[BruceZ]

[ QUOTE ]
I ask this mostly because I am curious and more experienced minds than me can address it.

I have at times read some discussions about how valid your BB/hr rate is for 10,000 hands. What I am curious about is why this is an insufficient sample size for reasonable calculations.

If I naively sit down and try to model my probabilities, I would probably start with a simple model. If I am dealt 2 cards in HoldEm, there is some unknown BB/hand for this combination of 2 cards. In theory this is modeled with some pmf, but for simplicity I will approximate with a pdf f(*).

Assuming this density isn't something truly ugly it has its first and second moments and therefore mean and variance are meaningful.

Now, since each hand is independent, my 10,000 samples give me a Sample Mean with Variance = Sigma^2/10000. That is what really bothers me.

Now, our Sample Mean has the same Expectation as f(*) but has 10,000th the variance. Additionally, the Central Limit Theorem assures that our Sample Mean has a nice normal distribution.

From this, one would naively think that any BB/hand estimate that we generated would have a pretty small Standard Error. We should have fairly small Confidence Interval's.

I can see a number of problems with this happy naive calculation, but it does make me curious. In a plain limit game I don't recall winning more than 30BB's in a single hand, nor losing much more than 10. Most of the time my results are pretty stable. That would suggest my variance has pretty strict bounds and shouldn't be too high.

What problem are people encountering that keeps this estimate from being useful?

- Lex

[/ QUOTE ]

The standard error is still a significant fraction of the win rate at 10,000 hands. Typical SD for 100 hands is around 17 big bets, so the standard error for 10,000 hands is 17/sqrt(10,000/100) = 1.7 big bets. This is a significant fraction of the win rate for 100 hands which is typically 2-4 big bets.

If you only want to know that you are a winning player to 5% significance after 10,000 hands, then you must have a win rate of 1.64*1.7 = 2.8 big bets/100 hands.