# Central limit theorem and poker

1. Nov 15, 2007

### JohanL

Can you use the central limit theorem to prove that different people have different amount of luck over time in poker?

Was a couple of years ago i studied probabilties and statistics so dont really remember how to use it.

The theorem most often called the central limit theorem is the following. Let X1, X2, X3, ... be a sequence of random variables which are defined on the same probability space, share the same probability distribution D and are independent. Assume that both the expected value μ and the standard deviation σ of D exist and are finite.

Consider the sum S_j_n = X1 + ... + Xn. Then the expected value of Sn is nμ and its standard error is σ n1/2. Furthermore, informally speaking, the distribution of Sn approaches the normal distribution N(nμ,σ2n) as n approaches ∞.

If X_i i=1,...,n is the outcome of all possible pokerhands. S_j j=1,.. is for different players. Could you then say that average S_j / n tends to the normal distribution?
If its not possible i guess its because the distribution, they are not the same, or are they?

Last edited: Nov 15, 2007
2. Nov 15, 2007

### mathman

The average tends to the mean. As a practical matter I suspect that the difference in average outcome for different players is a matter of skill, although luck plays a roll in the short run.

3. Nov 16, 2007

### JohanL

ok, so in the long run the luck evens out between different players and it does NOT tend to a normal distribution?

4. Nov 16, 2007

### mathman

I suggest you take a careful look at what the law of large numbers and the central limit theorem mean. You appear to have some confusion about the basics.

5. Nov 17, 2007

### JohanL

Dont have time for that, only reason i asked was a discussion about this in a poker forum. I thought someone here could give me a simple yes or no. About half in the forum argued that the luck was about the same for all people in the long run and the other half said there could be hugh differences.
And then i remembered the central limit theorem and thought that you maybe could prove that the luck for different players would tend to a normal distribution, even tho intuitively i think it should even out in the long run.

6. Nov 17, 2007

### CRGreathouse

Math people are funny about definitions, so using the term 'luck' makes it a little hard on us. But let's dispense with that -- let's say that luck is the percentage of good hands that a player gets. (You can choose whatever you like to be a good hand, as long as you don't vary it.) In fact you could even weight this so that getting a royal flush would add more to your luck than a full house.

Luck for a given player is does not determine future events, so having a straight flush one hand doesn't make it less likely that you'll get one next hand (provided the deck is shuffled in the meantime). And yes, using the Central Limit Theorem it can be shown that different players will have their lucks converge to the same number. Those who have played only a little will have more variation in their luck -- if you've only been dealt one hand and it's a straight flush, you're luck is a lot higher than other players'. But veteran players will all have about the same luck.

7. Nov 17, 2007

### JohanL

Hopefully this will end the discussion :)

8. Nov 27, 2007

### colby2152

As others have said, the CLT supports the luck balancing out among players over a large number of hands. However, that is simply skewed because each hand is weighted different. If a person comes off lucky winning a couple big hands, then it makes a big difference.

9. Nov 27, 2007

### colby2152

As others have said, the CLT supports the luck balancing out among players over a large number of hands. However, that is simply skewed because each hand is weighted different. If a person comes off lucky winning a couple big hands, then it makes a big difference.