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Since there is no house advantage, your expected ending amount is the same as your starting amount, B+D. If you end up with either 0 or X, you have X with probability (B+D)/X.
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very elegant... i love it. thanks for your answers.
now how do you incorporate the house edge into this argument? that is, some casinos don't offer the nearly fair games like BJ switch, but instead offer only games like "no peek" blackjack where the HA is not negligible (.55% in that case).
Would you first calculate the simple expected end amount of playing out the wagering requirement as
E_end_amount = (B+D) - WR*HA
And now set the EV of reaching your target X or bust as equal to that:
X*(P_reach_target) = B + D - WR*HA
Or perhaps, using the logic of another post you made about blackjack, only subtract of .9*WR*HA?
So that the chance of winning is:
(B + D - WR*HA)/X
This of course assumes that you can reach your target and WR requirement at approximately the same time, but from my understanding of
this bonuswhores.com post you can do this with high probability by using the following optimal bet size:
bet=[T-C]^2/(4*WR)
where T is the target, C is your current balance, WR is how much you have left to play.
Which brings me to my last question: How is this forumula derived and just how effective is the strategy of using its recommended bet sizes? Eg, what is the distribution of your projected distance from the target (or from the WR) those times you hit the target using this strategy?
Thanks again,
gm