Ciencia

Predicting the outcome of roulette
Michael Small1, 2, a) and Chi Kong Tse2
1) 2)

School of Mathematics and Statistics, The University of Western Australia Department of Electronic and Information Engineering Hong Kong Polytechnic University, Hong Kong. (Dated: 1 May 2012)

arXiv:1204.6412v1 [nlin.CD] 28 Apr 2012

There have been several popular reports of various groups exploiting thedeterministic 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 profit. To do this, we provide a very simple model for themotion 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 realisable 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 wheelto 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 beexploited 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. PACS numbers: 05.45.Tp, 01.65.+g Keywords: roulette, mathematical modeling, chaos “No one can possibly win at roulette unless he steals money from the tablewhen the croupier isn’t looking” (Attributed to Albert Einstein in1 ) 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 earlydevelopment of chaos theory 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. Whilenone of these systems are optimal, they all demonstrate positive expected return.
I. A HISTORY OF ROULETTE

a) Electronic

mail: michael.small@uwa.edu.au

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 attributed9 , perhaps erroneously1 , to the mathematician Blaise Pascal5 . Despitethe 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 mathematician149 . In any case, the device was introduced toParisian 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...