Download - Martingale Comparison
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Probability of win
Trial 1 Trial 2 Trial 3
Outcome Outcome Outcome
1 Win 0.5 1 Win 0.5 1 Win
1 Win 0.5 1 Win 0.5 1 Loss
1 Win 0.5 1 Loss -0.5 2 Win
1 Loss -0.5 2 Win 0.5 1 Win
1 Win 0.5 1 Loss -0.5 2 Loss
1 Loss -0.5 2 Win 0.5 1 Loss
1 Loss -0.5 2 Loss -0.5 4 Win
1 Loss -0.5 2 Loss -0.5 4 Loss
Total expectancy using Martingale
Trial 1 Trial 2 Trial 3
Outcome Outcome Outcome
1 Win 0.5 1 Win 0.5 1 Win
1 Win 0.5 1 Win 0.5 1 Loss
1 Win 0.5 1 Loss -0.5 1 Win
1 Loss -0.5 1 Win 0.5 1 Win
1 Win 0.5 1 Loss -0.5 1 Loss
1 Loss -0.5 1 Win 0.5 1 Loss
1 Loss -0.5 1 Loss -0.5 1 Win
1 Loss -0.5 1 Loss -0.5 1 Loss
Total expectancy using fixed size betting
Bet sizeb1 Probabilityp1 Bet sizeb2 Probabilityp2 Bet sizeb3
Bet sizeb1
Probabilityp1
Bet sizeb2
Probabilityp2
Bet sizeb3
The above illustrates, over 3 trials, that the Martingale offers no ege in eOnly by changing the win probability in the yellow cell, can we alter e!pect
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50
#inal e!pectancy
b1p1 $ b2p2 $ b3p3
0.5 1.5
-0.5 0.5
0.5 1.0
0.5 1.0
-0.5 -1.0
-0.5 0.0
0.5 0.5
-0.5 -!.5
0.0
#inal e!pectancy
b1p1 $ b2p2 $ b3p3
0.5 1.5
-0.5 0.5
0.5 0.5
0.5 0.5
-0.5 -0.5
-0.5 -0.5
0.5 -0.5
-0.5 -1.5
0.0
Probabilityp3
Probabilityp3
pectancy over flat betting"ancy"
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1 2 3 % & ' 1 2 3 % & ' 1 2 3 % & ' Perm(
) )))))) ) ) ) ) ) ) ) ) ) ) ) ) 1 2 3 % & ' 1
1 )))))1 ) ) ) ) ) 1 ) ) ) ) ) 1 1 2 3 % & & 2
2 ))))1) ) ) ) ) 1 ) ) ) ) ) 1 1 1 2 3 % % & 3
3 ))))11 ) ) ) ) 1 1 ) ) ) ) 1 2 1 2 3 % % % 4
% )))1)) ) ) ) 1 ) ) ) ) ) 1 1 1 1 2 3 3 % & 5
& )))1)1 ) ) ) 1 ) 1 ) ) ) 1 1 2 1 2 3 3 % % 6
' )))11) ) ) ) 1 1 ) ) ) ) 1 2 2 1 2 3 3 3 % 7
* )))111 ) ) ) 1 1 1 ) ) ) 1 2 3 1 2 3 3 3 3 8
+ ))1))) ) ) 1 ) ) ) ) ) 1 1 1 1 1 2 2 3 % & 9
))1))1 ) ) 1 ) ) 1 ) ) 1 1 1 2 1 2 2 3 % % 10
1) ))1)1) ) ) 1 ) 1 ) ) ) 1 1 2 2 1 2 2 3 3 % 11
11 ))1)11 ) ) 1 ) 1 1 ) ) 1 1 2 3 1 2 2 3 3 3 12
12 ))11)) ) ) 1 1 ) ) ) ) 1 2 2 2 1 2 2 2 3 % 1313 ))11)1 ) ) 1 1 ) 1 ) ) 1 2 2 3 1 2 2 2 3 3 14
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1% ))111) ) ) 1 1 1 ) ) ) 1 2 3 3 1 2 2 2 2 3 15
1& ))1111 ) ) 1 1 1 1 ) ) 1 2 3 % 1 2 2 2 2 2 16
1' )1)))) ) 1 ) ) ) ) ) 1 1 1 1 1 1 1 2 3 % & 17
1* )1)))1 ) 1 ) ) ) 1 ) 1 1 1 1 2 1 1 2 3 % % 18
1+ )1))1) ) 1 ) ) 1 ) ) 1 1 1 2 2 1 1 2 3 3 % 19
1 )1))11 ) 1 ) ) 1 1 ) 1 1 1 2 3 1 1 2 3 3 3 20
2) )1)1)) ) 1 ) 1 ) ) ) 1 1 2 2 2 1 1 2 2 3 % 21
21 )1)1)1 ) 1 ) 1 ) 1 ) 1 1 2 2 3 1 1 2 2 3 3 22
22 )1)11) ) 1 ) 1 1 ) ) 1 1 2 3 3 1 1 2 2 2 3 23
23 )1)111 ) 1 ) 1 1 1 ) 1 1 2 3 % 1 1 2 2 2 2 24
2% )11))) ) 1 1 ) ) ) ) 1 2 2 2 2 1 1 1 2 3 % 25
2& )11))1 ) 1 1 ) ) 1 ) 1 2 2 2 3 1 1 1 2 3 3 26
2' )11)1) ) 1 1 ) 1 ) ) 1 2 2 3 3 1 1 1 2 2 3 27
2* )11)11 ) 1 1 ) 1 1 ) 1 2 2 3 % 1 1 1 2 2 2 28
2+ )111)) ) 1 1 1 ) ) ) 1 2 3 3 3 1 1 1 1 2 3 29
2 )111)1 ) 1 1 1 ) 1 ) 1 2 3 3 % 1 1 1 1 2 2 30
3) )1111) ) 1 1 1 1 ) ) 1 2 3 % % 1 1 1 1 1 2 31
31 )11111 ) 1 1 1 1 1 ) 1 2 3 % & 1 1 1 1 1 1 32
32 1))))) 1 ) ) ) ) ) 1 1 1 1 1 1 ) 1 2 3 % & 33
33 1))))1 1 ) ) ) ) 1 1 1 1 1 1 2 ) 1 2 3 % % 34
3% 1)))1) 1 ) ) ) 1 ) 1 1 1 1 2 2 ) 1 2 3 3 % 35
3& 1)))11 1 ) ) ) 1 1 1 1 1 1 2 3 ) 1 2 3 3 3 36
3' 1))1)) 1 ) ) 1 ) ) 1 1 1 2 2 2 ) 1 2 2 3 % 37
3* 1))1)1 1 ) ) 1 ) 1 1 1 1 2 2 3 ) 1 2 2 3 3 38
3+ 1))11) 1 ) ) 1 1 ) 1 1 1 2 3 3 ) 1 2 2 2 3 39
3 1))111 1 ) ) 1 1 1 1 1 1 2 3 % ) 1 2 2 2 2 40
%) 1)1))) 1 ) 1 ) ) ) 1 1 2 2 2 2 ) 1 1 2 3 % 41
%1 1)1))1 1 ) 1 ) ) 1 1 1 2 2 2 3 ) 1 1 2 3 3 42
%2 1)1)1) 1 ) 1 ) 1 ) 1 1 2 2 3 3 ) 1 1 2 2 3 43
%3 1)1)11 1 ) 1 ) 1 1 1 1 2 2 3 % ) 1 1 2 2 2 44
%% 1)11)) 1 ) 1 1 ) ) 1 1 2 3 3 3 ) 1 1 1 2 3 45
%& 1)11)1 1 ) 1 1 ) 1 1 1 2 3 3 % ) 1 1 1 2 2 46
%' 1)111) 1 ) 1 1 1 ) 1 1 2 3 % % ) 1 1 1 1 2 47
%* 1)1111 1 ) 1 1 1 1 1 1 2 3 % & ) 1 1 1 1 1 48
%+ 11)))) 1 1 ) ) ) ) 1 2 2 2 2 2 ) ) 1 2 3 % 49
% 11)))1 1 1 ) ) ) 1 1 2 2 2 2 3 ) ) 1 2 3 3 50
&) 11))1) 1 1 ) ) 1 ) 1 2 2 2 3 3 ) ) 1 2 2 3 51
&1 11))11 1 1 ) ) 1 1 1 2 2 2 3 % ) ) 1 2 2 2 52
&2 11)1)) 1 1 ) 1 ) ) 1 2 2 3 3 3 ) ) 1 1 2 3 53
&3 11)1)1 1 1 ) 1 ) 1 1 2 2 3 3 % ) ) 1 1 2 2 54
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&% 11)11) 1 1 ) 1 1 ) 1 2 2 3 % % ) ) 1 1 1 2 55
&& 11)111 1 1 ) 1 1 1 1 2 2 3 % & ) ) 1 1 1 1 56
&' 111))) 1 1 1 ) ) ) 1 2 3 3 3 3 ) ) ) 1 2 3 57
&* 111))1 1 1 1 ) ) 1 1 2 3 3 3 % ) ) ) 1 2 2 58
&+ 111)1) 1 1 1 ) 1 ) 1 2 3 3 % % ) ) ) 1 1 2 59
& 111)11 1 1 1 ) 1 1 1 2 3 3 % & ) ) ) 1 1 1 60
') 1111)) 1 1 1 1 ) ) 1 2 3 % % % ) ) ) ) 1 2 61
'1 1111)1 1 1 1 1 ) 1 1 2 3 % % & ) ) ) ) 1 1 62
'2 11111) 1 1 1 1 1 ) 1 2 3 % & & ) ) ) ) ) 1 63
'3 111111 1 1 1 1 1 1 1 2 3 % & ' ) ) ) ) ) ) 64
(( ( ( ( ( ((
(( ( ( ( ( ((
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"a#e $ara#eters
Ma!imum number of trials -1.'/ '
%xpectancy
0verage win size 1
0verage loss size 1 4 or Profit #actor
Probability of win &)")5 6elly value
&etting syste# 'MM(
7tarting bet size 1
8pon a win, multiply prior bet size by ) an then a
8pon a loss, multiply prior bet size by 2 an then a
9uit after how many wins: ' -' never ;uit/
9uit after how many losses: ' -' never ;uit/
Trial 1 Trial 2
Outcomes Bet size Outcome !pectancy Bet size Outcome !pectancy Bet size
LLLLLL 1 Loss -0.50 2 Loss -0.50 4
LLLLLW 1 Loss -0.50 2 Loss -0.50 4
LLLLWL 1 Loss -0.50 2 Loss -0.50 4
LLLLWW 1 Loss -0.50 2 Loss -0.50 4
LLLWLL 1 Loss -0.50 2 Loss -0.50 4
LLLWLW 1 Loss -0.50 2 Loss -0.50 4
LLLWWL 1 Loss -0.50 2 Loss -0.50 4
LLLWWW 1 Loss -0.50 2 Loss -0.50 4
LLWLLL 1 Loss -0.50 2 Loss -0.50 4
LLWLLW 1 Loss -0.50 2 Loss -0.50 4
LLWLWL 1 Loss -0.50 2 Loss -0.50 4
LLWLWW 1 Loss -0.50 2 Loss -0.50 4
LLWWLL 1 Loss -0.50 2 Loss -0.50 4
LLWWLW 1 Loss -0.50 2 Loss -0.50 4
The table below illustrates, over ' -or less/ trials, that the Martingale oany betting system an playing strategy, an convince yourself that it!pectancy ivie by ave bet size> -gray bac=g
To simulate Martingale betting@ upon a win, multiply bet size by ), anTo simulate anti-Martingale betting@ upon a win, multiply bet size by 2To simulate d)*le#bert betting@ upon a win, multiply bet size by 1, anTo simulate flat betting@ upon a win, multiply bet size by 1, an then a
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LLWWWL 1 Loss -0.50 2 Loss -0.50 4
LLWWWW 1 Loss -0.50 2 Loss -0.50 4
LWLLLL 1 Loss -0.50 2 Win 0.50 1
LWLLLW 1 Loss -0.50 2 Win 0.50 1
LWLLWL 1 Loss -0.50 2 Win 0.50 1
LWLLWW 1 Loss -0.50 2 Win 0.50 1
LWLWLL 1 Loss -0.50 2 Win 0.50 1
LWLWLW 1 Loss -0.50 2 Win 0.50 1
LWLWWL 1 Loss -0.50 2 Win 0.50 1
LWLWWW 1 Loss -0.50 2 Win 0.50 1
LWWLLL 1 Loss -0.50 2 Win 0.50 1
LWWLLW 1 Loss -0.50 2 Win 0.50 1
LWWLWL 1 Loss -0.50 2 Win 0.50 1
LWWLWW 1 Loss -0.50 2 Win 0.50 1
LWWWLL 1 Loss -0.50 2 Win 0.50 1
LWWWLW 1 Loss -0.50 2 Win 0.50 1
LWWWWL 1 Loss -0.50 2 Win 0.50 1
LWWWWW 1 Loss -0.50 2 Win 0.50 1
WLLLLL 1 Win 0.50 1 Loss -0.50 2
WLLLLW 1 Win 0.50 1 Loss -0.50 2
WLLLWL 1 Win 0.50 1 Loss -0.50 2
WLLLWW 1 Win 0.50 1 Loss -0.50 2
WLLWLL 1 Win 0.50 1 Loss -0.50 2
WLLWLW 1 Win 0.50 1 Loss -0.50 2
WLLWWL 1 Win 0.50 1 Loss -0.50 2
WLLWWW 1 Win 0.50 1 Loss -0.50 2
WLWLLL 1 Win 0.50 1 Loss -0.50 2
WLWLLW 1 Win 0.50 1 Loss -0.50 2
WLWLWL 1 Win 0.50 1 Loss -0.50 2
WLWLWW 1 Win 0.50 1 Loss -0.50 2
WLWWLL 1 Win 0.50 1 Loss -0.50 2
WLWWLW 1 Win 0.50 1 Loss -0.50 2
WLWWWL 1 Win 0.50 1 Loss -0.50 2
WLWWWW 1 Win 0.50 1 Loss -0.50 2
WWLLLL 1 Win 0.50 1 Win 0.50 1
WWLLLW 1 Win 0.50 1 Win 0.50 1
WWLLWL 1 Win 0.50 1 Win 0.50 1
WWLLWW 1 Win 0.50 1 Win 0.50 1
WWLWLL 1 Win 0.50 1 Win 0.50 1
WWLWLW 1 Win 0.50 1 Win 0.50 1
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WWLWWL 1 Win 0.50 1 Win 0.50 1
WWLWWW 1 Win 0.50 1 Win 0.50 1
WWWLLL 1 Win 0.50 1 Win 0.50 1
WWWLLW 1 Win 0.50 1 Win 0.50 1
WWWLWL 1 Win 0.50 1 Win 0.50 1
WWWLWW 1 Win 0.50 1 Win 0.50 1
WWWWLL 1 Win 0.50 1 Win 0.50 1
WWWWLW 1 Win 0.50 1 Win 0.50 1
WWWWWL 1 Win 0.50 1 Win 0.50 1
WWWWWW 1 Win 0.50 1 Win 0.50 1
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Total overall e!pectancy 0.00
0verage bet size 2.25
1 !pectancy ivie by ave bet size 0.001")) Percentage of positive outcomes ,
)"))
1
)
Trial 3 Trial % Trial &
Outcome !pectancy Bet size Outcome !pectancy Bet size Outcome !pectancy
Loss -0.50 Loss -0.50 1 Loss -0.50
Loss -0.50 Loss -0.50 1 Loss -0.50
Loss -0.50 Loss -0.50 1 Win 0.50
Loss -0.50 Loss -0.50 1 Win 0.50
Loss -0.50 Win 0.50 1 Loss -0.50
Loss -0.50 Win 0.50 1 Loss -0.50
Loss -0.50 Win 0.50 1 Win 0.50
Loss -0.50 Win 0.50 1 Win 0.50
Win 0.50 1 Loss -0.50 2 Loss -0.50
Win 0.50 1 Loss -0.50 2 Loss -0.50
Win 0.50 1 Loss -0.50 2 Win 0.50
Win 0.50 1 Loss -0.50 2 Win 0.50
Win 0.50 1 Win 0.50 1 Loss -0.50
Win 0.50 1 Win 0.50 1 Loss -0.50
fers no ege in e!pectancy over flat betting" #eel welcome to e!periment by changiimpossible to alter e!pectancy, no matter what the probability of a win is, an no moun/" The table coul be e!tene from ' to any number of trials ? the result woul
then a 1 -reverts bet size bac= to 1/A upon a loss, multiply bet size by 2 an a, an then a ) -oubles your bet/A upon a loss, multiply bet size by ) an a 1 -ra .1 -reuces your bet/A upon a loss, multiply bet size by 1 an a 1 -increases
) -retains bet size/A upon a loss, multiply bet size by 1 an a ) -retains bet size
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Win 0.50 1 Win 0.50 1 Win 0.50
Win 0.50 1 Win 0.50 1 Win 0.50
Loss -0.50 2 Loss -0.50 4 Loss -0.50
Loss -0.50 2 Loss -0.50 4 Loss -0.50
Loss -0.50 2 Loss -0.50 4 Win 0.50
Loss -0.50 2 Loss -0.50 4 Win 0.50
Loss -0.50 2 Win 0.50 1 Loss -0.50
Loss -0.50 2 Win 0.50 1 Loss -0.50
Loss -0.50 2 Win 0.50 1 Win 0.50
Loss -0.50 2 Win 0.50 1 Win 0.50
Win 0.50 1 Loss -0.50 2 Loss -0.50
Win 0.50 1 Loss -0.50 2 Loss -0.50
Win 0.50 1 Loss -0.50 2 Win 0.50
Win 0.50 1 Loss -0.50 2 Win 0.50
Win 0.50 1 Win 0.50 1 Loss -0.50
Win 0.50 1 Win 0.50 1 Loss -0.50
Win 0.50 1 Win 0.50 1 Win 0.50
Win 0.50 1 Win 0.50 1 Win 0.50
Loss -0.50 4 Loss -0.50 Loss -0.50
Loss -0.50 4 Loss -0.50 Loss -0.50
Loss -0.50 4 Loss -0.50 Win 0.50
Loss -0.50 4 Loss -0.50 Win 0.50Loss -0.50 4 Win 0.50 1 Loss -0.50
Loss -0.50 4 Win 0.50 1 Loss -0.50
Loss -0.50 4 Win 0.50 1 Win 0.50
Loss -0.50 4 Win 0.50 1 Win 0.50
Win 0.50 1 Loss -0.50 2 Loss -0.50
Win 0.50 1 Loss -0.50 2 Loss -0.50
Win 0.50 1 Loss -0.50 2 Win 0.50
Win 0.50 1 Loss -0.50 2 Win 0.50
Win 0.50 1 Win 0.50 1 Loss -0.50
Win 0.50 1 Win 0.50 1 Loss -0.50
Win 0.50 1 Win 0.50 1 Win 0.50
Win 0.50 1 Win 0.50 1 Win 0.50
Loss -0.50 2 Loss -0.50 4 Loss -0.50
Loss -0.50 2 Loss -0.50 4 Loss -0.50
Loss -0.50 2 Loss -0.50 4 Win 0.50
Loss -0.50 2 Loss -0.50 4 Win 0.50
Loss -0.50 2 Win 0.50 1 Loss -0.50
Loss -0.50 2 Win 0.50 1 Loss -0.50
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Loss -0.50 2 Win 0.50 1 Win 0.50
Loss -0.50 2 Win 0.50 1 Win 0.50
Win 0.50 1 Loss -0.50 2 Loss -0.50
Win 0.50 1 Loss -0.50 2 Loss -0.50
Win 0.50 1 Loss -0.50 2 Win 0.50
Win 0.50 1 Loss -0.50 2 Win 0.50
Win 0.50 1 Win 0.50 1 Loss -0.50
Win 0.50 1 Win 0.50 1 Loss -0.50
Win 0.50 1 Win 0.50 1 Win 0.50
Win 0.50 1 Win 0.50 1 Win 0.50
Total o+erall expectancy
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Trial '
Bet size Outcome !pectancy
!2 Loss -0.50 ?31"&)
!2 Win 0.50 )"&)
1 Loss -0.50 )"))
1 Win 0.50 1"))
2 Loss -0.50 ?1"))
2 Win 0.50 1"))
1 Loss -0.50 )"&)
1 Win 0.50 1"&)
4 Loss -0.50 ?3"))
4 Win 0.50 1"))
1 Loss -0.50 )"&)
1 Win 0.50 1"&)
2 Loss -0.50 ?)"&)
2 Win 0.50 1"&)
g the values in the yellow cells to simulatetter what strategy you use"always be the same"
-oubles your bet/"verts bet size bac= to 1/" your bet/""
#inale!pectancy
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1 Loss -0.50 1"))
1 Win 0.50 2"))
Loss -0.50 ?*"))
Win 0.50 1"))
1 Loss -0.50 )"&)
1 Win 0.50 1"&)
2 Loss -0.50 ?)"&)
2 Win 0.50 1"&)
1 Loss -0.50 1"))
1 Win 0.50 2"))
4 Loss -0.50 ?2"&)
4 Win 0.50 1"&)
1 Loss -0.50 1"))
1 Win 0.50 2"))
2 Loss -0.50 )"))
2 Win 0.50 2"))
1 Loss -0.50 1"&)
1 Win 0.50 2"&)
1 Loss -0.50 ?1&"))
1 Win 0.50 1"))
1 Loss -0.50 )"&)
1 Win 0.50 1"&)2 Loss -0.50 ?)"&)
2 Win 0.50 1"&)
1 Loss -0.50 1"))
1 Win 0.50 2"))
4 Loss -0.50 ?2"&)
4 Win 0.50 1"&)
1 Loss -0.50 1"))
1 Win 0.50 2"))
2 Loss -0.50 )"))
2 Win 0.50 2"))
1 Loss -0.50 1"&)
1 Win 0.50 2"&)
Loss -0.50 ?'"&)
Win 0.50 1"&)
1 Loss -0.50 1"))
1 Win 0.50 2"))
2 Loss -0.50 )"))
2 Win 0.50 2"))
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1 Loss -0.50 1"&)
1 Win 0.50 2"&)
4 Loss -0.50 ?2"))
4 Win 0.50 2"))
1 Loss -0.50 1"&)
1 Win 0.50 2"&)
2 Loss -0.50 )"&)
2 Win 0.50 2"&)
1 Loss -0.50 2"))
1 Win 0.50 3"))
0.00