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˛ + pł + . . . + 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.