Computing the Expected Total Amount Wagered Given a

Fixed Percentage Return and Initial Bankroll

 

The numbers as in our example (100, 97, 94.09, 91.27, 88.53) represent a geometric series with a
common ratio of 0.97, the decimal equivalent to our 97 percent return. For the general case where:

p = Decimal Equivalent of Percent Return

B = Starting Bankroll

T = Total Amount Wagered after n sessions where each successive session starts

with the return from the previous session.

 

Then

T = B ( 1 + p + p + . . . + p n - 1 ) ( 1 )

 

Multiplying each side by p we have

 

p T = B ( p + p + p + . . . + p n ) ( 2 )

 

Subtracting ( 2 ) from ( 1 ) , we have

 

T p T = B ( 1 - p n ) or T ( 1 p ) = B ( 1 - p n )

 

and

 

T = B ( 1 - p n ) / ( 1 p )

 

If p < 1 and the number of sessions becomes very large ( n → ∞ ) , then p n → 0 and we are left with

 

T = B / ( 1 p )

 

Applying this formula to our previous example ( B = $ 100, p = 0.97 ), we find

 

T = $100 / (1 - 0.97) =  $100 / 0.03 = $ 3333.33

 

Given a starting bankroll of 100 credits, the table below shows how many one credit games one can

expect to play on average before losing all the credits.

 

Percent Return

Expected Number of Games

100

99

10,000

98

5,000

97

3,333

96

2,500

95

2,000

94

1,667

93

1,429

92

1,250

91

1,111

90

1,000

89

909

88

833

87

769

86

714

85

667

84

625

83

588

82

556

81

526

80

500

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