normal distribution

[jason1990]

I started a thread in "Poker Theory", but it was suggested it better belongs here. The original post was this:

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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?

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I think it was then misinterpreted, so I posted this:

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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.]

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Then, finally, I decided to be very specific and posted this:

"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."

Hopefully, people here will have some insight into this question.