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Online Poker Watchdog has carried out 'bad beat' tests on PokerStars hands. PokerStars is the largest independent online poker site in the world.
We started by downloading a sample of 1 million PokerStars hands from PTR*, importing to Poker Tracker*.
1) We separated those hands that were all-in pre-flop and looked at these hands from the perspective of the 'underdog' in every case ('all-in equity'<50%).
2) We calculated the difference between the number of hands that were expected to win in an ideal world, with the number of hands that actually won in reality - this is called the actual deviation.
If the actual deviation is positive it means that generally the underdog hands improved to win more often than expected - i.e. there were more bad beats than expected.
If the deal at PokerStars was fair the actual deviation should be very small.
3) But how small is very small? To answer this question we calculated the standard deviation of the sample so that we could compare the actual deviation to it.
If the actual deviation is less than 2 times the standard deviation then this is strong evidence of no bias in the sample.
If the actual deviation is greater than 5 times the standard deviation then this is strong evidence of a bias in the sample.
Sometimes results will occur where the actual deviation is slightly greater than 2 times the standard deviation (and occasionally greater than 3 standard deviations). This can occur from time to time due to variance. It does not necessarily mean that a bias has been found, although it may be grounds to run further tests on a particular poker site.
This bad beat test has been repeated for pre-flop dominated hands ('all-in equity' from 17-32%) and for hands that were all-in on the flop:
Data Group | Actual Deviation | Standard Deviation | Actual Deviation/Standard Deviation |
---|---|---|---|
Pre-flop: underdog | -78.6 | 122.2 | -0.64 |
Pre-flop: dominated | -58.2 | 71.5 | -0.81 |
Flop: underdog | +13.5 | 54.6 | +0.25 |
Turn: underdog | +41.5 | 31.1 | +1.34 |
From the table above we can see that the actual deviation is well within two standard deviations in all tests and we can conclude that there is no evidence of rigging.
For more information on interpreting deviation see this article.