Introduction to roulette Roulette-Bet

The game of roulette has a long, glamorous, inglorious history, and has been connected with several notable men of science. 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.

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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. By the turn of the century, the game was popular and wide-spread. Its popularity bolstered by its apparent randomness and inherent (perceived) honesty.

The game of roulette consists of a heavy wheel, machined and balanced to have very low friction and designed to spin for a relatively long time with a slowly decaying angular velocity. The wheel is spun in one direction, while a small ball is spun in the opposite direction on the rim of a fixed circularly inclined surface surrounding and abutting the wheel.
As the ball loses momentum it drops toward the wheel and eventually will come to rest in one of 37 numbered pockets arranged around the outer edge of the spinning wheel. Various wagers can be made on which pocket, or group of pockets, the ball will eventually fall into. It is accepted practice that, on a successful wager on a single pocket, the casino will pay 35 to 1. Thus the expected return from a single wager on a fair wheel is:

In the long-run, the house will, naturally, win. In the eighteenth century the game was fair and consisted of only 36 pockets. Conversely, an American roulette wheel is even less fair and consists of 38 pockets. We consider the European, 37 pocket, version as this is of more immediate interest to us. The image below illustrates the general structure, as well as the layout of pockets, on a standard European roulette wheel.

European roulette wheel.  One can see the arrangement of the 37 numbers 0 to 36 and the respective colors (red and black).The rotating roulette wheel is surrounded by a fixed track with eight deflectors. Although the motion of the wheel and the ball (in the outer track) are simple and linear, the addition of several metal deflectors on the stator (the fixed frame on which the rotating wheel sits) makes the game even more unpredictable. The sharp frets between pockets also introduce strong non-linearity as the ball slows and bounces between pockets.

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 chaos theory. 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 pro t. 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.

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We will 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 this 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.