I have compiled a chart that shows how big is the variance for the various win-rates.

Short explanation

• Each chart represent the graph for given EV. Find out your EV (bb/100 hands) and then look at the corresponding chart.
• For each such graph (for given EV , e.g. the one you have selected) one can see how the expectations change when increasing the number of played hands (the X-Axis).
• The red part is the probability that you will lose money after the corresponding number of hands.
• The yellow part is the probability that your profit will be between zero and 1/2 of your expected mean
• The dark green part is the probability that your profit will be between 1/2 and your expected mean
• The bright green part is the probability that your profit will be more than your expected mean

Some examples

• The bright green graph is always 50%, because the probability that you'll finish over (as well as under) your expected mean is 0.5 (50%).
• When your EV is 0 bb / 100 hands, there is 50% change that you'll lose money and 50% chance that you'll win money.
• If you run at 2 bb / 100 hands, there is 30% chance that you'll be under water (lose) after 20000 hands!
• If you run at 8 bb / 100 hands, there is less than 5% change that you'll lose money after only 10000 hands, and only after 5000 hands you may expect with >90% probablity that you'll win money.

Hope this helps (and I hope that I have no bugs .

Nice work... what game is this for though, and what standard deviation are you using? SD's obviously vary quite a bit depending on the game, and your play style.

also is this big blinds or big bets? important distinction.

This is big blinds.

The StD that I use is 45 bb / 100 hands. I think for NL-ring (deepstack) the StD is between 40-80 bb/100. Which means that the chart is a "lower bound" of the variance, i.e. we can expect that this is the lower bound of the % for negative results.

I'm not sure how many StDs are good assumption. If someone knows typical StDs for various games (or ranges) I can rerun the calculations and produce charts for different StDs too.

i play mostly 6max 100NL/200NL, as well as a bit of heads up, and my standard deviations are more like this:

6-max: 100 bb/100
HU: 145 bb/100


are yours from full ring? also are you fairly tight or something? i play around 25/19 at 6max and 48/40 at HU.

Here are some more charts, for different standard deviations. It seems that the original chart, although being a lower bound, was somewhat off the typical Stds (in short, this means you may expect even higher variance than shown):

• No limit, ring: Std 85 bb / 100h
• No limit, 6max: Std 100 bb / 100h
• No limit, headsup: Std 145 bb / 100h

Nice work. Maybe I'm not looking hard enough, but is there a program/site out there that shows examples of different graphs for the same win rate+ standard deviation? I'd like to see the swings that take place over a large number of hands.

that's a pretty cool illustration though - you can be an 8bb/100 winner at HU and still have a fairly good chance of being down after 50k hands.

Nice charts, but I wonder how they would look in case of PL Omaha. PLO's winrates and StDevs are bigger than those in NL Holdem. I know a guy who has winrate 10.8BB/100 (5.4ptBB/100) after 220k hands at PLO200/PLO400. Personally I have winrate 18BB/100 after 70k+ hands but at lower stakes. You can assume StdDev 125BB/100 for full ring games and 135 for 6-max.

Regards,
Jan

NL Fullring StD 45 - 85 BB/100h - looks like that you expect the average StD to be approx. 65 BB/100h, which means 6,5BB per hand.

Ok, I understand this and also the very nice charts. Now, I wonder how big the estimated probability is that a very good bot could be programmed in such a way that its longterm NL winrate is let's say at least 2 BB/100h or higher.

Do you think this probability is close to 100%?

I think not more than approx. 5% or so of all human poker players can have such high longterm winrates. Do you have a better overview or better estimates? Expectations for each chart you have posted would be very interesting.

For humans I guess we "only" need a very huge database of hands, and then we can determine the winrates pretty accurately. I guess someone that already have such (meaning e.g. billion of hands at given limit) should post. There is no theory here, it's simply sampling (or I have not understood your queston).

My problem is that I want to find out how far winning poker players are fooled by randomness on exactly this topic which you discuss here in this thread. Most poker players - and also winning poker players - do not know what variance (the title of your thread) actually is. If you discuss this seriously with a couple of winning poker players here in this forum, you risk a dead penalty because of your blasphemy.

My biggest concern is, that I do not know, how big the survivorship bias and the selection bias is, if I try to get meaningful statistical data. Poker game conditions change permanently and therefore, I do not believe also that some sort of accurate long term win rate actually exists. My point is that there is no such thing like independent, repeated, and identically distributed trials in poker which will result in the same expected gain per average unit wagered as that which applies to a single trial. Only with computer simulations and very huge samples you can find such indentically distributed trials.

Another problem. Nobody knows how the distribution of a winning player actually looks like. Is it a Normal distribution, some sort of Student T-distribution or something else like a Logistic distribution. Do we have a symmetric distribution with equal tails or a skewed distribution with longer tails on one side than the other. How big is the skewness, the average of the cubed deviations (scaled by the cube of the standard deviation)? How big is the kurtosis which relates to the fat of the tails or the peakedness of the center of a distribution?. If this considerations are relevant than we can assume that there is a 3th and 4th moment in poker which can matter long term results very easy.

Such things have already been discussed here in very old 2+2 forum threads. However, as far as I can see it, nobody found out the empiric truth. Such things, however, are important if I want to estimate the win rate of a "typical" winning player, whatever that actually means. As long as I do not know correct estimates of win rates no variance analysis makes real sense to me. It can be that we consistently overestimate the average win rate of a "typical" winning player. Therefore relative short descriptive statistics alone are only weak evidence without any statistical significance.

One estimate is imho relatively sure. The win rate of the "typical" average poker player correlates arround the average rake percentage he pays for his own input. What this means - especially if I assume an average SD like you and players with the same skill or edge level - is easy to calculate. However, it is very easy to make a wrong maximum likelyhood estimation with such parameters not knowing how the whole distribution looks like.

Another aspect. I know what you have done and how you got your results. I know also how experienced you obviously are with certain other things and where you come from. I do not want to open a certain campaign thread here in this forum and point to your web side and certain links posted there. I ask you as an very experienced programer to help the 2+2 community to find a better knowledge about probabilities of possible win rates in poker games, instead. Your work and your knowledge with computer simulations could be very valuable for all poker players arround the world.

Your two major points are both valid.

I assumed normal distribution, while in fact it might be a different one. Most of the things in poker we calculate with normal distribution, most of the threads or acrticles I've seen are using it. The distribution in fact can be different, but this (again) we can determine only from actual data.

Graphing that out will be fairly easy as far as I can think of it. We just need the profit/loss of each hand played, then we can dump the numbers in an excel, and excel can do the rest (graphing the things). All the stats properties we can get from there.

I'll get this into my todo list but I can't promise when I'll be able to get it running. I cannot imagine that noone did such analysys so far, maybe we just haven't searched enough.

The other question, that conditions may vary, is much more complex. I think that most winning pros should keep conditions similar, e.g. have defined table selection criterias, etc. Also I think winning players are much more selective in changing their game, they will usually change their play when they are quite confident what they are doing (some won't - and hence you get to the winners that don't know where they stand issue).

I can only suggest that the above gaph are applied to one type of game, one limit (i.e. don't combine hands from different limits), same poker network. The opponents may vary too, hence it might be a better idea to look at sessons, instead of hands: I've see this done, but I would question it. As opponents vary also within one session, usually in an hour > half of the opponents on the table are changed.

Thank you.
I try to give you some ideas for your todo list in form of a wish list:

What happens when 10 aritifical poker players play a huge sample size (say 1Mio. hands) together?
a) First test - 10 same artifical players
b) Second test - 10 different players with some sort of normal distributed skill levels.
c) Third test - 10 different random players

Are the differences between the players significant?
Are there major differences between the average distribution and the distribution of the best artifical poker player? Can we really assume Normal Distributions under such artifical conditions - with only different mean and SD as parameters or have all players the same distribution with same mean and SD?

What happens, if you repeat the hole test several times under same conditions?
Is it possible to manifest accurate and different win rates in case of such artifical conditions or is this construct a myth?

Please do not ignore the rake consideration in such huge tests. I think you know why.

I would like to see the inverse of these. That is, what do the charts look like for -1bb losers, -2bb, -4bb, etc. From that we could tell the probability that say a -4bb player is +$ after say 10k or 50k hands. Can these probabilities be inferred from the +bb charts by symmetry? If not, any chance you could create charts these for a couple of the "standard" standard deviations?

i'm pretty sure the charts can work symmetrically. just switch the < and > signs and it should work, remembering that "ExpEV" (expected expected value?) is negative

One example only in order to demonstrate why I want to know certain things:

Your are an average poker player. Your expectation correlates arround an average rake of -1,5BB/100h. You play 1.000.000 hands. Your average SD = 50BB/100h or 5BB per hand. We assume a Normal Distribution. You want to know your expectation with a confidence probability of 99.73%, that means z-score = 3.

EV = -0.015 * 1,000,000 = -15,000 BB

confidence intervall:
worst case scenario: -15,000BB - 3 * 5 * square root (1,000,000) = -30,000 BB
best case scenario: -15,000BB + 3 * 5 * square root (1,000,000) = 0

That means with a 99,73% probability you will be a loser after 1Mio. hands.

If I assume that all poker players have more or less the same distribution, than it becomes clear that out of 1,000,000 poker players only 1,000,000 * (1- 0,9973) = 2,700 poker players can be winning poker players after 1,000,000 hands. How big their winnings are is another question.

I want to find the truth and not to rub salt into a wound.

In another thread I tried to work out, why the typical poker player is confronted with randomness and why this randomness forces necessarily down- and upswings.

I tried to explain this as follows:

"Have you ever felt strange tachycardias and placed with fast fingers and too less deliberation huge bets in poker? That is the typical point were chance is involved. Chance can earn you a big victory or punish you with a huge loss. If you repeat such trials very often and long term you will come sooner or later to the point were your destiny as poker player is crucially affected. Some winning players can be suddenly losing players and the same can happen vice versa. Poker is at decisive points very often a game of luck and not a game of skills. I hope you know as poker player the house that luck build. One hand only can affect your destiny whether you are a famous poker player or not and whether you have a long or extended losing or winning streak or not."

Now, you are a winning poker player with a win rate > 2BB/100h. But you want to know your win rate in terms of 1BB/100h exactly in order to get a true picture. Your average NL SD is 65BB/100. You want to know how many hands you must play to get the true picture. You want to know the answer with 95% confidence and you think that only the winning side of the Normal Distribution is of any interest for you. Z-score is in this case z = 1.645. I use the formula I got from jay_shark to find the answer:

minimum sample size = 100 * (1,645 * 65/1) ^ 2 = 1,143,295 hands.

Now you want the true picture with 99,73% confidence and under a double sided consideration. Well, than you are confronted with the following calculation:

minimum sample size = 100 * (3 * 65/1) ^ 2 = 3,802,500 hands.

I think this says not everything but most of the truth.

This is another try of the same sort of math were I here no sound and fury:

http://forumserver.twoplustwo.com/sh...=256021&page=9

Yes, I can assume that the skill levels of all poker players are normal distributed in one distribution. And this is the problem of all poker players who are unable to understand the problem.

btw:
Jerrod Ankenman said in one of these MOP-study-treads sometimes to me forget the rake consideration of the average poker player. This is only blabla. Now I ask him to react and give a clear answer, why he has taken me for an idiot.

i just read your last 6 posts in this thread and i have to say none of them made any sense. what exactly are you trying to figure out?

I think that the so-called "average win-rate of a - typical - winning poker player" is a myth and want to prove this. If I get no assistance than I will do my own research with comupter simulations on this matter. I will find out the truth. It is only a question of time.

Most winning poker players are fooled by randomness.

what do you mean "it's a myth"?

Make the following test to understand me.

Ask 1,000 2+2 winning poker players - no losers - how big their average win rates are today. Ask the same people - no other ones - one year later again and in two, three, four and five years again.

My thesis is:
1) The average win rate of this 1,000 winning poker players will be much lower one year later and in two years you will see another significant reduction as an average. The average win rate of this winning players will correlate long term - slowly but surely - in the direction of -EV or the rake they pay.
2) The total number of winning poker players (the survivors) will become smaller and smaller.

If this thesis is true, than we have a true win rate of the "typical" winning poker player. But this win rate is something else than most winning poker players find in their PT-Database. Therefore the variance analysis as made by indiana in this threat makes no sense and most winning poker players are fooled with such considerations or by randomness.

If my thesis is true, than we have to realise and understand that even poker is a game were the winning poker player must expect -EV. I do not predicate with this thesis that nobody will survive. It would be interesting, however, to find out how the survivorship curve and the true win rate of the average winning poker player looks like.