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Hmm what heading to give this??? T-hopper comment please too!
I think part of the explanation is in the fable post I put up here. So much of the output of the Schlesinger Brain Trust on this is over-rated. All the Cummings hands survival formula is, is taking any of the usual derivations for ruin--see the Kriegman paper on bjmath.com-- and adding to the bankroll the expected profits (ev*h, h number of hands or trials) to comeup with a formula that gives the chance of being at or below zero bankroll AFTER so many hands. The SBT couldn't prove even this simple idea, except for a paper submitted from a University of Minnesota profesor, who equally missed the idea of demonstrating how it can apply to negative bankrolls. Showing how it can apply to negative bankrolls---ie what are my chances of winning enough to cancel a credit card debit is the example I used---provides a simple proof it works for AFTER so many hands, for ANY bankroll level.
Then you use what everyone has used for years as infinite goal ruin, except you treat it as it is derived, chances of being below zero WHILE playing forever, REALIZE that this infinite goal ruin is actually an open box formula, giving results short of the goal number of hands, to addin the probability of being at a bankroll at or below zero before the target number of trials, in the way that you compare a finite result, to an infinite result.
Then you have a formula that gives you the total probability of being at or below zero, while keeping on palying through h hands no matter what. Overall then, but ONLY AFTER THESE STEPS, you also have a prediction of the total probability of running out of money while attempting to play h hands.
The SBT only gave blather about how these terms were in some sort of tug of war, and one derivative equation that suposedly was posted somewhere by Don Schlesinger, that had 3 incorrect differential equation errors, and re-emailed to me, and reams of sims, but no theory whatsoever as to why this patchwork of ruin formulas should work at all.
It works only because the SBT had really stumbled across the insights (1)(on the closed box assumption being reversed for infinite goals) and (2) (on continuence in a random sampling process) and being totally clueless that they had done so.
I am glad you seem to have undestood this.
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