Let p = probability of winning (range = 0 to 1, 0=sure loss, 1= sure win)
EV(Single trial) = 2*p - 1
[u]Progressive bet and probability of bet upon each loss[/u]
Bet 2^0, 2^1, 2^2,.......,2^(n-1)
Prob(Bet) (1-p)^0, (1-p)^1, (1-p)^2,........(1-p)^(n-1)
EV(Overall) = Sum[Bet*Prob(Bet)*EV(Single trial)] = Sum[2^(n-1)*(1-p)^(n-1)*(2*p-1)]
as n varies from 1 to infinity
[u]This is overall expectation from martingale progression:[/u]
1) EV(Overall) = (2*p-1)*Sum[2^(n-1)*(1-p)^(n-1)]
[u]let S = Sum[2^(n-1)*(1-p)^(n-1)] & do some algebra[/u]
S = 2^0*(1-p)^0+2^1*(1-p)^1+2^2*(1-p)^2+...+2^(n-1)*(1-p)^(n-1)
[u]S*2*(1-p) = 2^1*(1-p)^1+2^2*(1-p)^2+...+2^(n-1)*(1-p)^(n-1)+2^n*(1-p)^n[/u]
S*(1-2*(1-p)) = 2^0*(1-p)^0 - 2^n*(1-p)^n
S*(2*p-1) = 2^0*(1-p)^0 - 2^n*(1-p)^n
S = (2^0*(1-p)^0 - 2^n*(1-p)^n)/(2*p-1) = (1-2^n*(1-p)^n)/(2*p-1)
Plug S into equation 1) above
EV(Overall) = 1-2^n*(1-p)^n
Results of martingale as trials vary from 1 to infinity depends upon p (prob of winning)
If p = 1/2, EV(Overall) = 0, (martingale breaks even)
If p > 1/2, As n approaches infinity EV(Overall) approaches 1 (martingale succeeds)
If p < 1/2, As n approaches infinity EV(Overall) approaches -infinity (martingale loses an infinite amount)