normal distribution

[jason1990]

Does anyone know, roughly, how many poker hands a person needs to play before their winnings become approximately normally distributed? For example, suppose someone has a winrate of 2BB/100 and a standard deviation of 15BB/100. They might think that if they play 100 hands, then they have a 95% chance of winning somewhere between -28BB and 32BB (a spread of 2 SD's). But this is only true if the result of playing 100 hands is (roughly) normally distributed. Is it?

[Dominic]

There are a lot more people on here who are better at the math than I, but I do know that 100 hands is about as reliable a sample as two hands are.

You need thousands upon thousands of hands to get a reliable sample.

[jason1990]

Yes, I fully agree. But that's not my question. Let me rephrase: suppose you play 1000 sessions of 100 hands each. You plot the result of each session on a graph. Will this graph have a bell-shaped curve?

For example, if you plot the result of each hand on a graph, that graph will not have a bell-shaped curve. It will have a huge spike at 0, because of all the folding, and will not be symmetrical about the mean, since (on any particular hand) it is very likely that you will lose a small amount and much less likely you will win a large amount. What this shows is that the result of a single hand is not normally distributed.

But what about the result of a 100 hand session?

[Edit: By the way, I should clarify for non-mathematicians: By "normally distributed", I mean that it has a Gaussian distribution which is characterized by the bell-shaped curve. I do not mean, in any way, that the result is "normal" (or "typical") in the ordinary sense of the word, i.e. I am not asking whether the result of 100 hands can be used as a reliable indicator of long-term results. I know that this is very far from the truth.]

[jason1990]

[ QUOTE ]
Does anyone know, roughly, how many poker hands a person needs to play before their winnings become approximately normally distributed? For example, suppose someone has a winrate of 2BB/100 and a standard deviation of 15BB/100. They might think that if they play 100 hands, then they have a 95% chance of winning somewhere between -28BB and 32BB (a spread of 2 SD's). But this is only true if the result of playing 100 hands is (roughly) normally distributed. Is it?

[/ QUOTE ]

It occured to me that I should probably be more specific. I am assuming that the winrate and SD are completely accurate. So...

Suppose a player plays a 100 million hands and, based on this, determines that his winrate is 2BB/100 and his SD is 15BB/100. Converting this to various different units, we get

winrate: 0.2BB/10 = 2BB/100 = 20BB/1000 = 200BB/10000 = 2000BB/100K
SD: 4.7BB/10 = 15BB/100 = 47BB/1000 = 150BB/10000 = 474BB/100K.

Now, after this player has already played these 100 million hands, he sits down at the poker table and asks himself 5 questions:

1. "Do I have about a 95% chance of winning between -9.2BB and 9.6BB on the next 10 hands I play?"

2. "Do I have about a 95% chance of winning between -28BB and 32BB on the next 100 hands I play?"

3. "Do I have about a 95% chance of winning between -74BB and 114BB on the next 1000 hands I play?"

4. "Do I have about a 95% chance of winning between -100BB and 500BB on the next 10000 hands I play?"

5. "Do I have about a 95% chance of winning between 1052BB and 2948BB on the next 100K hands I play?"

The point of these questions is this: you have a 95% chance of falling within 2 SD's of your mean provided the thing you're asking about has the bell-shaped property. People always say you need 100K hands to determine if you're a winning player. However, the bell-shaped property is going to emerge long before 100K hands. So the answer to questions 4 and 5 is a definite "yes". The answer to question 1 is an obvious no, as experience alone can tell us. This is because the bell-shaped property does not emerge after only 10 hands. But how about questions 2 and 3? Mathematical experience indicates to me that the answer to question 3 is very likely to be yes. So question 2 is the one that interests me most.

[RayGarlington]

I'll just point out that there are 169 starting hands, so you can't even get 1 instance of each in 100 tries, much less have a distribution.

[ceczar]

i believe that the distribution of sample means will always have a gaussian distribution, regardless of the distribution of the underlying. if you view 100k hands as your population, and 100 hand groups as samples, the winrates (means) should be distributed on a normal distrubition with mean equal to your true winrate and SD equal to your true SD (well, maybe not exactly, but it's related to it in some defined way). it's been a while, but if you only have your measured winrate and SD then it's a T-dist instead of a normal i think.

if you assume you stay at the same skill level and the games are the same level of toughness, then you can view each successive 100 hands as samples from a the population of all hands you will ever play (or play during a time period). if you knew your true winrate and SD, your winrate for the next 100 or 1000 hands would indeed be distributed normally.

i wouldn't be surprised if you got totally different and better answers if you posted this in the probability forum.

[Dov]

Jason,

Try cross posting this in the probability forum. Those guys will be able to help you much more. (That's where the 'math guys' generally hang out)

Dov

[jason1990]

Thanks to you and ceczar for this advice. I will do that.

[jason1990]

By "distribution", I mean a "probability distribution", which, informally, you can imagine as a table or a graph, giving the probabilities of all the various possible outcomes of the event in question. So in this sense, even a single hand has a distribution.

[axioma]

well, the more hands played per session, and the more sessions (points on the x-axis), the closer the resulting graph will be to a gaussian. but im sure you already knew that.

but yes, i suppose ultimately a poker lifetime (of course, given a constant playing skill...) will be gaussian in appearance.

[axioma]

interesting theoretical question.

what shape curve would the lifetime of a player who randomly choses his option (fold, check, call, bet, raise) look like, assuming he has a bankroll large enough that he wont go broke?

[Blarg]

[ QUOTE ]
"Do I have about a 95% chance of winning [...] on the next 10 hands I play?"


[/ QUOTE ]

Cards have no memory. You have no particular chance of winning or losing on the next hand based upon what happened on the last, whether the last one or last one billion, and no matter how you graph it.

You merely have very long term tendencies.

[jason1990]

I suppose this is an example of how trying to be more specific only obfuscated the issue. The previous 100 million hands that this hypothetical player had played were introduced only to establish the fact that he has a true winrate of 2BB/100 and a true SD of 15BB/100. Once that it is established, we can then speak of the probability of [...] happening on the next [...] hands. I am not suggesting that the past hands are influencing the future hands.

[CrisBrown]

Hi jason,

The short answer is: a very, VERY big number.

The reason that normal distribution curves apply to independent events. Poker results are not independent events. They are Markhov Progression events. That is, past events have some effect on future events. You learn how to play better. Your opponents learn how you play. You tilt. You move up in limits, thus encountering more difficult opposition. You go back and read and learn more....

Because your hand database isn't a set of independent events, it takes a VERY long time to reach a point of stability where you could expect a predictive normal distribution curve to emerge, if indeed it ever would.

Cris

[meow_meow]

Nobody seems to be answering your question in a satisfactory manner (no offense intended), and I'm interested in the answer to this as well.

Maybe I'll take another shot at rephrasing the question:

Are 100 hand samples (in terms of bb won or lost) normally distributed about the mean? (the mean being the true win rate in bb/100)

If the answer is no, then the SD in terms of bb/100 (what PT calculates for you for example) is not meaningful.

I don't know myself how to answer this question. One possible (but uninteresting) reason that they do not is that the amount won on any given hand (related to pot size) is limited to certain values, and the range is a discrete set on values, rather than being continuous (for example, pot cannot be smaller than sb+bb or larger than 12BB/player), although 100 hands may be enough to smooth this out considerably.

I personally do not think that 100 hand chunks are normally distributed, because I think there are far more outliers than would be expected from a normal distribution (just a feeling).

I wish PT would provide the actual 100 hand chunks so that this could be tested....

[jason1990]

I started a thread with the same title is the Probability area, if you want to look at that. There have been a couple responses, but nothing decisive. My guess is that 100 hands are fairly normal, only because I think the one-hand distribution (for a typical good player) is not terribly pathological. Sure, it's discrete, but it can take any integer value between -12 and 108, as well as half-integer values. I think there's enough there for it to be relatively "smooth" (albeit not normal in shape) -- at least "smooth" enough so that after 100 hands, the "smoothness" and normal shape are evident. But it seems that without more concrete information, we cannot answer this. It doesn't seem to be a situation where general probability theory gives an answer -- the answer depends on the data specific to poker.

Also, you are probably right about the outliers. But it sure would be nice to be able to quantify, at least roughly, the error we make when we assume N hands are normal.