[BruceZ]

[ QUOTE ]
Nobody? Well I was thinking about it.

Let's say you have to play 5,000 hands to clear the bonus, which would mean for you to not clear it your WR would have to be -5 BB/100 hands for you to not comlete it.

SO then the question becomes, what are the chances that a player with a WR of 1.5 BBs/100 hands and a sd of 17 BBs/100 hands runs at -5BBs/100 hands or worse in 5,000 hands.

Now that I think about it I think I have to find the chance that I will never be down 250 BBs during any block of X hands.

Thoughts? Links? Anybody?

Thanks,
Ben

[/ QUOTE ]

As a rule of thumb, your risk of going broke sometime during the 5000 hand interval is a little more than twice the probability of being broke at the end of the 5000 hand interval, assuming that you always play that long (see note below). On average you should be up 75 BB at the end of the 5000 hands, so being down 250 BB is 325 BB below average. The standard deviation for 5000 hands is 17*sqrt(50), so the probability of being down 325/(17*sqrt(50)) standard deviations is given by the Excel function =NORMSDIST(-325/(17*SQRT(50))) =~ 0.34%. So your probability of going broke sometime in the 5000 hands interval is a little more than twice this or a little more than 0.68%. The short term risk of ruin formula from Blackjack Attack by Don Schlesinger gives 0.89%.

NORMSDIST[(-250-1.5*50)/(17*SQRT(50))] +
EXP[-2*1.5*50*250/(17^2*50)]*NORMSDIST[(-250+1.5*50)/(17*SQRT(50))] =~ 0.89%.

So both methods agree that your risk of ruin is a little less than 1%.

EDIT: Fixed typo in first method. Also, this first method (rule of thumb) cannot be used for a general number of hands, and it is only a reasonable approximation in this case because 5000 hands is sufficiently small. To be specific, the number of hands must be much less than 100*[2.7*17/(2*1.5)]^2 =~ 23,400, which is the number of hands for which being 0.34% =~ 2.7 standard deviation below average gives a maximum loss.