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.

Also, if anyone has writen any code that parses hand histories and does statistical analysis on the results that wouldnt be easy to see using pokerTracker, please post a link to the code. For example, looking at hand results for how much your pre-flop expectation when HU differs from actual results of the hands, etc, etc. For example, I'd like to be able to run HH through some code that analyzes expected EV vs actual results, or that analyzes the probability of the river hitting opponent vs actual results, etc, etc. I have been working on some Perl code to start doing some of this, but time is tight... so if anyone has anything I could add to, that'd be great.

Of course they can't deal AA and KK according to odds...nobody would chase the flushes if they didn't have that suited A or K in their hands.

Next I'd like to see the amount of times you lose with a better hand (getting it all in preflop) to a larger stack... Seems to happen to happen all to often.

Update: it gets worse. I just looked at the other monster hand, AK (any suit). I had 2767 occurrances in 236,317 sample. It should occur on average 1 in 82 hands. Get this... my results show it is almost *exactly* as out of whack as AA and KK. 98.4% probability that a random sample would have more occurrances of AK. By itself, its a enough to perk up, but in combination with the AA and KK stats all being around 98.4% ALSO, its alarming. All three of the top 3 hands are way out of whack. By my calculations, the odds of this are 2.744 out of 1 million, though I would welcome others showing their calculation methods. With each new item I look at and run numbers for, it becomes farther and farther away from random probability and looks more and more strongly to be rigged. It is very concerning. And I have only begun looking at the more complex, multi-variable scenarios. But so far, a 99.99973%chance that its not random is pretty strong and disturbing.

you run good

are all these hands from one site or is it multiple sites?

super system says you should be dealt AK 1 in 110 hands...not sure where youre geting 1 in 82 and claims a pair to be 1 in 220...maybe im reading it wrong?

 

Originally Posted by MarkFSU1

super system says you should be dealt AK 1 in 110 hands...not sure where youre geting 1 in 82 and claims a pair to be 1 in 220...

Such is where Doyle makes his monies...

stars will see this thread, turn off doom switch, let him run good for 6 months and he'll forget about it.

A pair 1 in 220 hands? i find this VERY hard to believe...

The 1 in 82 figures can be seen at: http://www.sitandgoplanet.com/multit...obability.html
ALso, intuitively it makes sence, since we know there are several more cominations to make AK than a PP.

I should note that the vast majority of these hands are from hyper SNGs. MTTs are included as well, but my SNG volume is much higher. If I had to ballpark it I would say maybe 90% of these hands are from the hyper SNGs.
It would be interesting to extract out just the hyper SNG hands from the MTT hands, but tough to do. My intuition doesn't sense anything out of whack with the MTTs from a gut feel, but the hyper SNGs (source of most of this data) seem way out of whack, as I'm confirming.

I've quantified the significance of the deviation using the binomial distribution analysis I mentioned above. 5% off over a quarter mil hands is fairly significant... around 2 standard deviations is significant. In and of itself, it certainly still lies within the realm of "very unlucky, but not extreemly unlikely", but take it in combination with the fact that AA was also dealt 2 standard deviations from expectation, as well as KK. Then combine that with the fact/ pattern that this is only occurring for the premo hands (AA, KK, AK, etc), which to a probability equation hold no special value over say, 22, but we know that we'd rather not see these hands less. And then start combining the odds of all premo hands being dealt much less than others, and it heads out of the realm of "very unlikey" and into the realm of "extreemely highly unlikely without an influencing factor". Also notice that obviously someone wouldn't be allowed to be at the very far extremes such as having never had an AA dealt for 236,000 hands. And like I mention, there is more complex analysis underway for other factors, none of which are looking good, so far.

I've always been the type to kinda chuckle when someone mentions rigging, especially with no analysis... and was inclined to think i would find that it was just my perception of things looking highly unlikely, But I'm running my math, and finding what I am finding and its not good.

See my response above. It is being quantified. So far I've looked at 3 separate events: dealt AA, dealt KK, and dealth AK. Assuming random/fair, 98% of players get more AA, 98% also get more KK. To top it off, 98% of players ALSO get AK more often. Any of these in isolation is "very unlucky". In combination, much more significant.

The binomial distribution calculations take into account sample size and deviation you mention. For ex., if my sample size were half, but I had 5% off, that 98% would become a smaller number since the sample was smaller, thus expanding the likelyhood of variance as the main influencing factor. Plug it into the online calculator I linked to and play around with it if anyone is interested.

Its basic and a given that larger samples would be even better. But you are asserting that 1% deviation over 2M hands is not as bad as 3% deviation over 500K hands, and that is NOT the case. If I had 1 trillion hands and was deviating 0.5%, it would indicate a very highly unlikely case. As an example, try the calculator I linked to and enter the coin flip scenario:
0.5% expected successes (heads)
49900000 sucesses (heads)
100000000 trials
so it is off by just 1/500 (0.2%), yet so highly unlikely that the caclulator considers it virtually impossible (90 decimal places).

So yes the deviation as a percentage falls, but its more than made up for by the increasing sample size ruling out simple variance. The calculation is what it is, as it accounts for sample size when it tells how unlikely something is. Standard deviation ranges tighten / loosen as as sample sizes grow/shrink. So 2 standard deviations is 2 standard deviations, and thats significant over one factor but esp three. If the sample were 10K, the deviation bands would be so broad (like being dealt AA twice in 10K) that it would need to be an a very far extreeme to be equiavlent to where it is with my date and sample.