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 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.

1.
T. A.
Bass
,
The Newtonian Casino
(
Penguin
,
London
,
1990
).
2.
R. A.
Epstein
,
The Theory of Gambling and Statistical Logic
(
Academic
,
New York
,
1967
).
3.
E. T.
Bell
,
Men of Mathematics
(
Simon and Schuster
,
New York
,
1937
).
4.
The Italian mathematician, confusingly, was named Don Pasquale2, a surname phonetically similar to Pascal. Moreover, as Don Pasquale is also the name of a 19th century opera buff, this attribution is possibly fanciful.
5.
F.
Downton
and
R. L.
Holder
, “
Banker's games and the gambling act 1968
,”
J. R. Stat. Soc. Ser. A
135
,
336
364
(
1972
).
6.
B.
Okuley
and
F.
King-Poole
,
Gamblers Guide to Macao
(
South China Morning Post
,
Hong Kong
,
1979
).
7.
Three, if one has sufficient finances to assume the role of the house.
8.
C.
Kingston
,
The Romance of Monte Carlo
(
John Lane The Bodley Head Ltd.
,
London
,
1925
).
9.
E. O.
Thorp
, “
Optimal gambling systems for favorable games
,”
Rev. Int. Stat. Inst.
37
,
273
293
(
1969
).
10.
Life Magazine Publication, “
How to Win $6,500
,” Life, 46, December 8,
1947
.
11.

Alternatively, and apparently erroneously, reported to be from Californian Institute of Technology in Ref. 2.

12.
S. N.
Ethier
, “
Testing for favorable numbers on a roulette wheel
,”
J. Am. Stat. Assoc.
77
,
660
665
(
1982
).
13.
Time Magazine Publication
, “
Argentina—Bank breakers
,”
Time
135
,
34
, February 12,
1951
.
14.
The first, to the best of our knowledge.
15.
H.
Poincaré
,
Science and Method
(
Nelson
,
London
,
1914
). English translation by Francis Maitland, preface by Bertrand Russell. Facsimile reprint in 1996 by Routledge/Thoemmes, London.
16.
J. P.
Crutchfield
,
J.
Doyne Farmer
,
N. H.
Packard
, and
R. S.
Shaw
, “
Chaos
,”
Sci. Am.
255
,
46
57
(
1986
).
17.
E. O.
Thorp
,
The Mathematics of Gambling
(
Gambling Times
,
1985
).
18.
C. E.
Shannon
, “
A mathematical theory of communication
,”
Bell Syst. Techn. J.
27
,
379
423
, 623–656 (
1948
).
19.
N. H.
Packard
,
J. P.
Crutchfield
,
J. D.
Farmer
, and
R. S.
Shaw
, “
Geometry from a time series
,”
Phys. Rev. Lett.
45
,
712
716
(
1980
).
20.
T. A.
Bass
,
The Predictors
, edited by
A.
Lane
(
Penguin
,
London
,
1999
).
21.
J. D.
Farmer
and
J. J.
Sidorowich
, “
Predicting chaotic time series
,”
Phys. Rev. Lett.
59
,
845
848
(
1987
).
22.
BBC Online, “
Arrests follow £1m roulette win
,” BBC News March 22,
2004
.
23.
BBC Online, “
Laser scam” gamblers to keep £1m
,” BBC News December 5,
2004
.
24.
M.
Small
and
C. K.
Tse
, “
Feasible implementation of a prediction algorithm for the game of roulette
,” in Asia-Pacific Conference on Circuits and Systems (IEEE,
2008
).
25.
J.
Strzalko
,
J.
Grabski
,
P.
Perlikowksi
,
A.
Stefanski
, and
T.
Kapitaniak
,
Dynamics of gambling, Vol. 792 of Lecture Notes in Physics
(
Springer
,
2009
).
26.
Implementation on a “shoe-computer” should be relatively straightforward too.
27.
C. T.
Yung
, “
Predicting roulette
,” Final Year Project Report, Hong Kong Polytechnic University, Department of Electronic and Information Engineering, April
2011
.
28.
K. S.
Chung
, “
Predicting roulette II: Implementation
,” Final Year Project Report, Hong Kong Polytechnic University, Department of Electronic and Information Engineering, April
2010
.
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