Central limit theorem and poker

In summary, the central limit theorem states that luck tends to even out over a large number of hands.
  • #1
JohanL
158
0
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 don't 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:
Physics news on Phys.org
  • #2
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
ok, so in the long run the luck evens out between different players and it does NOT tend to a normal distribution?
 
  • #4
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
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
JohanL said:
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.

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
Thanks for your answer !

Hopefully this will end the discussion :)
 
  • #8
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
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.
 

1. What is the central limit theorem?

The central limit theorem is a statistical concept that states that the sampling distribution of the mean of a large number of independent and identically distributed random variables will be approximately normally distributed, regardless of the underlying distribution of the individual variables.

2. How does the central limit theorem apply to poker?

In poker, the central limit theorem applies to the distribution of hands dealt. As the number of hands dealt increases, the distribution of hands dealt will approach a normal distribution. This means that the most common hands dealt will be those in the middle of the distribution, such as pairs or suited connectors, while the least common hands will be those at the extremes, such as straight flushes or high card hands.

3. How does the central limit theorem affect a player's strategy in poker?

The central limit theorem can affect a player's strategy in poker by helping them better understand the likelihood of certain hands being dealt and the expected value of those hands. It also allows players to make more informed decisions based on the distribution of hands dealt and the probability of certain outcomes.

4. Is the central limit theorem applicable to all forms of poker?

Yes, the central limit theorem is applicable to all forms of poker, as long as the number of hands dealt is large enough. This includes both live and online poker games, as well as different variations of poker such as Texas Hold'em, Omaha, and Stud.

5. Are there any limitations to the central limit theorem in poker?

While the central limit theorem is a useful concept in understanding the distribution of hands dealt in poker, it does have limitations. For example, it assumes that the hands dealt are independent and identically distributed, which may not always be the case in certain poker games. Additionally, it does not take into account other factors such as player skill and strategy, which can also impact the distribution of hands dealt in a game of poker.

Similar threads

  • Set Theory, Logic, Probability, Statistics
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
285
  • Set Theory, Logic, Probability, Statistics
Replies
9
Views
3K
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
969
  • Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
19
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
4
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
7
Views
7K
  • Linear and Abstract Algebra
Replies
1
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
8
Views
2K
Back
Top