Poker and the Central Limit Theorem

This is something I've been thinking about for a while on my own. People would talk about making X BB/100 with some standard deviation. Or of being Y% confident they are a winning SNG player.

But I couldn't understand how it was possible to apply normal curve type analysis to poker results. Because in raw form poker results are of course not normally distributed. In a cash game a standard TAG player folds preflop most hands. Thus the result is heavily skewed by $0 entries where he just folded preflop and neither made nor lost money.

In a tournament, even a simpler structure such as a 50% 30% 20% SNG. In this case there would be a multi-modal distribution, dominated by $0 bust outs and a distribution of 3-2-1 finishes. So how can you apply normal curve analysis to input data which is not normal? I couldn't figure this out.

Then I got thinking about something simpler. Coin flips. Suppose with a fair coin you assign a score of 1 for flipping heads and 0 for flipping tails. Now over a trial of say 100 flips you'll see a bimodal distribution, a large number of 0 scores and 1 scores. So that's not normally distributed. However if you consider the aggregate score over 100 flips and do a series of trials of 100 flips then some things happen. Of course you'll see a bunching around scores of 50 with a small number outside the 40-60 range and very small numbers outside the 20-80 range.

Hmmm, so by aggregating the coin flips together the sum of the series behaves like a normal curve even though the raw underlying data is not normally distributued. It is bimodal, just a large number of 0s and 1s.

In poker that can be immediately applied to double or nothing sng tournaments. In that event half the players win double the buyin and half get nothing. Well that's the same as a coin flip. In fact a 10 player DoN tournament is the same as 5 heads up sng tournaments. So over time a DoN player could aggregate his results into groups of say 10, 30, whatever works and then pull out a mean and standard deviation and get a confidence level if he is really beating (or losing) in the event or possibly just running well/not so well.

Although you can also aggregate the DoN concept to larger tournaments. For example a 10 player 50/30/20 sng could be expressed as a series of HU sng's among the players. The same could be done with say a $4.40/180. So if individual DoN "coin flips" can be made normal then it would probably be valid to aggregate the coin flips into a tournament payout and then that could be treated in a DoN way.


But the part I wasn't sure about was if it was proper to "synthesize" a normal distribution out of non-normal underlying data by aggregating them together and examining the groups of aggregates. As raw poker results, both cash game and tournament are certainly non-normal. Is this valid or is it some kind of charlatanism, trying to create normalized data out of "thin air".

I wasn't sure. But then I made the link to the Central Limit Theorem. [video]. The Central Limit theorem shows that it is proper to group together the individual results of cash game hands played, or tournament results. Remarkably what emerges from the groupings is normally distributed data which has a mean and standard deviation. Awesome!

So for cash games you need to determine the grouping size at which the summary data starts to behave as a normal curve. We hear X BB/100 a lot so I guess the analysis has settled on groupings of 100 hands. So if your minimum grouping is 100 hands (and that may be optimistically small) then that would explain a lot of the conventional wisdom around determining if you are beating or losing at a given level. For example even if the data is normally distributed then you really need at least 30 data points as a bare minimum to do any type of analysis. For the cash game that means you would need to play 3,000 hands to get 30 100 hand samples. Even that is pretty small and hard to reliably make any conclusions from. At 10,000 hands at least you would have 100 sample points. Not great but you'd have a lot more data to work with and the graph should look much smoother.

But if someone is running well, or poorly over a 1,000 hand sample then really you can't say anything about it. It's just too small to comment on. It makes you understand why some say you need to play 25,000 - 100,000 hands at a level to really have any idea if you're doing well or not. That can be a discouraging thing to deal with for a player (unless you're a huge multi tabler), not really knowing if you're winning.

Now with tournaments for 50/30/20 SNG I'm not sure how many you need to aggregate together to get a normal distribution. Although I've read that you need to play around 500 SNG at a level to get a feel if you're winning. For larger tournament, say $4.40/180 it may take a very large aggregate size (90-200?) for things to start behaving normally. For really large MTT (1000+ runners) then the minimum aggregate size I think would be huge. I'd guess upwards of 1000 events for a 1000 player MTT. I've read that for large MTT you can't play enough of them to establish a normal distribution of results. I would believe it.

Maybe everyone else knew all this stuff all along. Still I was somewhat pleased with this result that I was able to figure out about "synthesizing" a normal type distribution out of raw poker results using the Central Limit Theorem. Before I figured it out myself I was thinking of contacting a former Math professor to ask him about the idea of synthesizing. I'm glad I figured it out on my own.