Bad Beats Post

Something has seemed out of whack for a while.  I have played well over 1 million hands, but rather than complain about a "feeling", I wanted to look at hard numbers and data.  There are many things I could look at to see if different probabilities are out of whack with expectation.  The very first thing I decided to look at, using my db with 236,317 tournament hands (a significant sample size), was the distribution of how many times I got dealt pocket pairs, particularlly premium pocket pairs.  A pretty simple thing to analyze.

At first, I looked at the breakdown by blind-level and was a bit concerned to find that at level after level, I got dealt fewer AA and KK than any other pocket pairs.  For example at the 50/100 blind level, the sample was 73,926, from which I could expect to be dealt a particular pocket pair, like AA, on average apx 336 times.  The two that were farthest below expectation were AA (-16) and KK (-21).  This repeated at each blind level I looked at.   

So next, I aggregated and looked at all 236,317 hands together, regarless of blind level.  What I found furthered my concern.  Again, the least dealt pocket pair was AA (1010), followed by KK (1040), furthing the suspicion.  So I ran some hard statistical analysis on this and found that on average for that sample, there is only a 2.5% chance of being dealt fewer than 1010 AA (and 97.5% chance of being dealt more).  Note that this doesnt give any weight to the value of AA vs other lower pairs.  The fact that this repeats itself with the other monster pair, KK, is again concerning.

I'll give a quick summary of the mathematical reasoning and tools used for review.
If we think of this as a binomial distribution problem, where success (=1) is being dealt an AA, and failure (=0) is not, we know that we have an expected probability of success in a single trial of 1/220, or .00454545..., the odds of being dealt a pocket pair.  From my data, we know we had 1010 successes in 236,317 trials.  After working the math on my own, I also found a caclulator online you can use by plugging in these variables:  http://stattrek.com/Tables/Binomial.aspx   You can see the conclusion is that for this sample, we would expect more AAs 97.4% of the time.  I would also expect to see KK significantly more than I did.  When you look at both together (use values of .009090909090909090909, 2050, and 236,317 in the online calculator), we see my distribution become even more unlikely (98.3%).

What is more concerning yet is that when I look only at the most recent 100,000 hands, the pattern exacerbates itself more dramatically.

Note that if there is rigging that damps the number of premium hands, it would be foolish to damp them to fall much more than 1 or 2 standard deviations away from the norm.  So the 98.3% number I see above would be the maximum one could reasonably expect under that scenario.  In other words, its the max range that remains "unprovable" due to variance.

More to come.... note that this is the very first thing I have looked at after suspecting suspicious activity.  I am starting to look at other aspects that are looking troubling as well, but will post only after I work out numbers.  I would like to keep this an ongoing discussion, and invite others to use their db to plug numbers into the calculator I linked to.