I am forced to give a name to these two different probabilities-methods of prooving cheating:
1.) The one is the
“ending up that low or below” which is the hypothesis test using the Normal Distribution I did in a previous post.
2.) And the other is the
“risk of ruin” ,which is the application of the risk of ruin formula
when it is applicable to prove cheating. The definition of this
when is exactly the definition of this second method.
Besides the misconception one might fall (as I did), that made me use the risk of ruin probability where it was not applicable (and got the wrong result of 18.74%), one could also make the below (and I think reverse) misconception:
When one decides to finish taking down the stats at a point where his balance is at the lowest point it was ever observed during the hands of these stats, (e.g. because the balance dropped to zero and he would need to redeposit to continue), then the probability of risk of ruin and of the probability of “ending up that low or below”, are not the same, as one might expect with a first, shallow thought. Why is that?
The probability of “ending up that low or below” presupposes that the number of hands of the sample is not depended on the condition that the player can decide to stop counting his stats when his balance is lower than any other time during the history of these hands. Such an independency is assured when e.g. the number of the hands of the sample is predetermined before the hands start happening, e.g. until meeting a bonus wagering requirement.
Therefore if the player stops counting his stats when his balance is at its lowest point, this creates a bias which makes the probability of “ending up that low or below”, not applicable and false if applied to prove cheating. But in this case, the probability of risk of ruin IS appropriate to use in order to prove cheating! E.g. in the case that one loses all his deposited bankroll, he can claim as a proof of cheating, the very low risk of ruin probability of losing X units of bankroll in Y number of hands (for this I use the calculator
)
One might claim that the only valid method-probability for proving cheating is the “ending up that low or below”. But I disagree: Suppose one plays 10,000 hands at a game of 0.3% house edge, placing 1$ on each hand. And suppose that the final results do not indicate cheating with more than a 75% certainty, eg. he finally loses only 140$. But before these 10,000 hands were completed, at the 6459th hand his balance had fallen to an amazing -2000$ below the starting balance – initial bankroll (!!!), and after that, it raised back to the point of -140$ below the starting balance, which is not extremely far from expectation. Now the method of “ending up that low or below” ignores that fall of -2000$!!!
Therefore, the risk of ruin probability can also be used for proving cheating, by referring to the lowest point that the account balance (current bankroll) fell below the starting account balace (initial bankroll), or referring to a very large downward swing of the account balance, that it was met somewhere among the history of hands. So, we have 2 weapons to identify cheating by observing the profit/loss data. I think though, that there might be some additional considerations to apply this, and I am still thinking about it. It might have nothing to do with the risk of ruin probability. Oh, yes, that's it: "What is the probability that the bankroll-account balance to have such a large fall from its highest point to its lowest point (a fall which happens in
x hands), after playing a total of
y hands? (the highest and lowest points observed in
y hands)"
You might have ended up close to the expected loss, but still you might have been cheated. How can one investigate this?
I just remembered that the results of this "wrong" calculator were matching with the "risk of busting" tables of your site! I am getting nuts now! I have to make some tests.
I just found the answer! The "risk of ruin" probability IS NOT the "ending up below expectation" probability. The "risk of ruin" SHOULD BE HIGHER than the "ending up below expectation", as the "risk of ruin" is the probability reaching the point of -123 units for the first time, whereas the "ending up" implies that one has passed that -123 point many times before ending up there. And therefore, it is much more ... lets say "probable" for the bankroll to pass for the first time the -123 point, that to ... pass it many times!
