QFT

Agreed. Nice work Holdem & Pokerboy.
Mods - Please sticky.

I couldnt get my thick head around this maths until now.

+1

Isn't it likely that the curve representing a poker players win rate per 100 hands is not normally distributed. There are other distributions besides normal distributions, and the "skill" factor should mean that, on average, a skilled player who has the same normal distribution of starting hands and flops (normal distribution based on RnGs allocation of cards) will win more and lose less than a less skilled player. The card and hand distribution should be normal, but the amount of wins and losses might not be.

Is there a way to plot several winning and losing player PT results regarding win rate/100 distributions over 200K hands to see if the "curve" looks normal vs some other, skewed distribution curve? It is likely, I think, that the curves are skewed with the bad players losing more money more often with the same cards thand the good ones.

Nice work. I knew some math but this refreshed it a little. Seems i'm 99.3% a winning player at 3.6 +/- 1.5 ptBB/100 at 25NL (40k hands / MT ratio = 4)

Thanks for all the work Holdem, great thread.

Please Sticky.

Will need to read this a few times (and the rest) to take it in.

.......


Interesting view bottomset, I've always tried to approach poker in the "long term" view. Yet you believe there to be no long term game.

Has my mind been corrupted by poker books?

Absolutely true. There are facets of variance we often don't even consider. For instance when we run into set over set it's a cooler and we move on confident that it will happen in our favor and all will be even.

But what happens if we lose a full BI twice when we hit the wrong end of set over set but when it goes in our favor twice it's against some short stack? Same with AA vs KK.

For some of these kinds of variance the "long run" may be too long to even out.

No, there is a theoretical long term. It just may be longer than you play cards for.

if i didn't mess up the math, here are some simulations with a winrate of 3 and standard deviation of 35 over 50k hands:


100k hands:

100% ACK.

This is something I have in my mind for sometime now when thinking over these "in the long run it is all skill"-debates sometimes happening.

Still so many people seem not to realize/consider it at all, that I always thought it might be my more naive pennytable/pokernewbie mentality.

Yes, the distribution function for a given player's results over a given sample is not a normal distribution, but it probably is somewhat close if the sample size is at least hundreds of hands. This isn't to say that all players have the same normalish distribution, each one (if they always played the exact same strategy in the same types of games) would have their own "true" winrate and SD... using fiaca's graphs as an illustration, better players would have a larger slope for the pink line representing their expectation, and players with a larger SD are likely to have more variation about this line.

Different players have different winrates and SDs, so they have different distributions for their results, so I don't think that having lots of players submit results for 200K hands and plotting them would answer the question you're interested in. What would be effective (if I understand your question right) would be having one player record their result every 100 hands, and plotting, say, 1000 of these results and seeing how close they approximate a normal distribution (or if we want the result to wind up looking a little more like a normal distribution then we'd use samples of maybe 500 hands instead of 100). The key is that we want each of these 100 hand samples to be generated by the same distribution (so from the same player, with the same playing style, etc. Ideally even the same opponents/etc.), so that when we plot enough of them we start to see the shape of that underlying distribution.

bottomset's point is just that you don't reach the "long term" (as we saw above, if you define reaching the long term as earning within .5 ptBB/100 of your true winrate, it'll take you a couple million hands to be 95% sure you're there, and your game and the game conditions change way before you reach that). Of course he still agrees that what you should be worrying about is making good decisions that maximize your expectation (or best satisfy whatever your goals are - maybe you intentionally sacrifice a teeny amount of EV in order to lower your variance) since you can't control your luck and it should be irrelevant to your decisions.

That isn't the only reason PokerTracker underestimates our true variance. PT uses the old B&M methodology for calculating s.d. Before the internet it wasn't possible to easily track every poker hand. Therefore the session by session method was used. Sessions tend to flatten true variance. Our quitting strategy affects the variance.
Today online it's possible to calculate s.d. using the hand by hand method. This usually produces a variance about 20% higher than the other method.

Bump for awesomeness.