[irchans]

Thanks for the interesting post BruceZ. I won't have time to post my derivation until Monday at earliest, but I would like to point out that the two formulas are very similar. (I will be travelling soon for the weekend.)

Notation

n = starting bank roll
B = winning bankroll
sigma = 1 hour standard deviation
e = earn per hour


BruceZ-Schlesinger Formula (with a little algebra):

pwin = (1 - r^(n/sigma)) / (1 - r^(B/sigma))
r = (sigma - e) / (sigma + e)

Irchans Formula :

pwin = (1 - R^n)) / (1 - R^B)
r = (sigma^2 - e) / (sigma^2 + e)


If {e/sigma, e/sigma^2} are small and {n, n/sigma} are large then our formulas produce very similar answers. For the example given ( sigma = 13.11, n = 50, B = 300, e = 1.05 ) we get the following results:

BruceZ-Schlesinger Formula pwin = 0.469788

Irchans Formula pwin = 0.469158


Both Formulas are similar to

Simplest Formula:

pwin = (1 - Exp[ - n q] ) / (1 - Exp[ - B q] )
q = 2 e / sigma^2

which gives

pwin = 0.469155