[ QUOTE ]
now how do you incorporate the house edge into this argument? ... Would you first calculate the simple expected end amount of playing out the wagering requirement as
E_end_amount = (B+D) - WR*HA
[/ QUOTE ]
No. An important point for analyzing bonuses is that busting out relieves you of the wagering requirement. You also have to worry about completing the wagering requirement while you are not close to busting out or reaching your target.
[ QUOTE ]
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.
[/ QUOTE ]
First, that formula is wrong. Look at what it suggests when the wagering requirement is 0! It doesn't take into account the house advantage or your balance (which is an upper bound on the amount you can bet).
Second, the idea behind it may be that after n fair bets, the standard deviation is sqrt(n)*bet size (assuming coin-tosses, not blackjack). If you use a constant bet size of (T-C)^2/(4WR), it is about a 2-standard deviation result to be above the target after you reach the wagering requirement, which happens after (4 WR^2)/(T-C)^2 bets.
I think a better strategy is to aim for T+WR'*HA, where WR' is the remaining wagering requirement after betting. Either bet everything, or bet just what you need to reach T+WR'*HA. Once you reach T+WR'*HA, grind out the wagering requirement with tiny bets.
I believe my strategy decreases the average amount wagered, hence increases the probability of success. The exact performance is messy, but should be easy to test by experiment.