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http://wizardofodds.com/software/absolutepoker.html
Michael Shackleford's website. I wondered when he would chime in.
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He has done some great analysis here.
In particular, he calculates the probability that someone would run this good in a multi-table tournament: 1 in in 1.88 × 10^44, or, in long form, 1 in 188,000,000,000,000,000,000,000,000,000,000,000,00 0,000,000. He put it at 14 standard deviations above the mean.
This is broadly consistent with my earlier calculations that put the cheaters' cash win rates at 15 standard deviations above the mean. The only thing I had off was the number of zeroes at the end of the number - I had guess there would've been around 30, when there should be 44.
I had previously thought that the cheaters had about the same chance of running this good as winning six consecutive one-in-a-million lotteries. In reality, the cheaters actually have the same chance of running this good as winning
seven consecutive one-in-a-million lotteries.
<font color="red">To summarise: either these are now
two of
literally the most incredibly miraculous occurences in the
history of the universe, or they are cheating.</font>
I have not found any other sample of
anything with a result of fourteen or fifteen standard deviations above the mean. If anyone is aware of any, can they please PM or email me?
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While I like the article and appreciate that he has weighed in, drawing conclusions from this stat is ridiculous:
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In particular, he calculates the probability that someone would run this good in a multi-table tournament: 1 in in 1.88 × 10^44
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This is based on the assumption that all players play the same and should win equally. This is a ridiculous assumption and the most simplistic analysis possible. A loose aggressive player can run over a table for many hands, and get these kind of mind bending deviations that the Wizard provides (though probably not nearly as large).
The real proof and the proper analysis is found by looking at his decision making ability. For each hand, estimate the odds that, given no hole card knowledge, he could make the correct play (folding when beat, raising against a weaker hand, reraising a bluffer, betting into a weaker hand, not betting into a weaker hand). Someone needs to do this analysis properly, giving generous allowances to the null hypothesis - and as every experienced player intuitively recognizes, they'll get an astronomical number that proves hole card knowledge. That will be the statistical smoking gun.
But the Wizard's number is a really bad way of approaching it (the most simplistic way possible), and one that can generate huge deviations even for people that can't see hole cards.