[jason1990]

Thanks for the input on N. Unfortunately, without p_f and p_c, it is not very useful. So far, no one has contacted me about quantifying their reasoning.

Let me expand a little on what is needed. Whatever we agree to use for p_0, it will undoubtedly be very small. So the probability of cheating, given the HHs, can be simplified to roughly

p_0*p_c/[p_0*p_c + p_f].

Dividing numerator and denominator by p_0*p_c gives

1/[1 + p_f/(p_0*p_c)].

So what we are trying to argue is that p_f/(p_0*p_c) is very small. What matters is not the exact value of this quantity, but rather its order of magnitude. If we take my suggestion for p_0 and Analyst's estimate for N, this says that p_0 is roughly 10^{-6} or 10^{-7}. Hence, we must argue that p_f/p_c is much smaller than 10^{-7}. Suppose we aim at showing it is of order 10^{-9}.

In other words, we want to argue that the likelihood of the HHs looking the way they do without cheating is a billion times smaller than the likelihood of them occurring with cheating.

Again, it is only orders of magnitude that matter. I am not asking anyone to give exact probabilities, only estimates up to order of magnitude. For example, what is the probability a non-cheater would make that 10 high call on the river? 1/10? 1/100? 1/1000? 1/10000? Or even smaller? Similarly, what is the probability that someone who could see hole cards would make that call? 1/10? 1/100? Etc... The answer to this question is not supposed to be some hard-core factual answer based on megabytes of statistical analysis. It is just supposed to be the opinion of the person answering it -- namely, the high-stakes player who believes cheating occurred. Your experience in playing these games is what gives credence to your answers. If you answer these questions for every street of every hand in the HHs, then it is certainly conceivable that the ratio p_f/p_c will come out to be smaller than 1 in a billion. Maybe much smaller.

Let me also say something about my motivations. What I see here is a number of people who are obviously not prone to making exaggerated claims about online poker. Yet here they are, completely convinced that cheating took place. Their arguments are being ignored by AP and ridiculed by other authors. I would like to see their claims given fair and full consideration, rather than being dismissed out of hand. I do not have the experience to judge their arguments for myself. But I do have the skills to help formulate their arguments in the language of probability, or at least try to do so. It is my hope that, if successful, this would increase the chances that their arguments are taken seriously.

Also, to anyone who wants me to work with them on this, we can do the analysis in private by email if you like. In the end, if you are not happy with the analysis -- if you feel it does not fairly represent your argument and your opinions on the matter -- then we will discard it completely. I am not interested in "tricking" anyone into making claims they do not intend to make. I simply want to help you, if you are interested, to translate your arguments in support of cheating into the language of probability.