testing the roulette wheel · with this roulette betting system, you will bet on the almost even...

TESTING THE ROULETTEWHEEL

Mihael Perman

University of Ljubljana

Osijek, June 4th, 2009

SOME QUOTATIONS

The generation of random numbers is too important to

be left to chance.

Robert R. Coveyou, Oak Ridge National Laboratory,

ZDA

The only way to win at roulette is to steal chips when

the croupier looks the other way.

Albert Einstein

With this roulette betting system, you will bet on the

almost even money bets. You will bet on black or red or

even or odd. You will not bet on the high-odds choices.

All you need to win with this system is to win 7 out of

20 times.

Internet, 1995

Roulette is the most glamorous of all the casino games.

An air of elegance surrounds the roulette table, and its

spinning wheel seems to be a perfect agent by which the

goddess of fortune may intevene in the affairs of mor-

tal men. How much superior is this unapproachable

mechanistic device to those games like dice and cards,

where human hands may tamper with fate.

But more than glamour, the game presents to me a

certain irresistible challenge. The roulette is intended

to be a symmetrical gambling device, the odds for which

always favour the house. In the long run, it would

appear that a player must inevitably lose. But due to a

certain degree of asymmetry in the wheel’s production,

or due to its later wear, the odds may shift enough to

favor a player on certain bets. The shrewd observer

may spot such a case and actually be able to play a

winning game. Herein lies the challenge.

Allan N. Wilson, Tha Casino Gambler’s Guide.

Unless we inconvenience ourselves by staying a long

time in the casino to increase the sample of spins,

how may we distinguish statistically a true weak bi-

ased number from the false random winners that even-

tually fluctuate all over the place and through which

we lose? This questions I have put to two professors

of mathematics, experts in roulette theory and play,

whom I quote elsewhere in this book, and they both de-

clare sadly that they have thus far no statistical method

to offer as a practical solution. For the greater success

of biased-wheel play, let’s hope that some day a solu-

tion may be found.

Russel T. Barnhart, Beating the Wheel

THE PROBLEM

The roulette wheel in principle generates random num-

bers uniformly distributed on the set {0, 1, 2, . . . , 36}.

Mechanical imperfections or wilful manipulation can lead

to deviations from uniformity in various ways. Gambling

houses are interested in statistics that would detect such

deviations as soon as possible with the smallest probabil-

ity of false alarms.

The reasons why “quality control” is desirable are the

following:

• The odds offered by the house should be those ad-

vertised.

• Skilled enough groups could take advantage of devi-

ations if they notice them before the supervisors of

the house. Relatively small deviations can “nudge”

the expected gain into the positive.

• Quality control should include the “human factor”.

Croupiers are human and could potentially cheat in

collusion with gamblers.

MATHEMATICAL FORMULATION

In statistical terms the problems is formulated as fol-

lows:

• We have observations X1, X2, . . . from the roullette

wheel taking values in {0, 1, 2, . . . , 36}. We will as-

sume that the observations are independent.

There are two main objectives:

• We would like to test the hypothesis (possibly se-

quentially)

H0 : X1, X2, . . . ∼ Uniform{0, 36} against

H1 : X1, X2, . . . ∼/ Uniform{0, 36} .

The question is what test statistics to choose and

how to decide whether to reject or accept the null–

hypothesis.

• We would like to detect a “change point”. The ob-

servations can start out as uniform but change to

another distribution as we are collecting the obser-

vations. How does one detect such a change?

THE CLASSICAL χ2 TEST

The usual χ2 test is the first idea to try.

Notation:

• Nkn is the frequency of outcome k after n spins of the

wheel.

• p0 is the probability of each outcome under the null–

hypothesis.

• m = 37.

We compute

χ2 =m−1∑

k=0

(Nk − np0)2

np0

.

0 5 10 15 20 25 30 35 400

0.005

0.01

0.015

0.02

0.025

0.03Probability distribution over cells

Fig. 1 Probability distribution for a “hanging” wheel

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 1000010

20

30

40

50

60

70

Number of spins

Trajectory of CHI statistic

Fig. 1a Behaviour of χ2–statistics for a “hanging” wheel

0 5 10 15 20 25 30 35 400

0.005

0.01

0.015

0.02

0.025

0.03

0.035


Fig. 2 Probability distribution for a “dented” wheel

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

20

40

60

80

100

120

140

Number of spins

Trajectories of CHI statistic and the likelihood ratio statistic

Fig. 2a Behaviour of χ2–statistics for a “dented” wheel

0 5 10 15 20 25 30 35 400

0.005

0.01

0.015

0.02

0.025

0.03


Fig. 3 Probability distribution for a “nicked” wheel

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

10

20

30

40

50

60

Number of spins

Trajectories of CHI statistic and the likelihood ratio statistic

Fig. 3a Behaviour of χ2–statistics for a “nicked” wheel

0 5 10 15 20 25 30 35 400

0.005

0.01

0.015

0.02

0.025

0.03Perfect probability distribution over cells

Fig. 4 Probability distribution for a perfect wheel

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

0

10

20

30

40

50

60

Number of spins

Trajectories of CHI statistics and likelihood ratio statistic

Fig. 4a Behaviour of χ2–statistics and the likelihood ratio

statistic for a perfect wheel

CENTRAL LIMIT THEOREM

The distributions of statistics to be used are all derived

from a simple observations based on the central limit the-

orem.

The vector

1√

np0

(N1

n−np0, N2

n−np0, . . . , Nmn −np0)

d→ Nm(0,Σ)

where Σ = I − 11T/m.

Remark: Multivariate normal vectors with the above

distribution are easy to simulate on the computer. One

only needs to simulate

(Z1 − Z̄, Z2 − Z̄, . . . , Zm − Z̄)

where the Z1, Z2, . . . , Zm are independent standard nor-

mals.

MAIN ALTERNATIVE HYPOTHESES

The alternative hypotheses we will consider:

0 5 10 15 20 25 30 35 400

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Celica

Probability distribution over cells

Fig. 5 Dented wheel. The payoff for betting on triplets

is 1:11. In the case shown betting on the first three cells

gives an expected payoff of 0.294.

0 5 10 15 20 25 30 35 400

0.005

0.01

0.015

0.02

0.025

0.03

Cells

Probability distribution over cells

Fig. 6 Hanging wheel. The payoff for betting on triplets

is 1:11. In the case shown betting on the best three cells

gives an expected payoff of 0.044.

CHOICE OF STATISTICS

We would like to devise statistics which would be bet-

ter traps for the given types of faults.

The χ2–test for the multinomial are based on the ex-

pression

χ2 =∑ (Observedi − Expectedi)

2

Expectedi

where the index i refers to cell number i. Normally the

cells would be non–overlapping but that is because only

in that case one can obtain analytical expressions for the

limit distribution.

• We want to monitor sectors of three, five or seven. A

plausible choice to monitor, say, triplets would be

CHI3 =m−1∑

k=0

(Nkn + Nk+1

n + Nk+2n − 3np0)

2

3np0

.

Here we interpret k + 1 and k + 2 modulo m − 1.

• Another plausible statistic is the maximum devia-

tion from the expected frequency of a “cell” which

can also be a sector of three, five or seven adjacent

pockets on the wheel.

MAX3 = max0≤k<m

Nkn + Nk+1

n + Nk+2n − 3np0√

np0

.

• Yet another alternative is the likelihood ratio test.

One looks at the quantity

LRATIO = log(supp Pp(Observed values)

Pp0(Observed values)

) .

As we observe more and more outcomes Wilks’ the-

orem asserts that 2 × LRATIO converges in distri-

bution to a χ2(36).

REMARK: As we do everything sequentially we actu-

ally observe stochastic processes of various statistics and

need to keep that in mind. So is there a “process” version

of Wilks’ theorem?

DISTRIBUTIONS OF STATISTICS

The distributions were obtained by simulation. Here

are the distributions of a sample of test statistics.

• CHI3 is the χ2–like statistic for triplets.

• MAX1 is a suitably standardised maximal positive

deviation from the expected frequences of cells.

• MAX3 is a suitably standardised maximal positive

deviation from the expected frequences for triplets.

0 50 100 150 200 250 300 350 4000

2000

4000

6000

8000

10000

12000

14000

16000

18000

CHI3

Den

sity

Distribution of CHI3

Fig. 7 Distribution of CHI3 statistic

0 1 2 3 4 5 60

2000

4000

6000

8000

10000

12000

14000

16000

18000

MAX1

Den

sity

Distribution of MAX1

Fig. 8 Distribution of MAX1 statistics’

1 2 3 4 5 6 7 8 90

2000

4000

6000

8000

10000

12000

14000

16000

MAX3

Den

sity

Distribution of MAX3

Fig. 9 Distribution of MAX3 statistics’

TRAJECTORIES FOR THE CHOSEN STATISTICS

The next few slides show various trajectories of CHIx

and MAXx statistics:

• Trajectories for an honest wheel.

• Trajectories for a dented wheel.

• Trajectories for a hanging wheel.

• Trajectories in two cases of real data.

LEGEND:

• CHI1 or MAX1

• CHI3 or MAX3

• CHI5 or MAX5

• CHI7 or MAX7

Perfect wheel

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

0

50

100

150

200

250

300

350

400

450

500

Number of spins

Trajectories of CHIx statistics with 95% tresholds

Fig. 10 Trajectories of CHIx statistics for a prefect wheel.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

1

2

3

4

5

6

7

8

Number of spins

Trajectories of MAXx statistics with 95% tresholds

Fig. 10a Trajectories of MAXx statistics for a prefect wheel.

Dented wheel

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

200

400

600

800

1000

1200

1400

1600

1800

2000

Number of spins


Fig. 11 Trajectories of CHIx statistics for a dented wheel.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

2

4

6

8

10

12

14

16

18

Number of spins


Fig. 11a Trajectories of MAXx statistics for a dented wheel.

Hanging wheel

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

200

400

600

800

1000

1200

1400

Number of spins


Fig. 12 Trajectories of CHIx statistics for a hanging wheel.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100001

2

3

4

5

6

7

8

9

10

Number of spins


Fig. 12a Trajectories of MAXx statistics for a hanging wheel.

Wheel AR04

0 1000 2000 3000 4000 5000 6000 7000 8000 90000

50

100

150

200

250

300

350

400

450

500

Number of spins

Trajectories of CHI statistics

Fig. 13 Trajectories of CHIx statistics for wheel AR04.

0 1000 2000 3000 4000 5000 6000 7000 8000 90001

2

3

4

5

6

7

8

Number of spins

Trajectories of MAX statistics

Fig. 13a Trajectories of MAXx statistics for wheel AR04.

Wheel HISPAR02

0 1000 2000 3000 4000 5000 6000 7000 8000 90000

200

400

600

800

1000

1200

1400

1600Statistike CHIx za cilinder HISPAR02 s 95% pragom

Fig. 14 Trajectories of CHIx statistics for wheel HISPAR02.

0 1000 2000 3000 4000 5000 6000 7000 8000 90000

2

4

6

8

10

12

14Statistike MAXx za cilinder HISPAR02 s 95% pragom

Fig. 14a Trajectories of MAXx statistics for wheel HISPAR02.

−5 0 5 10 15 20 25 30 35 400

0.005

0.01

0.015

0.02

0.025

0.03

0.035Empirical probability distribution over cells

Fig. 14c Empirical probability distribution over cells for wheel HISPAR02.

SEQUENTIAL TESTS

It is a luxury to assume a fixed number of observations.

There are several reasons for that:

• We never now why data collection has stopped. Was

that independent of teh outcomes? Usually that is

not the case.

• The house wants to stop a table as soon as there is

enough evidence that something is wrong.

MARTINGALES

The idea of a sequential test is that we reject the null-

hyposthesis as soon as possible given the significance level

α. But when is as soon as possible?

• One possible solution is to observe that the trans-

forms

χ̂2

k = k(χ2

k − mx(1 − x/m)) ,

where x is the width of the sector are MARTIN-

GALES under the null-hypothesis.

• For martingales one has MAXIMAL INEQUALITIES.

Under the null-hypothesis we can say

P ( max1≤k≤n

k

n(χ̂2

k − mx(1 − x/m))+ ≥ a) ≤

E[

(χ̂2n)

q+

]

aq,

where x+ is the positive part, and q ≥ 1.

TESTS

The test now does the following: choose an appropriate

q ≥ 1 and a > 0, and reject the null as soon as the

maximal inequality is violated.

One needs to calibrate the tests. As an example one

gets when α = 0.01

- For sectors of width x = 1 choose q = 8 and a =

29.97.


143.49.


316.00.


523.98.

The constant q is choosen in such a way that it mini-

mizes a.

EXAMPLES

Hanging wheel

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000−100

0

100

200

300

400

500

600

700

800

Stevilo iger

Trajektorije transformiranih statistik TCHIx za cilinder, ki visi

Wheel AR04

0 1000 2000 3000 4000 5000 6000 7000 8000 9000−50

0

50

100

150

200

250

300

350

Stevilo igerv

Potek transformiranih statistik TCHIx

THE KLOTZ STRATEGY

If the wheel is biassed there may be winning strategies.

One possible way is to maximize the expected logarithm

of your winnings. This idea from economics produces an

interesting strategy called the Klotz strategy.

The Klotz strategy is then combined with a Baysian

estimate of probabilities of certain outcomes in the sense

that

p̂i =ni + α

n + nα.

The parameter α may be interpreted as “caution”. The

higher it is, the less we are inclined to get exited by seem-

ingly more probable outcomes.

EXAMPLES

Here are some simulated and some real examples. In

all cases we take α = 100 and α = 200.

Slightly biassed wheel

0 1000 2000 3000 4000 5000 6000 7000 8000 90000

500

1000

1500

2000

2500

3000

3500

4000Potek kapitala pri previdnosti 100

Igra

Ka

pita

l

0 1000 2000 3000 4000 5000 6000 7000 8000 90000

500

1000

1500

2000

2500

3000

3500

4000Potek kapitala pri previdnosti 200

Igra

Ka

pita

l

More seriously biassed wheel

0 1000 2000 3000 4000 5000 6000 7000 8000 90000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

5 Potek kapitala pri previdnosti 100

Igra

Kap

ital

0 1000 2000 3000 4000 5000 6000 7000 8000 90000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

5 Potek kapitala pri previdnosti 200

Igra

Kap

ital

Real wheel AR04

0 1000 2000 3000 4000 5000 6000 7000 80000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Kap

ital

Igra

Potek kapitala na cilindru 0, previdnost=100

0 1000 2000 3000 4000 5000 6000 7000 80000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000Potek kapitala na cilindru 0, previdnost=200

Igra

Kap

ital

Real wheel HISPAR02

0 1000 2000 3000 4000 5000 6000 7000 80000

1

2

3

4

5

6

7

8

9

10x 10

4 Potek kapitala na cilindru 1, previdnost=100

Igra

Kap

ital

0 1000 2000 3000 4000 5000 6000 7000 80000

1

2

3

4

5

6

7

8

9

10x 10


Igra

Kap

ital

Real wheel Cammegh

0 1000 2000 3000 4000 5000 6000 7000 80000

1

2

3

4

5

6

7

8

9

10x 10

4

Kap

ital

Igra


0 1000 2000 3000 4000 5000 6000 7000 80000

1

2

3

4

5

6

7

8

9

10x 10

4

Kap

ital

Igra


Real wheel HISPAR04

0 1000 2000 3000 4000 5000 6000 7000 80000

1

2

3

4

5

6

7

8

9

10x 10


Kap

ital

Igra0 1000 2000 3000 4000 5000 6000 7000 8000

0

1

2

3

4

5

6

7

8

9

10x 10

4

Igra

Kap

ital


TESTING

One possible idea is to use the Klotz strategy as a test

statistics. If the optimal player starts winning too much

we reject the null-hypothesis. But what is too much?

Again we observe a few facts:

• Under the null-hypothesis the current capital of the

player is a non-negative supermartingale so it con-

verges to a finite limit.

• The supremum of the entire capital trajectory is a

finite random variable.

• One can either try to find an analytic estimate of the

distribution of the maximum or simulate.

• Here is the simulated distribution.

• The advantage is that the p-values have the meaning

in terms of money. It is not easy to get across simple

statistical ideas to the end-user.

The distribution of the maximum

0 10 20 30 40 50 60 700

50

100

150

200

250

300

350

Maximum

CONCLUDING REMARKS

Main points?

• One has to focus on certain types of alternative hy-

potheses. The entire space is just to big.

• The classical χ2–test does a poor job.

• If one takes marginal distributions of trajectories as

approximations to the “right” critical values one has

to proceed by simulation.

Remaining questions?

• Are the statistics chosen the right ones?

• What are the rules for deciding? In particular, what

are the right critical values for individual statistics?

Is it correct to just look at the marginal distribution?

Or does one have to consider the entire trajectory?

• If one were to test sequentially what is the right de-

cision rule?

• Is there a “process version” of Wilks’ theorem?

• What can one say about the asymptotic behaviour

of the test statistics? Do they converge under the

null-hypothesis?

testing the roulette wheel · with this roulette betting system, you will bet on the almost even...

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