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1

A Statistical Analysis of the Roulette Martingale System:

Examples, Formulas and Simulations with R1

Peter Pflaumer

Department of Statistics, Technical University of Dortmund

Abstract

Some gamblers use a martingale or doubling strategy as a way of improving their chances of winning.

This paper derives important formulas for the martingale strategy, such as the distribution, the

expected value, the standard deviation of the profit, the risk of a loss or the expected bet of one or

multiple martingale rounds. A computer simulation study with R of the doubling strategy is presented.

The results of doubling to gambling with a constant sized bet on simple chances (red or black

numbers, even or odd numbers, and low (1–18) or high (19–36) numbers) and on single numbers

(straight bets) are compared. In the long run, a loss is inevitable because of the negative expected

value. The martingale strategy and the constant bet strategy on a single number are riskier than the

constant bet strategy on a simple chance. This higher risk leads, however, to a higher chance of a

positive profit in the short term. But on the other hand, higher risk means that the losses suffered by

doublers and by single number bettors are much greater than that suffered by constant bettors.

1. Introduction

The martingale system is a popular betting strategy in roulette: Each time a gambler loses a bet, he

doubles his next bet, so that the eventual win leaves him with profit equal to his original stake.

However, the martingale system only works safely in casinos without table limits and where the

gambler has unlimited money. Both assumptions are not very likely. Therefore, the martingale

strategy is considered extremely risky. High losses are possible, although the probability of such a

loss is low. Various senses of the word “martingale” are reviewed by Mansuy (2009),

2. Martingale as a two-point distribution

It is assumed that the reader knows the casino game roulette. We regard an unbiased roulette, that is,

we assume that each number of the roulette wheel is equally likely. We restrict our analysis to the

European version of roulette with 37 numbers (with a single zero). However, the results can be

easily transferred to the American version of roulette with 38 numbers (with double zeros). Further

we first assume that if zero appears all bets on simple chances (red or black, even or odd, low or

high) are lost. They are not halved (à partager) or imprisoned (en prison) according to the rule of

some European casinos (see also Ethier, pp. 463–465). Finally, we assume that the gambler shall

risk a finite capital.

Denoting by g the profit or gain from a one-unit bet on a simple chance and by 19 / 37p the

probability of losing the bet, the expected value for the gambler is

( ) 1 1 1 1 2 0.027027Eg p p p .

And the variance is

22

22

( ) 1 1 1 1 2 4 4 4 1 0.99927Var g p p p p p p p .

Let N=1,2,3…,n be the number of coups or spins needed to achieve the first win in a martingale. A

martingale round consists of a number of N=1,2,…,n coups of consecutive losses followed by either

a win, or the loss of the total bet after n coups or n-1 doublings when the table limit has been

1 Paper presented at the 17th International Conference on Gambling & Risk Taking. May 27-30, 2019, Las

Vegas, NV.

2

reached. After a win or the total loss, the gambler starts a new martingale. Table 1 shows an

illustration of the martingale system with n=10 (table limit=512 units). The player wagers on red.

The probability of losing is p=19/37. The amount of the initial bet shall be one unit. On each loss,

the bet is doubled.

Table 1: Illustration of a martingale with n=10, p=19/37 (r=red, b=black & zero)

I II III IV V VI VII VIII IX X XI XII XIII

coup m black prob1 bet cum. bet gain G E(G) E(bet) E(m) p(m-1) prob2 E(betc)

IV × VII IV × VI IV × I V × XII

1 0 r 0.48649 1 1 1 0.48649 0.48649 0.48649 1.00000 0.48711 0.48711

2 1 br 0.24982 2 3 1 0.24982 0.74945 0.49963 0.51351 0.25014 0.50027

3 2 bbr 0.12828 4 7 1 0.12828 0.89799 0.38485 0.26370 0.12845 0.51379

4 3 bbbr 0.06588 8 15 1 0.06588 0.98814 0.26350 0.13541 0.06596 0.52768

5 4 bbbbr 0.03383 16 31 1 0.03383 1.04867 0.16914 0.06954 0.03387 0.54194

6 5 bbbbbr 0.01737 32 63 1 0.01737 1.09439 0.10423 0.03571 0.01739 0.55659

7 6 bbbbbbr 0.00892 64 127 1 0.00892 1.13288 0.06244 0.01834 0.00893 0.57163

8 7 bbbbbbbr 0.00458 128 255 1 0.00458 1.16808 0.03665 0.00942 0.00459 0.58708

9 8 bbbbbbbbr 0.00235 256 511 1 0.00235 1.20201 0.02117 0.00484 0.00236 0.60295

10 9 bbbbbbbbbr 0.00121 512 1023 1 0.00121 1.23570 0.02483 0.00248 0.00121 0.61924

10 10 bbbbbbbbbb 0.00128 -1023 -1.30435 1.30435

sum 1

-0.3056311.30815 2.05293 2.05293 1 5.50829

Remarks:

1

11

m

prob p p

;

1

21

21..

m

n

p

prob

p

pp

.

E(G)=expected gain, E(bet)=expected bet, E(m)=expected number of coups of a martingale round;

E(betc)=expected bet per coup of a martingale round; n=10: after 10 coups the table limit of 512 has

been reached.

From Table 1 follows the presentation with formulas in Exhibit 1, where p≥0.5 is the probability of

losing and the gambler might bet 1 unit on the first spin on red.

Exhibit 1: Illustration of a martingale with formulas (r=red, b=black & zero);

______________________________________________________________________________

coup

m colour probability bet cumulative bet gain

1 r

1

p

2 0 1

21

1

2 br

1

p

p 2

1 2

21

1

3 bbr

21

p

p 2

2 3

21

1

n bb…br 1(1 )

n

p

p

2

n-1 21

n

1

n bbb…b n

p

(21)

n

(r does not show up)

_______________________________________________________________________________

The sum of the probabilities in the third column is

1

1

1(1)1

ninnn

ippp pp

.

The probability that the gambler will lose all n bets is pn. When all bets lose, the total loss is 21

n

.

3

The probability that the gambler does not lose all n bets is 1-pn. In all other cases, the gambler wins

one unit. Thus, the expected profit or gain per martingale is

1

1

()1 121 1 1 21 1(2)

n

nnn i nn n

i

EG p p p p p p

.

If 19 / 37p, then the expectation is

10

19

( ) 1 2 -0.3056

37

EG

.

The distribution of the gain or profit Gi in the i-th martingale round follows a two-point distribution,

i.e.

21

nn

i

PG p and

11 n

i

PG p

with the expected value and the variance

() 2 1 11 1 2n

nn n

i

E

Gppp ,

2

() 4 2

nn

i

Var G p p .

With p=19/37 and n=10, one calculates the expected value and the variance as ( ) -0.3056

i

EG

and ( ) 1335.7

i

Var G . The standard deviation is ( ) 36.54

i

G

.

The variance reduces to

()2 1

n

i

Var G if p=0.5. If n=10 then we obtain 10

( ) 2 1 1023

i

Var G .

After M martingale rounds, the total profit will be

1

M

i

i

WG

. Expected value and variance of the

total profit are

() 1 2 n

E

WM p and

2

() 4 2

nn

Var W M p p .

If i

X

i=1,2,…,M is a random variable with a Bernoulli distribution with (0)

n

i

PX p and

(1)1

n

i

PX p , then the sum of M independent Bernoulli trials

1

M

i

i

X

X

has a binomial

distribution

,,1 n

B

IN M x p.

The linear transformation

21 2

nn

WMX XXM M

has the same binomial distribution. X is the number of martingale rounds 0,1, 2......

x

M which

were successful and ended with a win of one unit. If we define the number of busts Y=M-X, where

the martingale rounds are counted which ended with a loss, then we obtain 2n

WM Y.

The normal distribution can be used as an approximation to the binomial distribution of X or W if

the following rule of thumb holds:

19

nn

Mpp .

The Poisson distribution can be used as an approximation of X or W with n

M

p

if n is not too

small.

The probability that the total profit will be less than w is

()

()

wEW

PW w Var W

, where

is

the distribution function of the standard normal distribution. Specifically, the probability for a loss is

given by

()

0()

EW

PW Var W

.

The probability of a loss is a function of the number of played martingales if n and p are given.

4

1

1

2

1

12

()

0(1 ) () 42

n

nn

Mp

EW

PW u

Var W Mpp

,

where 1

u

is the (1-)-quantile of the standard normal distribution. Solving for M1- yields the

required number of martingale games corresponding to a given loss or win probability

2

222

11 1

2

42

21

nn

n

pp

M

uuCV

p

with the coefficient of variation ()

()

Var G

CV EG

. In

particular, 2

0.84

CV M.

With 10n and 19

37

p we obtain:

Loss probability 1- Win probability Quantile u1- Number of martingales

M1-

0.8 0.2 0.8416 10,125

0.84 0.16 1 119.562=14,295

0.9 0.1 1.2816 23,480

0.95 0.05 1.6449 38,680

0.99 0.01 2.3263 78,026

0.999 0.001 3.0902 136,508

E.g., we recognise that the probability of a positive profit is only 1 percent after playing 78,026

martingale rounds or 160,182 expected coups or spins (see section 3).

Figure 1 shows the distribution of the total profit after M=10,000 martingale rounds (more detailed

results are listed in the Appendix). The expected profit is E(W)=-3056.27, and the standard

deviation is 3655.66

W

. The probability of a positive profit is 18.28 percent if calculated with

the binomial distribution. The approximation with the normal distribution yields about 20 percent.

From Fig. 1 or more accurately from the table in the Appendix, we can observe that the probability

of a loss of 10,480 units is about 1.5 percent. The probability that the loss is 10,480 units or higher

is 3.55 percent. On the other side, the probability of a (positive) gain of 3,856 units or higher is only

about 3 percent.

p(W)

0.0000

0.0100

0.0200

0.0300

0.0400

0.0500

0.0600

0.0700

0.0800

0.0900

0.1000

0.1100

0.1200

-18000 -16000 -14000 -12000 -10000 -8000 -6000 -4000 -2000 0 2000 4000 6000 8000 10000 12000

W

Figure 1: Profit distribution after 10,000 martingale rounds

Exhibit 2 shows the results for only 20 martingale rounds or about 41 coups. The win probability is

very high. But the possible loss of 1,004 units considerably exceeds the possible total profit of only 20

5

units. Gamblers should be aware that the martingale play is a very risky strategy which could produce

extremely high losses.

Exhibit 2: Results of playing 20 martingales

____________________________________________________________________________________

M 20 E(G) -0.30562581

n 10 Var(G) 1335.6552

S(G) 36.5466168

p 0.51351351

1-pn 0.99872497

pn 0.00127503 E(W) -6.11251612

Var(W) 26713.104

n

M

p

0.0255005 S(G) 163.441439

x

(successes)

y (busts) W Binomial Cum.

Binomial

Poisson

20 0 20 0.9748 0.9748 0.9748

19 1 -1004 0.0249 0.9997 0.0249

18 2 -2028 0.0003 1.0000 0.0003

17 3 -3052 0.0000 1.0000 0.0000

. . . . . .

____________________________________________________________________________________

3. Expected number of coups and expected gain and bet of a martingale round

The gambler shall put his bet always on red. If, e.g., red appears after two black colours (bbr), the

martingale ends after 3 coups. The probability for this event is

21

p

p

, and the total amount bet

is 3

211247 . The maximum number of coups is n because of the table limit.

From Table 1 and Exhibit 2, we conclude that the expected number of coups is given by

1

11 1

11

1

() 1 1 1 1

n

nn

innin

ii

p

Em i p p n p p p i p p n p

p

.

Note that

1

0

1

1

n

ni

p

p

p

.

After calculating the second moment 2

()Em

, we find for the variance

2

2

() 2 11

1

nn

p

pp

Var m n

p

p

.

If n is large, we can use the following approximations for the expected value and the variance

(parameters of the geometric distribution):

1

() 1

Em

p

,

2

()

1

p

Var m

p

.

Table 2 shows the parameters of the number of coups m within a martingale game as a function of

the maximum rounds n due to the table limit. (p=19/37). Without a table limit, the expected value is

2.0555 and the variance is 2.17.

6

Table 2: Parameters of the number of coups

n Expectation Variance

Standard

deviation

1 1 0 0

2 1.514 0.250 0.500

3 1.777 0.701 0.837

4 1.913 1.149 1.072

5 1.982 1.504 1.226

6 2.018 1.754 1.324

7 2.036 1.918 1.385

8 2.046 2.021 1.421

9 2.050 2.083 1.443

10 2.053 2.120 1.456

Let 1, 2, . .

i

mn be the number of coups of the i-th martingale round. Then the total number of

coups of a roulette game is

1

M

i

i

Nm

with () ()EN M Em

and () ()Var N M Var m . E(N) is

the number of martingale rounds multiplied by the expected number of coups within a martingale

round.

The expected total amount bet within a martingale round is given by (see Exhibit 1)

1

1

12 ()

() 21 1 21 12 ()

n

nii nn

i

pEG

Ebet p p p

p

Eg

.

If p=0.5, we get

0.5

12

()lim

12

n

p

p

Ebet n

p

.

Ethier (2010, p. 279) remarks that the ratio () () (1 2)

()

EG Eg p

Ebet

corresponds to the expected

profit from a single-unit bet. This is not coincidental. He shows that all systems have this property

(see Ethier, 2010, p. 298 ff). “All betting systems lead ultimately to the same mathematical

expectation of gain per unit amount wagered” (Epstein, 2009, p. 52).

The variance is given by

2

(2 ) ( 1)

32 2 (2 1) 1 (2 )

4 1 1 2 (2 1)( 4 1) 1 2

nn n

npp

Var bet p p ppp p

0.5p

or

2

32 2 3

n

Var bet n n if 0.5p

.

Since a martingale consists on average of 1

() 1

n

p

Em

p

coups, we can conclude that the expected

value of a bet per coup is

12

1()

()

112 ()()

n

cn

p

pEG

Ebet

p

pEgEm

0.5p

.

7

Table 3: Parameters of the bet per martingale round as a function of n

p=19/37 p=0.5

n

Expected

value Variance

Standard

deviation

Expected

value Variance

Standard

deviation

1 1.000 0.000 0.000 1 0 0.000

2 2.027 0.999 1.000 2 1 1.000

3 3.082 6.158 2.482 3 6 2.449

4 4.165 22.140 4.705 4 21 4.583

5 5.278 62.813 7.925 5 58 7.616

6 6.420 156.854 12.524 6 141 11.874

7 7.594 363.378 19.062 7 318 17.833

8 8.799 804.021 28.355 8 685 26.173

9 10.037 1728.871 41.580 9 1 37.868

10 11.308 3651.859 60.431 10 2949 54.305

An alternative approach to calculating the expected value ()

c

Ebet , the second moment 2

()

c

E

bet ,

and thereby the variance ()

c

Var bet uses a modification of the geometric probability distribution of

the bet per coup which is seen in the following scheme in Exhibit 3, where n is the maximum

number of coups.

The expected value and the second moment of this distribution are given by

11

1

00

0

111(2)

() 2 2

1112

in

nn

iii

cnnn

i

ii

i

p

ppp

Ebet p

p

pp

p

,

1

22

0

111(4)

() (2)

1114

n

nii

cnn

i

p

pp

Ebet p

p

pp

,

2

2

()( ) ()

cc c

Varbet Ebet Ebet .

Exhibit 3: Derivation of the expected bet per coup

________________________________________________________________________________

coup i betc probability probability without table limit

since

1

0

1

lim 1

ni

iip

p

1 0

2 21

1

1..

n

p

pp

1

p

2 1

2 21

1..

n

p

p

pp

1

p

p

3 2

2

2

21

1..

n

p

p

pp

21

p

p

n 1

2n

1

21

1..

n

n

p

p

pp

11

n

p

p

________________________________________________________________________________

Simplifications of the above formulas arise if p=0.5:

8

1

2

()

21 2

n

cnn

Ebet n

21

()2

n

c

Ebet

2

1

12

()2 21

n

n

cn

Var bet n

.

4. Roulette simulations with R and its results

A simulation with R was carried out for 20,529 coups wagering on a simple chance. We chose this

number because we wanted to simulate about 10,000 martingale rounds. The simulation was

repeated 1,000 times. The initial bet was 1 unit on red. The probability of losing was 19/37. After

each loss, the bet was doubled until reaching the table limit of 512 units. Table 4 shows important

parameters (mean, standard deviation, and percentiles). The series length shows the maximum

number of times the colour red appeared in a row. Schilling (2012) provides approximation

formulas for the longest run of red or black. Other simulations of a roulette wheel can be found, e.g.,

in Turner (1998), Croucher (2005) or Kendall (2018).

Table 4: Simulation results

mean sd 0% 25% 50% 75% 100% skew kurtos

total profit W -2948.206 3693.94 -17753 -5385 -3181.5 -262 6995 -0.168 -0.014

bet per coup 5.51415 0.28 4.729699 5.317843 5.504676 5.694895 6.471723

max. series length 14.694 1.90 11 13 14 16 24

no. of martingales 9998.387 69.24 9808 9953 9998.5 10045 10194

The probability of a negative total profit is 80 percent.

The calculated and simulated values of the total profit and the bet per coup are more or less identical

as the following formulas show. Expected value and variance of the total profit after 10,000

martingale rounds or after 20,529 coups are

10

19

( ) 1 2 10, 000 1 2 -3,056.26

37

n

EW M p

and

10 20

219 19

( ) 4 2 10, 000 4 2 3654.66

37 37

nn

WMp p

.

The expected bet per coup is

10

10

19

19 12

1

12

137

37

( ) 5.508

19

112 19 12

137

37

n

cn

p

p

Ebet pp

.

Using the normal approximation, we get the probability of a negative total profit

3056

0 0.84 0.8

3655

PW

.

The distribution of the outcome is skewed to the left even after more than 20,000 coups (see Fig. 2).

In one simulation, the colour red appeared 24 times in a row. The series length record was registered

in 1943, when the colour red came up 32 times in a row (www.casino-games-

online.biz/roulette/record-series.html).

9

Figure 2: Histograms of the output or total profit (V1), the bet per coup (V2), the maximum series

length of red (V3), and the number of martingales (V4)

The next simulation presents one possible trend of the profit W playing around 100,000 martingales

(see Figure 3). The loss of the player with an initial wealth of zero and an initial bet of one will

0 50000 100000 150000 200000

-25000 -15000 -5000

coup numbe r

profit W

Figure 3: Trend of the total profit W of 205,301 simulated coups (roughly 100,000

martingales)

V2

frequency

5.0 5.5 6.0 6.5

0 50 100 150 200 250

V1

frequency

-15000 -10000 -5000 0 5000

0 50 100 150 200

10

amount to 24,097 after 205,301 coups. The highest profit of the series is 1,773, and the highest loss

is 25,437.

In the case of the absence of a table limit, the player would win all martingales in this

simulation if he were able to bet a maximum of 262,144 units (see Table 5).

Table 5: Distribution of the bets of 200,305 coups in the absence of a table limit

m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

bet 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 131072 262144

freq. 99687 51238 26338 13601 6983 3599 1901 964 491 245 121 63 33 17 10 4 3 2 1

5. Comparison of different roulette strategies

We will investigate how well a player using a doubling strategy would compete compared to a player

betting a comparable amount on a simple chance and a player betting a comparable amount on single

number each time (plein). See also Turner (1998) or Croucher (2005).

Assumptions:

a) Martingale player: Initial bet is one unit, doubling after each loss up to a table limit of 512 (a

maximum of 9 doublings). He plays 10,000 martingale rounds or expected 20,529 coups. We know

from the results of the previous chapter that the expected bet per coup is ( ) 5.508

c

Ebet .

b) Simple chance player: He plays 20,529 times and wagers 5.508 units each time on a simple

chance.

c) Plein player: He plays 20,529 times and wagers 5.508 units each time on a single number (plein).

Expected values and standard deviations of the three strategies are given in Table 6.

Table 6: Parameters of the total profit after 10,000 martingale rounds or 20,529 coups

E(W) ()W

Simple

chance

-3,056 0.027 5.508 20529 788.9 5.508 0.9993 20529

Single

number

-3,056 0.027 5.508 20529 4,607.1 5.508 34.08 20529

Martingale

n=10

-3,056 3,654.7

Figure 4 shows the density and distribution functions of the total profit for the chosen strategies using

the normal approximation. The riskiest strategy with the highest standard deviation is betting on a

single number. The highest risk yields also the highest probability of about 25 percent for a positive

total profit. The selected martingale strategy is comparable to the single number strategy with slightly

less risk. The probability of a positive profit is around 20 percent. With the simple chance strategy, it

is practically impossible to have a positive profit after 20,529 coups. High risk increases the

probability of a positive profit, but it also increases the risk of severe losses, as can been seen clearly

in Figure 4.

11

-20000 -5000 5000

0.0000 0.0002 0.0004

W

density

-20000 -5000 5000

0.00.20.40.60.81.0

W

P(W<w)

Figure 4: Density and distribution functions of the total profit for different roulette strategies after

20,529 coups (red: single number, blue: martingale, black: simple chance)

Next, we compare only 100 martingale rounds (same assumptions as above) with 205 coups betting

5.508 units on a single number. We calculate the following parameters:

Martingale betting

Straight betting

Expectation -30.56

-30.52

Standard deviation 365.41 460.41

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-3500 -3000 -2500 -2000 -1500 -1000 -500 0 500 1000 1500 2000

profit

Figure 5: Distributions of the total profit after 205 coups (martingale betting vs. single number

betting)

12

The probability that the martingale player wins 100 units is 88 percent. The probability of a positive

profit of a player betting on straight (single number) is only 48 percent. In contrast to the long-term

view, the probability of a positive profit is here lower for the single number player. But betting on

straight avoids extreme losses and often provides higher earnings compared to martingale betting

(see Figure 5).

The standard deviation does not reflect sufficiently the extreme risk of a martingale strategy. The

maximum loss of the straight betting strategy is around 1,000, whereas the maximum loss of the

martingale strategy is three times as high.

6. American roulette and consideration of en prison and la partage rules

The probabilities in American and European roulette are different because American roulette has an

extra green number (the double zero, 00). The probability of losing one bet on a simple chance is

20 0.5263

38

p . The expected gain decreases in this case form E(g)= -0.027 to E(g)= -0.0526.

However, all our derived formulas of the martingale strategy can be further used if we replace

19

37

p by 20

38

p.

Expected value and variance of the American roulette with absence of the special zero rules are

18 20 1

( ) 1 1 0.052632

38 38 19

Eg ,

2

1360

() 1 19 361

Var g

,

0.998614

g

.

The case is more complicated if we consider European roulette casinos using the “en prison” rule or

the “la partage” rule.

With the “la partage” rule, the player loses half the bet on a simple chance when the zero turns up.

Expected value and variance are no longer calculated by using a two-point distribution:

18 18 1 1 1

( ) 1 1 0.013514

37 37 2 37 74

Eg ,

2

2

22

18 18 1 1 145

()1 1

37 37 2 37 148

Eg

.

And the variance and the standard deviation are

2

145 1 1341

() 148 74 1369

Var g

,

0.989721

g

.

In order to use above martingale formulas, we propose using 18.75

37

p, being aware that the

results are now approximations. In this case, we obtain

13

18.25 18.75 1

( ) 1 1 0.013514

37 37 74

Eg ,

2

1 5475

() 1 74 5476

Var g

,

0.9999

g

.

With the “en prison” rule, the player leaves the bet (en prison = in prison) for the next spin of the

roulette wheel. If the subsequent spin is again zero, then the whole bet is lost. Otherwise the player's

money is returned.

Expected value and standard deviation are (derivation see Ethier, 2010, p. 464, Feldman/Fox, 1991,

p. 109)

1

( ) 0.013701 73

Eg ,

0.993220

g

.

In order to use above martingale formulas for approximation results, we should put 18.5

36.5

p,

where we obtain

18 18.5 1

( ) 1 1 0.01370

36.5 36.5 73

Eg ,

2

1 5328

() 1 73 5329

Var g

,

0.9999

g

.

A more sophisticated approach based on the appearance of zeros and colours is found in a

publication of Schneider (1997, p. 68) with

22

18

911754

37

1 0.49305

1849195

118 1 18

137 37 37 37

p

.

In this case, we should put 937441 0.50695

1849195

p

in order to use the above martingale formulas.

7. Conclusion

Methods for teaching introductory statistics are often considered ineffective because they do not show

a clear context between statistics and their use in the real world. A nice and instructive example of

illustrating statistical distributions in statistics courses is the application of the roulette martingale

strategy.

14

References

Croucher, J (2005): A Comparison of Strategies for Playing Even Money Bets in Roulette, Teaching

Statistics 27(1), 20–23.

Epstein, R A (2009): The Theory of Gambling and Statistical Logic, 2nd ed., Burlington.

Ethier S N (2010): The Doctrine of Chances: Probabilistic Aspects of Gambling, Berlin Heidelberg.

Feldman D; Fox M (1991): Probability: The Mathematics of Uncertainty, New York.

Hannum R (2007): The Partager Rule at Roulette: Analysis and Case of a Million Euro Win. In

Optimal Play: Mathematical Studies of Games & Gambling, Ethier, S., & Eadington, W., eds.

Reno, NV: Institute for the Study of Gambling & Commercial Gaming, University of Nevada

Kendall, G (2018): Did a roulette system “break the bank”? Significance, December, 26-29.

Mansuy, R (2009): The Origins of the Word “Martingale”, Electronic Journa@l for History of

Probability and Statistics, 5.1, June, 1-10.

Schilling, M F (2012): The Surprising Predictability of Long Runs, Math. Mag. 85, 141–149.

Schneider, R (1997): Roulette: Strategien und Gewinnchancen: Eine wahrscheinlichkeitstheo-

retische Analyse, Berlin.

Turner N E (1998): Doubling vs. Constant Bets as Strategies for Gambling, Journal of Gambling

Studies 14(4), 413–429.

15

Appendix: Profit distribution after 10,000 martingale rounds

M=10,000 martingales. n=10. p=19/37

M 10,000 EG -0.30562581

n 10 Var(G) 1335,6552

S(G) 36.5466168

p 0.51351351

1-pn 0.99872497

Pn 0.00127503 E(W) -3056,25806

Var(W) 13,356,552

lambda 12.750252 S(G) 3654,66168

rule npq>9 12.7339951 12.750252

y (busts) W Binomial Cum.

Binomial

Distribution

Poisson Cum.

Normal

Distribution

0 10,000 0 0 0 0.0002

1 8,976 0 0 0 0.0005

2 7,952 0.0002 0.0003 0.0002 0.0013

3 6,928 0.001 0.0013 0.001 0.0031

4 5,904 0.0032 0.0045 0.0032 0.0071

5 4,880 0.0081 0.0126 0.0081 0.0149

6 3,856 0.0173 0.0299 0.0173 0.0293

7 2,832 0.0315 0.0614 0.0315 0.0536

8 1,808 0.0502 0.1116 0.0503 0.0916

9 784 0.0712 0.1828 0.0712 0.1467

10 -240 0.0908 0.2736 0.0908 0.2205

11 -1,264 0.1053 0.3789 0.1052 0.3119

12 -2,288 0.1119 0.4908 0.1118 0.4168

13 -3,312 0.1097 0.6005 0.1097 0.5279

14 -4,336 0.0999 0.7004 0.0999 0.6369

15 -5,360 0.0849 0.7854 0.0849 0.7358

16 -6,384 0.0677 0.8531 0.0677 0.8187

17 -7,408 0.0507 0.9038 0.0507 0.8831

18 -8,432 0.0359 0.9397 0.0359 0.9293

19 -9,456 0.0241 0.9638 0.0241 0.96

20 -10,480 0.0154 0.9792 0.0154 0.9789

21 -11,504 0.0093 0.9885 0.0093 0.9896

22 -12,528 0.0054 0.9939 0.0054 0.9952

23 -13,552 0.003 0.9969 0.003 0.998

24 -14,576 0.0016 0.9985 0.0016 0.9992

25 -15,600 0.0008 0.9993 0.0008 0.9997

P(W<0)=0.7985; P(W>0)=0.2015

16

M=20 martingales, n=10, p=19/37

M 20 EG -0.30562581

n 10 Var(G) 1335.6552

S(G) 36.5466168

p 0.51351351

1-pn 0.99872497

pn 0.00127503 E(W) -6.11251612

Var(W) 26713.104

lambda 0.0255005 S(G) 163.441439

x

(success)

y (busts) W Binomial Cum.

Binomial

Poisson

20 0 20 0.9748 0.9748 0.9748

19 1 -1004 0.0249 0.9997 0.0249

18 2 -2028 0.0003 1.0000 0.0003

17 3 -3052 0.0000 1.0000 0.0000

16 4 -4076 0.0000 1.0000 0.0000

15 5 -5100 0.0000 1.0000 0.0000

14 6 -6124 0.0000 1.0000 0.0000

13 7 -7148 0.0000 1.0000 0.0000

12 8 -8172 0.0000 1.0000 0.0000

11 9 -9196 0.0000 1.0000 0.0000

10 10 -10220 0.0000 1.0000 0.0000

9 11 -11244 0.0000 1.0000 0.0000

8 12 -12268 0.0000 1.0000 0.0000

7 13 -13292 0.0000 1.0000 0.0000

6 14 -14316 0.0000 1.0000 0.0000

5 15 -15340 0.0000 1.0000 0.0000

4 16 -16364 0.0000 1.0000 0.0000

3 17 -17388 0.0000 1.0000 0.0000

2 18 -18412 0.0000 1.0000 0.0000

1 19 -19436 0.0000 1.0000 0.0000

0 20 -20460 0.0000 1.0000 0.0000