Should you think that statistics predicts the future you are sadly mistaken. Statistics actually predicts the possibility of a result as a future history, looking backwards mover results after a definite number of trials, hands or samples. This is known sometimes as the closed box assumption, or closed box problem.
Yet infinite goal ruin is never at the limits it is based on, never completes a definite number of trials, but always has been shown to be a reasonable limit to finite range results. But close examination never the less shows that so-called infinite goal ruin is actually an open box!
(1) It shows results before a limit or barrier is reached. Its results are within the box limits, but visable and calculable never the less.
(2) That it involves giving any ruin prediction, shows a subtle but important point about all statistical analysis: that all statistical predictions involve continuing to the end of a sample, EVEN WITH NEGATIVE bankrolls.
You always assume that you have kept on trucking if you will. So what is called ruin must really be considered the total probability of ranging below a bankroll level of zero. Those are actually important modifications to the usual understanding of the Central Limit Theorom!
All of the postings and writing of Schlesinger, Kim Lee, ML, and even the host here Reid, show an understanding of (2) above, but very little about (1). Chris Cummings excelant endpoint trip ruin formula can be proven without (2) (as it is in the recent Gambling Conference paper Schlesinger so often reffers to) but this understanding makes the process much easier.
Under assumption (2) the Cummings endpoint formula can be demonstrated with a negative bankroll---such as trying to play off a credit card balance before new interest charges kick-in--- and thus generallized. I am going to show a modified version of this, to keep the formulas in exponential form as well as explain this within the context of Samuelson's general infinite goal ruin.
Samuelson's formula is that infinite goal ruin is:
e^(-2*alpha), where alpha is the ratio between an optimal bankroll and your actual bankroll. This would make alpha then =B/EKB, where B is your actual bankroll and EKB the optimal number of units. EKB= (sd^2)/ev, where ev is the expected value. Then ruin becomes = e^(-2*B*ev/(sd^2))
If you then somehow wished to approximate the probability of ending below a bankroll of zero, but only AFTER playing a given number of hands you would add in the expected value of playing that number of hands. Ruin then becomes= e^(-2*(B+(h*ev))*ev/(sd^2).
Schlesinger just moves the formula to a normal distribution form to be easier with modern spreadsheets.
The correlation between infinite goal results and finite results is sometimes called Euler's limit or E=(1-(1/h)). If we consider that the infinite goal formula times E is the probability of crossing ruin within the box, since this so-called infinite goal formula is actually the open box estimate within the target number of samples, and that the Cummings formula gives the result at or after the target sample, then the total probability of crossing zero within h hands, or being ruined before h hands overall is:
E*[e^(-2*B*ev/(sd^2))+e^(-2*(B+(h*ev))*ev/(sd^2)) which is another form of Schlesinger trip ruin.
As far back as 1982 I had played with this WITHOUT trying to comeup with anything like the Cummings formula. I instead used the back and forth tendency of a given game directly, trading a narrower range of accurate estimates for some other important modeling. I called the ratio of the overall probability of crossing a barrier, to the endpoint probability, the RWI, or random-walk-index. I have had DD misquote some of his own results on this, and ML attack this because this approach and Don's approach, did not predict ruin where the number of hands was too small to actually have ruin possible. That is the false range of hands I mentioned in another thread here. That is the level of attack I have faced from the so-called, "Schlessing brain trust."
This back and forth model is admittedly not accurate for a wide range of hands, but is important for generalizing how to combine ruin formulas for different ruin versus success or survival criteria, what I have tried to get called "ruin flavors."
A general combination of ruin flavors would estimate overall ruin to be=(r1+r2+....rn)*(1-r1)*(1-r2)....*(1-rn). If you wished to combine flavors of ruin that involved one goal excedeing another along the same line of results, you have to determine and isolate the terms that involve this back and forth tendency and not include them more than once. You would not adjust for going back and forth, which is the real precise cause for why finite goal ruin excedes infinite goal ruin---each win or loss puts you forward or bakcwards in the spectrum of risk---more than once. This is easier to see with the RWI approach.
Now that I have shown however a derivation of Schlesinger trip ruin and how it can be extended, it is only fair to point out what went wrong with their understanding on these questions. It is also useful to have derived an exponential version simply because it is less risky to need a scientific calculator along, on your casino trips, than your laptop: rooms do get invaded when you are pulled up often!
But not realizing that you have significantly modified the Central Limit Theorom, by coming up with trip ruin, is appaling and only proves the clones to be credentialed empiracists rather than true theorists!
It is for example the closed box assumption of the CLT that has always been used to challenge Chebayev polynomial analysis, where exact results are predicted, from analysis of repetitive patterns in normal fluctuation, when a given level of fluctuations is known. Normal stock trading where a given group of trades is known will trigger certain "trading circuit breakers," is a well-known example of where fluctuations are known.
Well the clones as I call them missed out on scratching out the fundamental disproof of this group of theories.
I think the blackjack possibilities are clear too.