trajectory. This will necessarily be in the region to the left of the point at which the ball's trajectory is tangent to Eq. (6), and this is highlighted in the figure as a green solid. Typically a crooked table will only be slightly crooked and hence this region will be close to but biased toward the approaching ball. The width of that region depends on , which in turn can be determined from Eq. (6).

The case of the crooked table. The blue curve denotes the stability criterion (6), while the red solid line is the (approximate) trajectory of the ball with indicating two successive times of complete revolutions. The point at which the ball leaves the rim will therefore be the first intersection of this stability criterion and the trajectory. This will necessarily be in the region to the left of the point at which the ball's trajectory is tangent to Eq. (6), and this is highlighted in the figure as a green solid. Typically a crooked table will only be slightly crooked and hence this region will be close to but biased toward the approaching ball. The width of that region depends on , which in turn can be determined from Eq. (6).

Hand-measurement of ball and wheel velocity for prediction. From two spins of the wheel, and 20 successive spins of the ball we logged the time (in seconds) *T*(*i*) for successive passes past a given point (*T*(*i*) against *T*(*i* + 1)). The measurements *T*(*i*) and *T*(*i* + 1) are the timings of successive revolutions—direct measurements of the angular velocity observed over one complete rotation. To provide the simplest and most direct indication that handheld measurements of this quantity are accurate, we indicate in this figure a deterministic relationship between these quantities. From this relationship, one can determine the angular deceleration. The red (slightly higher) points depict these times for the wheel, the blue (lower) points are for the ball. A single trial of both ball and wheel is randomly highlighted with crosses (superimposed). The inset is an enlargement of the detail in the lower left corner. Both the noise and the determinism of this method are evident. In particular, the wheel velocity is relatively easy to calculate and decays slowly, in contrast the ball decays faster and is more difficult to measure.

Hand-measurement of ball and wheel velocity for prediction. From two spins of the wheel, and 20 successive spins of the ball we logged the time (in seconds) *T*(*i*) for successive passes past a given point (*T*(*i*) against *T*(*i* + 1)). The measurements *T*(*i*) and *T*(*i* + 1) are the timings of successive revolutions—direct measurements of the angular velocity observed over one complete rotation. To provide the simplest and most direct indication that handheld measurements of this quantity are accurate, we indicate in this figure a deterministic relationship between these quantities. From this relationship, one can determine the angular deceleration. The red (slightly higher) points depict these times for the wheel, the blue (lower) points are for the ball. A single trial of both ball and wheel is randomly highlighted with crosses (superimposed). The inset is an enlargement of the detail in the lower left corner. Both the noise and the determinism of this method are evident. In particular, the wheel velocity is relatively easy to calculate and decays slowly, in contrast the ball decays faster and is more difficult to measure.

Predicting roulette. The plot depicts the results of 700 trials of our automated image recognition software used to predict the outcome of independent spins of a roulette wheel. What we plot here is a histogram in polar coordinates of the difference between the predicted and the actual outcome (the “Target” location, at the 12 o'clock position in this diagram, indicating that the prediction was correct). The length of each of the 37 black bars denote the frequency with which predicted and actual outcome differed by exactly the corresponding angle. Dotted, dotted-dashed, and solid (red) lines depict the corresponding 99.9%, 99%, and 90% confidence intervals using the corresponding two-tailed binomial distribution. Motion forward (i.e., ball continues to move in the same direction) is clockwise, motion backwards is anti-clockwise. From the 37 possible results, there are 2 instances outside the 99% confidence interval. There are 7 instances outside the 90% confidence interval.

Predicting roulette. The plot depicts the results of 700 trials of our automated image recognition software used to predict the outcome of independent spins of a roulette wheel. What we plot here is a histogram in polar coordinates of the difference between the predicted and the actual outcome (the “Target” location, at the 12 o'clock position in this diagram, indicating that the prediction was correct). The length of each of the 37 black bars denote the frequency with which predicted and actual outcome differed by exactly the corresponding angle. Dotted, dotted-dashed, and solid (red) lines depict the corresponding 99.9%, 99%, and 90% confidence intervals using the corresponding two-tailed binomial distribution. Motion forward (i.e., ball continues to move in the same direction) is clockwise, motion backwards is anti-clockwise. From the 37 possible results, there are 2 instances outside the 99% confidence interval. There are 7 instances outside the 90% confidence interval.

Parameter uncertainty. We explore the effect of error in the model parameters on the outcome by varying the three physical parameters of the wheel (a) and introducing uncertainty in the measurement of timing events used to obtain estimates of velocity and deceleration (b). In (a), we depict the effect of perturbing the estimated valuesof (green—affine, increasing steeply) (red—affine, decreasing) and (blue—affine and increasing slowly) from 90% to 110% of the true value. In (b), we add Gaussian noise of magnitude between 0.5% and 10% the variance of the true measurements to initial estimates of all positions and velocities. Horizontal dotted lines in both plots depict error corresponding to one whole pocket in the wheel. The vertical axis is in radians and covers —half the wheel. In the upper panel, least variation in outcome is observed with errors in estimation of .

Parameter uncertainty. We explore the effect of error in the model parameters on the outcome by varying the three physical parameters of the wheel (a) and introducing uncertainty in the measurement of timing events used to obtain estimates of velocity and deceleration (b). In (a), we depict the effect of perturbing the estimated valuesof (green—affine, increasing steeply) (red—affine, decreasing) and (blue—affine and increasing slowly) from 90% to 110% of the true value. In (b), we add Gaussian noise of magnitude between 0.5% and 10% the variance of the true measurements to initial estimates of all positions and velocities. Horizontal dotted lines in both plots depict error corresponding to one whole pocket in the wheel. The vertical axis is in radians and covers —half the wheel. In the upper panel, least variation in outcome is observed with errors in estimation of .

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