I Think I Am Probably A Winning Player After 4k Hands...

4k hands seems intuitivly too small to figure out if you are probably a winning player, can some one check my math?

My standard deviation is 29 PTBB/100 hands (I play no limit)

Therefore, the standard deviation for my winrate after 4k hands should be:

29/sqrt(40) = 4.6 PTBB/100 hands

My winrate so far is 7.6 PTBB/100 hands, so I conclude that a breakeven player would have to be running on the good side over 1.65 standard deviations to be running as good as I am

Therefore, I claim I am probably (probably meaning ~85% sure) I am a winning player

eh?

[jtr]

Your calculations look about right. If you're happy with a 15% chance of getting the same result if your true win rate was zero, then more power to you. (Not trying to be rude, just pointing out that an alpha level of 0.15 is unusually high for a statistical test.)

The problem I have with this is that the top end of your confidence interval goes higher than what is actually sustainable. I think that many of these winrate estimates should take a more Bayesian approach. As a more extreme example, suppose I am up 25 BB/100 in 900 hands of short-handed limit, with a SD of 27 BB/100. (I am using limit in my example because I don't know what reasonable NL numbers are, and the SD is high because I am running hot). The standard deviation for my winrate is 9 BB/100, so +/- 2 SD gives a range from 7 to 43 BB/100. But my prior beliefs are that 7 BB/100 is unsustainable, so all I would take my results to show is that I am running really hot. To make an estimate of my winrate, I would probably make up some reasonable Bayesian prior and do things that way.

With your numbers, 7.6 PTBB/100 hands is very good, but doable. As it is on the high end of sustainable results, it means that you probably (but not necessarily) are running somewhat hot, and I would guess that a Bayesian analysis would have a slightly higher chance of being breakeven or worse than what you got.

Yeah this crossed my mind a bit

Maybe I should start out with the assumption that without any data, the probability density of my true winrate is uniformly distributed over [-10,10] PTBB/100 hands

But I'm too lazy for that

I'm too lazy to do that as well, which is why I didn't give any numbers in my post.

Some easier numbers to compute: if your true winrate is uniform over [-8,8], then your chance of being breakeven or worse almost doubles. Uniform over [-12,12] is only slightly higher than what you computed.