Predicting the outcome of rouletteMichael Small1, 2, a) and Chi Kong Tse21)School of Mathematics and Statistics, The University of Western Australia2)Department of Electronic and Information EngineeringHong Kong Polytechnic University, Hong Kong.

(Dated: 1 May 2012)

There have been several popular reports of various groups exploiting the deterministic nature of thegame of roulette for profit. Moreover, through its history the inherent determinism in the game ofroulette has attracted the attention of many luminaries of chaos theory. In this paper we providea short review of that history and then set out to determine to what extent that determinism canreally be exploited for profit. To do this, we provide a very simple model for the motion of aroulette wheel and ball and demonstrate that knowledge of initial position, velocity and accelerationis 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 wheel to measurethe relevant parameters. By applying this techniques to a standard casino-grade European roulettewheel we demonstrate an expected return of at least 18%, well above the −2.7% expected of arandom bet. With a more sophisticated, albeit more intrusive, system (mounting a digital cameraabove the wheel) we demonstrate a range of systematic and statistically significant biases which canbe exploited to provide an improved guess of the outcome. Finally, our analysis demonstrates thateven a very slight slant in the roulette table leads to a very pronounced bias which could be furtherexploited to substantially enhance returns.

PACS numbers: 05.45.Tp, 01.65.+gKeywords: roulette, mathematical modeling, chaos

“No one can possibly win at roulette unless hesteals money from the table when the croupier

isn’t looking” (Attributed to Albert Einstein in1)

Among the various gaming systems, both cur-rent and historical, roulette is uniquely de-terministic. Relatively simple laws of motionallow one, in principle, to forecast the pathof the ball on the roulette wheel and to itsfinal destination. Perhaps because of this ap-pealing deterministic nature, many notablefigures from the early development of chaostheory have leant their hand to exploiting thisdeterminism and undermining the presumedrandomness of the outcome. In this paperwe aim only to establish whether the deter-minism in this system really can be profitablyexploited. We find that this is definitely pos-sible and propose several systems which couldbe used to gain an edge over the house in agame of roulette. While none of these sys-tems are optimal, they all demonstrate posi-tive expected return.

a)Electronic mail: michael.small@uwa.edu.au

I. A HISTORY OF ROULETTE

The game of roulette has a long, glamorous, in-glorious history, and has been connected with sev-eral notable men of science. The origin of the gamehas been attributed9, perhaps erroneously1, to themathematician Blaise Pascal5. Despite the roulettewheel becoming a staple of probability theory, thealleged motivation for Pascal’s interest in the de-vice was not solely to torment undergraduate stu-dents, but rather as part of a vain search for perpet-ual motion. Alternative stories have attributed theorigin of the game to the ancient Chinese, a Frenchmonk or an Italian mathematician149. In any case,the device was introduced to Parisian gamblers inthe mid-eighteenth century to provide a fairer gamethan those currently in circulation. By the turn ofthe century, the game was popular and wide-spread.Its popularity bolstered by its apparent randomnessand inherent (perceived) honesty.

The game of roulette consists of a heavy wheel,machined and balanced to have very low friction anddesigned to spin for a relatively long time with a

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slowly decaying angular velocity. The wheel is spunin one direction, while a small ball is spun in theopposite direction on the rim of a fixed circularly in-clined surface surrounding and abutting the wheel.As the ball loses momentum it drops toward thewheel and eventually will come to rest in one of 37numbered pockets arranged around the outer edge ofthe spinning wheel. Various wagers can be made onwhich pocket, or group of pockets, the ball will even-tually fall into. It is accepted practise that, on a suc-cessful wager on a single pocket, the casino will pay35 to 1. Thus the expected return from a single wa-ger on a fair wheel is (35+1)× 1

37 +(−1) ≈ −2.7%8.In the long-run, the house will, naturally, win. In theeighteenth century the game was fair and consistedof only 36 pockets. Conversely, an American roulettewheel is even less fair and consists of 38 pockets. Weconsider the European, 37 pocket, version as this isof more immediate interest to us19. Figure 1 illus-trates the general structure, as well as the layout ofpockets, on a standard European roulette wheel.

Despite many proposed “systems” there are onlytwo profitable ways to play roulette15. One can ei-ther exploit an unbalanced wheel, or one can ex-ploit the inherently deterministic nature of the spinof both ball and wheel. Casinos will do their utmostto avoid the first type of exploit. The second exploitis possible because placing wagers on the outcomeis traditionally permitted until some time after theball and wheel are in motion. That is, one has anopportunity to observe the motion of both the balland the wheel before placing a wager.

The archetypal tale of the first type of exploitis that of a man by the name of Jagger (varioussources refer to him as either William Jaggers orJoseph Jagger, or some permutation of these). Jag-ger, an English mechanic and amateur mathemati-cian, observed that slight mechanical imperfectionin a roulette wheel could afford sufficient edge toprovide for profitable play. According to one incar-nation of the tale, in 1873 he embarked for the casinoof Monte Carlo with six hired assistants. Once there,he carefully logged the outcome of each spin of eachof six roulette tables over a period of five weeks12.Analysis of the data revealed that for each wheelthere was a unique but systematic bias. Exploitingthese weaknesses he gambled profitably for a weekbefore the casino management shuffled the wheelsbetween tables. This bought his winning streak to asudden halt. However, he soon noted various distin-guishing features of the individual wheels and wasable to follow them between tables, again winningconsistently. Eventually the casino resorted to re-distributing the individual partitions between pock-

ets. A popular account, published in 1925, claims heeventually came away with winnings of £65, 00012.The success of this endeavour is one possible inspira-tion for the musical hall song “The Man Who Brokethe Bank at Monte Carlo” although this is stronglydisputed12.

Similar feats have been repeated elsewhere. Thenoted statistician Karl Pearson provided a statisticalanalysis of roulette data, and found it to exhibit sub-stantial systematic bias. However, it appears thathis analysis was based on flawed data from unscrupu-lous scribes25 (apparently he had hired rather lazyjournalists to collect the data).

In 1947 irregularities were found, and exploited,by two students, Albert Hibbs and Roy Walford,from Chicago University1316. Following this lineof attack, S.N. Ethier provides a statistical frame-work by which one can test for irregularities in theobserved outcome of a roulette wheel10. A similarweakness had also been reported in Time magazinein 1951. In this case, the report described varioussyndicates of gamblers exploiting determinism in theroulette wheel in the Argentinean casino Mar delPlata during 194827. The participants were colour-fully described as a Nazi sailor and various “fruithucksters, waiters and farmers”27.

The second type of exploit is more physical (thatis, deterministic) than purely statistical and has con-sequently attracted the attention of several math-ematicians, physicists and engineers. One of thefirst17 was Henri Poincare5 in his seminal work Sci-ence and Method21. While ruminating on the natureof chance, and that a small change in initial condi-tion can lead to a large change in effect, Poincare il-lustrated his thinking with the example of a roulettewheel (albeit a slightly different design from themodern version). He observed that a tiny change ininitial velocity would change the final resting placeof the wheel (in his model there was no ball) suchthat the wager on an either black or red (as in amodern wheel, the black and red pockets alternate)would correspondingly win or lose. He concludedby arguing that this determinism was not importantin the game of roulette as the variation in initialforce was tiny, and for any continuous distributionof initial velocities, the result would be the same:effectively random, with equal probability. He wasnot concerned with the individual pockets, and hefurther assumed that the variation in initial velocityrequired to predict the outcome would be immea-surable. It is while describing the game of roulettethat Poincare introduces the concept of sensitivityto initial conditions, which is now a cornerstone ofmodern chaos theory7.

2

FIG. 1. The European roulette wheel. In the left panel one can see a portion of the rotating roulette wheel andsurrounding fixed track. The ball has come to rest in the green 0 pocket. Although the motion of the wheel andthe 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 strongnonlinearity as the ball slows and bounces between pockets. The panel on the right depicts the arrangement of thenumber 0 to 36 and the colouring red and black.

A general procedure for predicting the outcomeof a roulette spin, and an assessment of its utilitywas described by Edward Thorp in a 1969 publica-tion for the Review of the International StatisticalInstitute25. In that paper, Thorp describes the twobasic methods of prediction. He observes (as othershave done later) that by minimising systematic biasin the wheel, the casinos achieve a mechanical per-fection that can then be exploited using determinis-tic prediction schemes. He describes two determin-istic prediction schemes (or rather two variants onthe same scheme). If the roulette wheel is not per-fectly level (a tilt of 0.2◦ was apparently sufficient —we verified that this is indeed more than sufficient)then there effectively is a large region of the framefrom which the ball will not fall onto the spinningwheel. By studying Las Vegas wheels he observesthis condition is meet in approximately one third ofwheels. He claims that in such cases it is possibleto garner a expectation of +15%, which increasedto +44% with the aid of a ‘pocket-sized’ computer.Some time later, Thorp revealed that his collabo-rator in this endeavour was Claude Shannon26, thefounding father of information theory22.

In his 1967 book9 the mathematician Richard A.Epstein describes his earlier (undated) experiments

with a private roulette wheel. By measuring the an-gular velocity of the ball relative to the wheel he wasable to predict correctly the half of the wheel intowhich the ball would fall. Importantly, he noted thatthe initial velocity (momentum) of the ball was notcritical. Moreover, the problem is simply one of pre-dicting when the ball will leave the outer (fixed rim)as this will always occur at a fixed velocity. How-ever, a lack of sufficient computing resources meantthat his experiments were not done in real time, andcertainly not attempted within a casino.

Subsequent to, and inspired by, the work of Thorpand Shannon, another widely described attempton beating the casinos of Las Vegas was made in1977-1978 by Doyne Farmer, Norman Packard andcolleagues1. It is supposed that Thorp’s 1969 pa-per had let the cat out of the bag regarding prof-itable betting on roulette. However, despite the as-sertions of Bass1, Thorp’s paper25 is not mathemat-ically detailed (there is in fact no equations givenin the description of roulette). Thorp is sufficientlydetailed to leave the reader in no doubt that thescheme could work, but also vague enough so thatone could not replicate his effort without consider-able knowledge and skill. Farmer, Packard and col-leagues implemented the system on a 6502 micro-

3

processor hidden in a shoe, and proceeded to applytheir method to the various casinos of the Las VegasStrip. The exploits of this group are described indetail in Bass1. The same group of physicists wenton to apply their skills to the study of chaotic dy-namical systems20 and also for profitable trading onthe financial markets2. In Farmer and Sidorowich’slandmark paper on predicting chaotic time series11

the authors attribute the inspiration for that workto their earlier efforts to beat the game of roulette.

Less exalted individuals have also been employ-ing similar schemes, in some cases fairly recently.In 2004, the BBC carried the report of three gam-blers (described only as “a Hungarian woman andtwo Serbian men”3) arrested by police after winning£1, 300, 000 at the Ritz Casino in London. The triohad apparently been using a laser scanner and theirmobile phones to predict the likely resting place ofthe ball. Happily, for the trio but not the casino,they were judged to have broken no laws and allowedto keep their winnings4. The scheme we describe inSection II and implement in Section III is certainlycompatible with the equipment and results reportedin this case. In Section IV we conclude with some re-marks concerning the practicality of applying thesemethods in a modern casino, and what steps a casinocould take (or perhaps have taken) to circumventthese exploits. A preliminary version of these resultswas presented at a conference in Macao23. An inde-pendent and much more detailed model of dynamicsof the roulette wheel is discussed in Strzalko et al.24.Since our preliminary publication23, private commu-nication with several individuals indicates that thesemethods have progress to the point of independentin situ field trials.

II. A MODEL FOR ROULETTE

We now describe our basic model of the motionof the roulette wheel and ball. Let (r, θ) denote theposition of the ball in polar co-ordinates, and letϕ denote the angular position of the wheel (say, theangular position of the centre of the green 0 pocket).We will model the ball as a single point and so letrrim be the farthest radial position of that point (i.e.the radial position of the centre of the ball when theball is spinning with high velocity in the rim of thewheel). Similarly, let rdefl be the radial distance tothe location of the metal deflectors on the stator. Fornow, we will assume that drdefl

dθ = 0 (that is, thereare deflectors evenly distributed around the statorat constant radius rdefl < r). The extension to themore precise case is obvious, but, as we will see, not

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FIG. 2. The dynamic model of ball and wheel. Onthe left we show a top view of the roulette wheel (shadedregion) and the stator (outer circles). The ball is movingon the stator with instantaneous position (r, θ) while thewheel is rotating with angular velocity ϕ (Note that thedirection of the arrows here are for illustration only, theanalysis in the text assume the same convention, clock-wise positive, for both ball and wheel). The deflectorson the stator are modelled as a circle, concentric withthe wheel, of radius rdefl. On the right we show a crosssection and examination of the forces acting on the ballin the incline plane of the stator. The angle α is theincline of the stator, m is the mass of the ball, ac is theradial acceleration of the ball, and g is gravity.

necessary. Moreover, it is messy. Finally, we supposethat the incline of the stator to the horizontal is aconstant α. This situation, together with a balanceof forces is depicted in figure 2. We will first considerthe ideal case of a level table, and then in section II Bshow how this condition is in fact critical.

A. Level table

For a given initial motion of ball (r, θ, θ, θ)t=0 andwheel (ϕ, ϕ, ϕ)t=0 our aim is to determine the timetdefl at which r = rdefl. After launch the motion ofthe ball will pass through two distinct states whichwe further divide into four cases: (i) with sufficientmomentum it will remain in the rim, constrainedby the fixed edge of the stator; (ii) at some pointthe momentum drops and the ball leaves the rim;(iii) the ball will gradually loose momentum while

travelling on the stator as θ drops, so will r; and(iv) eventually r = rdefl at some time tdefl. At timet = tdefl we assume that the ball hits a deflector onthe stator and drops onto the (still spinning) wheel.

4

The exact position of the wheel when the ball reachesthe deflectors will be random but will depend onlyon ϕ(tdefl).

(i) Ball rotates in the rim

While travelling in the rim r is constant and theball has angular velocity θ. Hence, the radial accel-

eration of the ball ac = v2

r = 1r (rθ)2 = rθ2 where v

is the speed of the ball. During this period of mo-tion, we suppose that r is constant and that θ decaysonly due to constant rolling friction: hence r = 0 andθ = θ(0), a constant. This phase of motion will con-tinue provided the centripetal force of the ball on therim exceeds the force of gravity mac cosα > mg sinα(m is the mass of the ball). Hence at this stage

θ2 >g

rtanα. (1)

(ii) Ball leaves the rim

Gradually the speed on the ball decays until even-tually θ2 = g

r tanα. Given the initial acceleration

θ(0), velocity θ(0) and position θ(0), it is trivial tocompute the time at which the ball leaves the rim,trim to be

trim = − 1

θ(0)

(θ(0)−

√g

rtanα

). (2)

To do so, we assume that the angular accelerationis constant and so the angular velocity at any timeis given by θ(t) = θ(0) + θ(0)t and substitute intoequation (1). The position at which the ball leavesthe rim is given by∣∣∣∣∣ ( gr tanα)− θ(0)2

2θ(0)

∣∣∣∣∣2π

where | · |2π denotes modulo 2π.

(iii) Ball rotates freely on the stator

After leaving the rim the ball will continue (inpractise, for only a short while) to rotate freely onthe stator until it eventually reaches the various de-flectors at r = rdefl. The angular velocity continuesto be governed by

θ(t) = θ(0) + θ(0)t,

but now that

rθ < g tanα

the radial position is going to gradually decrease too.The difference between the force of gravity g sinαand the (lesser) centripetal force rθ2 provides inwardacceleration of the ball

r = rθ2 cosα− g sinα. (3)

Integrating (3) yields the position of the ball on thestator.

(iv) Ball reaches the deflectors

Finally, we find the time t = tdefl for whichr(t), computed as the definite second integral of(3), is equal to rdefl. We can then compute the in-stantaneous angular position of the ball θ(tdefl)) =

θ(0) + θ(0)tdefl + 12 θ(0)t2defl and the wheel ϕ(tdefl)) =

ϕ(0) + ϕ(0)tdefl + 12 ϕ(0)t2deflto give the salient value

γ = |θ(tdefl)− ϕ(tdefl)|2π (4)

denoting the angular location on the wheel directlybelow the point at which the ball strikes a deflec-tor. Assuming the constant distribution of deflectorsaround the rim, some (still to be estimated) distribu-tion of resting place of the ball will depend only onthat value γ. Note that, although we have described(θ, θ, θ)t=0 and (ϕ, ϕ, ϕ)t=0 separately, it is possibleto adopt the rotating frame of reference of the wheeland treat θ − ϕ as a single variable. The analysis isequivalent, estimating the required parameters maybecome simpler.

We note that for a level table, each spin of theball alters only the time spent in the rim, the ballwill leave the rim of the stator with exactly the samevelocity θ each time. The descent to the deflectorswill therefore be identical. There will, in fact, besome characteristic duration which could be easilycomputed for a given table. Doing this would cir-cumvent the need to integrate (3).

B. The crooked table

Suppose, now that the table is not perfectly level.This is the situation discussed and exploited byThorp25. Without loss of generality (it is only anaffine change of co-ordinates for any other orienta-tion) suppose that the table is tilted by an angle δsuch that the origin ϕ = 0 is the lowest point on the

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FIG. 3. The case of the crooked table. The bluecurve denotes the stability criterion (6), while the redsolid line is the (approximate) trajectory of the ball withθ(t1) + 2π = θ(t2) indicating two successive times ofcomplete revolutions. The point at which the ball leavesthe rim will therefore be the first intersection of thisstability criterion and trajectory. This will necessarilybe in the region to the left of the point at which the ball’strajectory is tangent to (6), and this is highlighted in thefigure as a green solid. For typical physical realisations,this is close to θ = 0 but biased toward the approachingball. The width of that region depends on θ(t1) − θ(t2),which in turn can be determined from (6).

rim. Just as with the case of a level table, the timewhich the ball spends in the rim is variable and thetime at which it leaves the rim depends on a stabil-ity criterion similar to (1). But now that the tableis not level, that equilibrium becomes

rθ2 = g tan (α+ δ cos θ). (5)

If δ = 0 then it is clear that the distribution ofangular positions for which this condition is firstmet will be uniform. Suppose instead that δ > 0,then there is now a range of critical angular ve-locities θ2

crit ∈ [ gr tan (α− δ), gr tan (α+ δ)]. Once

θ2 < gr tan (α+ δ) the position at which the ball

leaves the rim will be dictated by the point of inter-section in (θ, θ)-space of

θ =

√g

rtan (α+ δ cos θ) (6)

and the ball trajectory as a function of t (modulo2π)

θ(t) = θ(0) + θ(0)t (7)

θ(t) = θ(0) + θ(0)t+1

2θ(0)t2. (8)

If the angular velocity of the ball is large enough thenthe ball will leave the rim at some point on the halfcircle prior to the low point (ϕ = 0). Moreover, sup-pose that in one revolution (i.e. θ(t1)+2π = θ(t2) ),

the velocity changes by θ(t1) − θ(t2). Furthermore,suppose that this is the first revolution during whichθ2 < g

r tan (α+ δ) (that is, θ(t1)2 ≥ gr tan (α+ δ)

but θ(t2)2 < gr tan (α+ δ)). Then, the point at

which the ball will leave the rim will (in (θ, θ)-space)be the intersection of (6) and

θ = θ(t2)− 1

2π

(θ(t1)− θ(t2)

)θ. (9)

The situation is depicted in figure 3. One can ex-pect for a tilted roulette wheel, the ball will sys-tematically favour leaving the rim on one half ofthe wheel. Moreover, to a good approximation, thepoint at which the ball will leave the rim follows auniform distribution over significantly less than halfthe wheel circumference. In this situation, the prob-lem of predicting the final resting place is signifi-cantly simplified to the problem of predicting theposition of the wheel at the time the ball leaves therim.

We will pursue this particular case no further here.The situation (5) may be considered as a generali-sation of the ideal δ = 0 case. This generalisationmakes the task of prediction significantly easier, butwe will continue to work under the assumption thatthe casino will be doing its utmost to avoid the prob-lems of an improperly levelled wheel. Moreover, thisgeneralisation is messy, but otherwise uninteresting.In the next section we consider the problem of im-plementing a prediction scheme for a perfectly levelwheel.

III. EXPERIMENTAL RESULTS

The problem of prediction is essentially two-fold.First, the various velocities must be estimated accu-rately. Given these estimates it is a trivial problemto then determine the point at which the ball willintersect with one of the deflectors on the stator.Second, one must then have an estimate of the scat-ter imposed on the ball by both the deflectors andpossible collision with the individual frets. To apply

6

0 10 20 30 40 50 60 700

5

10

15

20

25

30

35

T(i) sec.

T(i+

1) se

c.Duration of successive revolutions (ball blue, wheel red)

0 1 2 3 40

0.5

1

1.5

2

2.5

3

3.5

4

FIG. 4. Hand-measurement of ball and wheel ve-locity 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 red points depict these timesfor the wheel, the blue points are for the ball. A singletrial of both ball and wheel is randomly highlighted withcrosses (superimposed). The inset is an enlargement ofthe detail in the lower left corner. Both the noise andthe determinism of this method are evident. In partic-ular, the wheel velocity is relatively easy to calculateand decays slowly, in contrast the ball decays faster andis more difficult to measure. Using these (admittedlynoisy) measurements we were able to successfully pre-dict the half of the wheel in which the ball would stopin 13 of 22 trials (random with p < 0.15), yielding anexpected return of 36/18×13/22−1 = +18%. This trialrun included predicting the precise location in which theball landed on three occasions (random with p < 0.02).

this method in situ, one has the further complica-tion of estimating the parameters r, rdefl, rrim, αand possibly δ without attracting undue attention.We will ignore this additional complication as it isessentially a problem of data collection and statis-tical estimation. Rather, we will assume that thesequantities can be reliably estimated and restrict ourattention to the problem of prediction of the mo-tion. To estimate the relevant positions, velocitiesand accelerations (θ, θ, θ, ϕ, ϕ, ϕ)t=0 (or perhaps just

(θ−ϕ, θ− ϕ, θ− ϕ)t=0) we employ two distinct tech-niques.

A. A manual implementation

Our first approach is to simply record the timeat which ball and wheel pass a fixed point. This isa simple approach (probably that used in the earlyattempts to beat the wheels of Las Vegas) and istrivial to implement on a laptop computer, personaldigital assistant, embedded system, or even a mo-bile phone18,6,28. Our results, depicted in figure4 illustrate that the measurements, although noisy,are feasible. The noise introduced in this measure-ment is probably largely due to the lack of physicalhand-eye co-ordination of the first author. Simpleexperiments with this configuration indicate that itis possible to accurately predict the correct half ofthe wheel in which the ball will come to rest (statis-tical results are given in the caption of Fig. 4).

B. Automated digital image capture

Alternatively, we employ a digital cameramounted directly above the wheel to accurately andinstantaneously measure the various physical param-eters. This second approach is obviously a littlemore difficult to implement incognito. Here, we aremore interested in determining how much of an edgecan be achieved under ideal conditions, rather thanthe various implementation issues associated withrealising this scheme for personal gain. In all our tri-als we use a regulation casino-grade roulette wheel (a32” “President Revolution” roulette wheel manufac-turer by Matsui Gaming Machine Co. Ltd., Tokyo).The wheel has 37 numbered slots (1 to 36 and 0)in the configuration shown in figure 1 and has a ra-dius of 820 mm (spindle to rim). For the purposes ofdata collection we employ a Prosilica EC650C IEEE-1394 digital camera (1/3” CCD, 659×493 pixels at90 frames per second). Data collection software waswritten and coded in C++ using the OpenCV li-brary.

Figure 5 illustrates the results from 700 trials ofthe prediction algorithm on independent rolls of afair and level roulette wheel. Several things are clearfrom Fig. 5. First, for most the wheel, the probabil-ity of the ball landing in a particular pocket — rel-ative to the predicted destination — does not differsignificantly from chance: observed populations in30 of 37 pockets is within the 90% confidence inter-val for a random process. Two particular pockets —the target pocket itself and a pocket approximatelyone-quarter of the wheel prior to the target pocket— occur with frequencies higher than and less than

7

Target

FIG. 5. Predicting roulette. The plot depicts the re-sults of 700 trials of our automated image recognitionsoftware used to predict the outcome of independentspins of a roulette wheel. What we plot here is a his-togram in polar coordinates of the different between thepredicted and the actual outcome (the ”Target” locationindicating that the prediction was correct). The lengthof each of the 37 black bars denote the frequency withwhich predicted and actual outcome differed by exactlythe corresponding angle. Dot, dot-dashed and solid (red)lines depict the corresponding 99/9%, 99% and 90% con-fidence intervals using the corresponding two-tailed bino-mial distribution. Motion forward (i.e. ball continues tomove in the same direction) is clockwise, motion back-wards in anti-clockwise. From the 37 possible resultsthere are 2 instances outside the 99% confidence inter-val. There are 7 instances outside the 90% confidenceinterval.

(respectively) the expected by chance: outside the99% confidence interval. Hence, the predicted targetpocket is a good indicator of eventual outcome andthose pockets immediate prior to the target pocket(which the ball would need to bounce backwards toreach) are less likely. Finally, and rather specula-tively, there is a relatively higher chance (althoughmarginally significant) of the ball landing in one ofthe subsequent pockets — hence, suggesting that thebest strategy may be to bet on the section of thewheel following the actual predicted destination.

IV. EXPLOITS AND COUNTER-MEASURES

We would like to draw two simple conclusions fromthis work. First, deterministic predictions of theoutcome of a game of roulette can be made, andcan probably be done in situ. Hence, the tales ofvarious exploits in this arena are likely to be basedon fact. Second, the margin for profit is quite slim.Minor manipulation with the frictional resistanceor level of the wheel and/or the manner in whichthe croupier actually plays the ball (the force withwhich the ball is rolled and the effect, for exam-ple, of axial spin of the ball) have not been exploredhere and would likely affect the results significantly.Hence, for the casino the news is mostly good —minor adjustments will ameliorate the advantage ofthe physicist-gambler. For the gambler, one can restassured that the game is on some level predictableand therefore inherently honest.

Of course, the model we have used here is ex-tremely simple. In Strzalko et al.24 much more so-phisticated modeling methodologies have been pre-sented. Certainly, since the entire system is aphysical dynamical system, computational model-ing of the entire system may provide an even greatadvantage24. Nonetheless, the methods presented inthis paper would certainly be within the capabilitiesof a 1970’s “shoe-computer”.

ACKNOWLEDGEMENT

The first author would like to thank Marius Ger-ber for introducing him to the dynamical systemsaspects of the game of roulette. Funding for thisproject, including the roulette wheel, was providedby the Hong Kong Polytechnic University. Thelabors of final year project students, Yung ChunTing and Chung Kin Shing, in performing many ofthe mechanical simulations describe herein is grate-fully acknowledged.

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4BBC Online. ’Laser scam’ gamblers to keep £1m. BBCNews, December 5 2004.

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7J. P. Crutchfield, J. D. Farmer, N. H. Packard, and R. S.Shaw. Chaos. Scientific American, 255:46–57, 1986.

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11J. D. Farmer and J. J. Sidorowich. Predicting chaotic timeseries. Physical Review Letters, 59:845–848, 1987.

12C. Kingston. The Romance of Monte Carlo. John LaneThe Bodley Head Ltd., London, 1925.

13Life Magazine Publication. How to win $6, 500. Life,page 46, December 8 1947.

14The Italian mathematician, confusingly, was named DonPasquale9, a surname phonetically similar to Pascal. More-over. As Don Pasquale is also a 19th century opera buffa,this attribution is possibly fanciful.

15Three, if one has sufficient finances to assume the role ofthe house.

16Alternatively, and apparently erroneously, reported to befrom Californian Institute of Technology in9.

17The first, to the best of our knowledge.18Implementation on a “shoe-computer” should be relatively

straightforward too.19B. Okuley and F. King-Poole. Gamblers Guide to Macao.

South China Morning Post Ltd, Hong Kong, 1979.

20N. H. Packard, J. P. Crutchfield, J. D. Farmer, and R. S.Shaw. Geometry from a time series. Physical Review Let-ters, 45:712–716, 1980.

21H. Poincare. Science and Method. Nelson, London,1914. English translation by Francis Maitland, prefaceby Bertrand Russell. Facsimile reprint in 1996 by Rout-ledge/Thoemmes, London.

22C. Shannon. A mathematical theory of communication.The Bell Systems Technical Journal, 27:379–423, 623–656,1948.

23M. Small and C. K. Tse. Feasible implementation of aprediction algorithm for the game of roulette. In Asia-Pacific Conference on Circuits and Systems. IEEE, 2008.

24J. Strzalko, J. Grabski, P. Perlikowksi, A. Stefanski, andT. Kapitaniak. Dynamics of gambling, volume 792 of Lec-ture Notes in Physics. Springer, 2009.

25E. O. Thorp. Optimal gambling systems for favorablegames. Review of the International Statistical Institute,37:273–293, 1969.

26E. O. Thorp. The Mathematics of Gambling. GamblingTimes, 1985.

27Time Magazine Publication. Argentina — Bank breakers.Time, 135:34, February 12 1951.

28C. T. Yung. Predicting roulette. Final Year Project Re-port, Hong Kong Polytechnic University, Department ofElectronic and Information Engineering, April 2011.

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