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A Statistical Analysis of the Roulette Martingale System: Examples, Formulas and Simulations with R

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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.
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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 p0.5 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 2
2 3
21
1
n bb…br 1(1 )
n
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
, 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
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
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