January 04, 2011

Embracing the Variance - Part 3

I thought, I'd spend this post clearing up some stuff about my last 2 posts and responding to comments I got in the thread, but also from other sources.

First of all, let me say that I was very surprised by one result in the tourney: that if you are the best player in the tourney, in fact have a large edge over the rest of the field, THEN is the time to take marginal EV spots. That playing nitty then will not yield you the most money, while when you are barely a winner, you actually stand to gain to see others go down in flames while tightening up or just playing slow - it seems the EV is just too low to really make a difference to the fast increasing blinds that is a turbo tourney.

A few notes:

a) This is a Rush tourney. Your opponents constantly change. Hence you have an edge "over the field" which is a very wide concept and we can't really know that beyond of what mistakes we see at our table. In a normal tourney, where you are at your table for a long time, it is my feeling that we can increase our theoretical edge over the table by accumulating good reads and that the rest of the field is fairly unimportant to us and our standing in the tourney.

b) I am somewhat worried about the smoothness of my simulation. In the end, poker results are fairly discreet and I am afraid there are some problems at the edge, i.e. when we bust from the tourney. Since it is a limiting case, I don't quite know how to deal with that mathematically.

c) I am not a tourney expert. There are probably miriads of better insights written before this and I know for many people, that I'm reinventing the wheel. Still, I find it interesting and again, I found the turning point quite high as mentioned above, so it was very shocking to me.

The whole series of post was inspired by Jesse, his poker lunch and his post on 2p2. Thanks Jesse! http://forumserver.twoplustwo.com/17/small-stakes-limit/no-woman-no-cry-925446/

That post also brought up some questions from thesilverbail, which I will answer here:

" 1. I'm not sure I understand your methodology in part 2. You populate an LHE tourney with players with different winrates. 4 different categories (10, 5, 2, 0, -2). Do these winrates balance out ? Otherwise your conclusions might not be accurate."

No, I play one tourney, where I have a xBB/100 edge over the rest of the field throughout the tourney. Obviously that is a flawed assumption, since your edge over the field doesn't stay the same over the whole tourney. In fact, in the  next post I will run some sims mostly with decreasing edge over the field throughout the tourney, to see how this affects especially later stage risk approach.

"Do you mind sharing the code you wrote for the simulation?"

Oh boy, this is embarrassing, since I'm such a bad programmer (spaguetti code, here I come). I know there are much quicker ways of doing this, like just input a sheet of the tourney structure instead of playing each level out one by one by hand, but for now here it is:

"2. I think assigning each line to hands per hour and labeling the x axis in number of minutes per level is confusing. The difference is really number of hands per tournament level which is what allows the edge of the good players to stand out."

Yeah, it was just a starting point. Remember, the argument started with the discussion if you should play fast or slow in a game where you are constantly overbetting you bankroll and the blinds go up fast, but where you do have quite the edge over the field. After I posted this many people suggested I should start to vary the speeds, maybe proportional to the edge to get a sense of how this would affect it throughout the tourney, but I don't want to tweak to many things at the same time. For now I will do the varying edge thing, plus some slower structures, let the blinds increase less quickly.

"3. It's interesting to me that the avg #BB /person slopes very gently upwards in all the curves after level 10-13. why would this be so?"

It's just the function of the edge just finally catching up with the increasing blinds. It is dramatized or magnified due to the fact that I left constant edges over the field throughout the entire tourney, which of course doesn't happen

"4. I think you are going to argue based on these results that a player with > 3BB/100 winrate should widen his range to increase his "effective" hands/hr. The problem is that the hands you add at the bottom of your range won't have anywhere near ~3BB/100.
In fact this is almost an argument for tightening up in the tournament ! ( I think albacorela pointed this out). "

Besides the varying edge thing, the next thing I will try out is non-rush poker tourneys. At the end of the day, this is a turbo and an extremely fast structure. My feeling is that with a slower structure, you don't have to be so conservative, and can take more EV shots, but I still have no idea. It was quite eye opening to me that it took so large edges >3BB/100 for the EV to even make a dent and that perhaps playing nitty when having small edges in turbos and passing up some spots, does work, but only if the EV difference is minimal! In NL, this would be akin to passing up 22 vs AK (51:49), but not QQ vs AK (57:43).

On to copoka's comments:

"Bella, don't take it the wrong way, but "absolutely sure of your winrate" is border line silly. The only way to use "absolutely sure" and "winrate" in one sentence would be "one can be absolutely sure that he can never be absolutely sure about his winrate"
WR is nothing more than suggestive and convenient number and should not be used for any kind of "predictions" or forward looking analyses. Any simulation made based on assumed WR is basically useless since the assumption made is HUGE and never accurate."

I see this all the time, that one can never be sure of their winrate, that you have to play 17 million hands to know for sure, etc. Yes, you don't know your absolute winrate for a given limit or something like x number of hands. But you can fairly quickly assess if you are a winner in a certain lineup, estimate your edge and even pin down you approximate winrate based on that after a few hundred hands (loser, breakeven, marginal winner, winner, crusher).

Copoka, you and I have maybe played 20-50 hands total lifetime, yet at the time when you were playing a lot of HUHU, you constantly denied me action. Why did you do that if you didn't know what winrate you had against me and you could never be absolutely sure of it?

Notice how I am not deriving the winrate from past events. Take, for example the stockmarket, everybody will agree that forecasting the success of a company based on its past performance alone is very iffy and should be taken with a grain of salt. However, when you look at the fundamentals of a company, if it is based on a good product and not just hype or a bubble, you will win money. The more information you have about the company, the better you will be able to determine its likelihood to succeed and for you to make money in the long run.

If I see a game where I see constant mistakes and things I can exploit, I will know I can win in that game and I can fairly easy estimate my winrate in that situation. Otherwise you'd be truly lost, wouldn't you?

"So, although number of AA dealt, sets flopped or straights we make on the river can not effect difference in skill for a long time, there is another thing that can, is and will.
There are monies involved."

You are talking about poker being a game of people, that some of us tilt, that nobody can every play their best. Yes, those things all exist, but for the purpose of this calculation, they should not be important. If you sit down at a game, but are tired, you should factor that into your winrate estimation. Of course we can't be 1BB/100 (or whatever number) over long periods of times, there are fluctuations, even with the same lineup, sure. Then again, this is just a benchmark, something to hold on to. Otherwise let's just all throw our hands up in the air and proclaim that nothing can be predicted and it's all chance.

"First, those things are fairly rare comparing to the number of hands dealt so they converge much much slower in "calendar terms" so to speak. Nasty thing is that although they are rare, the effect on your earn is huge."

For every one of those you nasty ones, if you are a winner you will have an even better super-duper high winrate one statistically speaking. Variance goes both ways, not just negative!

"Second, even if we are lucky enough to see them balancing out, deck is still stacked against us due to measures we have to take to survive in this game."

Your argument makes no sense. If your theoretical winrate is the same it is exactly just as likely that you will have coolers at 10/20 as at 5/T. You are running into a huge gambler's fallacy here. Coolers don't even out, every hand is its own in the vacuum. It is just that heaters are a teeny bit more likely to happen than coolers if you are a good winning player.

If you lose money at the stockmarket and lose half your worth, of course you need to double your worth to be exactly at the starting point again, but if you have a positive expectation and bet proportional to your edge,  in the long run you WILL win money and the bets you make will be bigger and bigger proportionally, provided your edge keeps existing. I don't understand what is so hard to graps about that. The only thing that will make you lose more pots at higher stakes is that you might not have the edge there.

"Third. There is so much human and/or social factors involved in giving and getting action in poker, that the argument can be made that bell curve simply cant govern this process for the same reasons it is useless in any other field where human nature runs wild, like stock market, for example.So what convergence are we talking about in a first place? Where would that come from?

You don't believe poker follows the Central Limit Theorem, you believe in fat tails or black swans. Fine. I have never seen any evidence that suggests black swans for poker, but who knows, I have not seen any evidence that denies it either. For a while I tried to prove this by gathering about 1.5 million hands from winning players and describing their deviations from the normal distribution, I didn't see any, but I could not prove that there weren't any either. I don't know 100% what happens at the extreme edges, it might be that your disastrous sessions are even more disastrous. Fat tails is not my territory and it is very disputed, for the discussions in Mathematics of Poker it is safe to assume that the Central Limit Theorem holds, as most results are determined by something that is really truly random and has no psychology - the deck of cards.

"I'd say you are looking at graphs of skilled poker players who where lucky enough positive cooler balance for a long time. The proportion of skills vs luck is unknown. Prove me wrong.

Oh, btw, there is a pretty good HU poker player named Matt Hawrilenko. Have you seen his graph lately?

There is another one. Very talented and very naive. Talented enough to break the game of HULNE down to the science. Naive enough to declare in one of his blogs that "making 300k playing poker is a child's play" His name is is Bryce Paradis. Take a look how much he made over past 12 month."

Both players are playing in games where they don't have clear edges anymore, their winrates, while positive, are not the 4-5BB/100 they were used to when playing in softer games in 2006-08.
AFAIK both Hoss and The_Bryce have been playing a lot of 6max lately and not that much HU. Bryce himself has been going through tough times with his mind being on his real estate losses and his depression. I hardly think that you can be a clear winner during those times.

Both said that they played HUHU LHE at a time where there was no clear theory on how to play it and they had just extreme winrates, just because the skill difference is so big. That is not the case anymore, any 3/6 HUHU reg is much better than the 2006 50/100 players, where these guys made their money off.

What I was mostly referring to, are the bumhunters that refuse action, but to the biggest of fish. They have smooth graphs, no matter how bad they run, because their winrate over the fish smooths any graph. I took those HUHU players as an example, because HUHU is almost the only place where such extreme skill difference is shown and it has a direct impact on your winrate. You might be right though in today's games, where the games are tougher and even HUHU edges are small, so my comparison was for bigger edges :)


Anyway, enough for now, even though I could talk for hours about this, I need to go to sleep.

Posted By bellatrix at 07:37 AM


Tags: theory tourneys


Hood posted on January 12, 2011 at 17:12 PM


I have engaged Copoka twice in the past on exactly this topic, and I put forth very similar arguments, once on 2+2 and another time on a private forum.

Both times he didn't respond and the debate died.

He consistently puts forth the idea that the central limit theorem is invalid, quotes a big of Taleb then disappears. I wouldn't suggest you spend much more time engaging any back-and-forth as I think there's a good chance he doesn't even return to respond.

Anyway, congrats on your new spot on DC :)

bellatrix posted on January 16, 2011 at 08:27 AM


Heh! I know about Copoka, still, don't wanna seem like I don't get into things with comment-makers :)

Thanks on your congrats and just wanna say I loved your time-lapse series!

copoka posted on January 25, 2011 at 23:27 PM


“I see this all the time, that one can never be sure of their winrate, that you have to play 17 million hands to know for sure, etc. Yes, you don't know your absolute winrate for a given limit or something like x number of hands. But you can fairly quickly assess if you are a winner in a certain lineup, estimate your edge and even pin down you approximate winrate based on that after a few hundred hands (loser, breakeven, marginal winner, winner, crusher).

Copoka, you and I have maybe played 20-50 hands total lifetime, yet at the time when you were playing a lot of HUHU, you constantly denied me action. Why did you do that if you didn't know what winrate you had against me and you could never be absolutely sure of it?”

We only played 20-50 hands b/c assessing relative skill level as fast as possible is all I did. I had no idea what you WR was or mine. I had no clue How to calculate it even after assessing relative skill level. I don’t think anyone can do that. All I knew and all I cared about was the fact that you were not bad enough to justify playing giving the rake we are charged and possible bad variance.
To do what you are talking about, estimate edge and even pinpoint approximate winrate,
one has to have a very sophisticated way to quantify observed difference in skill level the same way they do it in a game of Black Jack for example ( playing 1deck instead of 6 decks adds x% to your edge, not doubling down when you should costs you y% or choosing table when dealer draws to soft 17 takes z% from your edge).
Impossible in poker!!!
Yes you can say to yourself, I play better, but you can never say or be sure that you play between 2 and 3 BB/100 better.
And you certainly can never put your assumed WR into any kind of simulation, forward looking analysis or winnings projection. The assumption is HUGE and never accurate thus rendering such theoretical exercises basically useless.
Sure you can do it for your own amusement, but that’s about it.

“Notice how I am not deriving the winrate from past events. Take, for example the stockmarket, everybody will agree that forecasting the success of a company based on its past performance alone is very iffy and should be taken with a grain of salt.”

No, its not “iffy” its bogus. Not only it should not be taken with a grain of salt, it should not be considered at all.

“However, when you look at the fundamentals of a company, if it is based on a good product and not just hype or a bubble, you will win money. The more information you have about the company, the better you will be able to determine its likelihood to succeed and for you to make money in the long run.”

No, you will not. And the simple reason why you will never make money on “funny mentals” is that all available information, fundamental, technical, whatever, is discounted by the market immediately. Unless you have insider information and act on it ( aka committing crime), or employ principals that apparently don’t even being considered by you, you’ll never be able to have an edge large enough to insure making money in a long run.

Indeed, stock market and poker are very similar in a lot of ways. Sickly similar in fact.
Again, nothing personal, but your naive references to stock market shed the light on why you are falling for so many common fallacies.
And you are not alone. Neither in poker world nor in the markets.
Its extremely common on every level. From college dropouts who learned how to semibluff turns, were lucky enough to run good while doing so, made some money and got confused in a process,
to “best of the bread”, “cream of the crop” and Nobel Prize laureates.

People, who ran LTCM had unlimited resources and access to foremost authorities in statistical and mathematical fields. Credentials and qualifications of experts used by that miserably failed organization were maybe even higher that those of authors of “Mathematics of pokers” or anyone on 2+2 who ever touched statistical subjects in any way.

Armed to the teeth with knowledge and expertise, they’ve created a model that put the chance of LTMC portfolio to lose 50% of its value in one day at 1 in 1 with thirty 0s.
That’s like a few billion lives of Universe.
They lost everything in 1 week!
All because they felt to fallacies that are so common among market participants and poker players alike.

Well, gotta run. Will come back with a few thoughts on fat tails and black swans.
GL at tables

copoka posted on February 04, 2011 at 21:15 PM


Since I’ve mentioned that imo your arguments are full of common fallacies, I think I should make some examples.

Let’s start with most obvious. Survivorship bias.

What you are doing in your analysis is that you simply concentrate on those who “survived” the process, on those who were on a right side of the variance and were lucky enough to win at least same amount of money in heaters as they lost in coolers thus allowing “WR” to do its “job” of creating nice smooth graph.
You inadvertently overlook those countless good players who for tens of thousands of hands were raped by 2outers in a huge pots only to make back peanuts when 2outers went their way (just one of many many examples of possible misfortune) and finally had to stop after loosing bankrolls or just simply in frustration.

Basically, you simply do what proverbial Texas “sharpshooter” did when he fired bunch of shots at the side of the barn and then went to paint a target centered on the biggest cluster of hits and then claimed to be a sharpshooter.

Survivorship bias, or counting only successes that benefit your theory and ignoring any failures that may prove your theory wrong, leads you to overly optimistic belief that coolers must eventually be canceled by heaters not only in terms of actual hands (btw, I have no doubts that’s the case) but also in amount of money won and lost.

The first part is the function of deck ONLY and thus virtually guaranteed by normal distribution.
The second part has unknown amount of unknown variables in it and thus can not be normally distributed. At least, you simply can’t prove that it can be which leaves you the only choice to blindly believe that its normally distributed. As Taleb calls it, you are fooled by randomness.

Here is my personal example.
In 2156 hands HU I was dealt 100 AA.
Out of those 100 times, 78 times my opponents either folded to my PF raise or folded PF from SB, thus converting what should be my biggest moneymaker into one of the smallest ones. Almost 50k hands later I’m still behind my “long term” AA “WR”
Try to fit it in your Bell Curve, please.

You see, when you admire those smooth graphs of skilled and lucky “winners” you simply ignore, or rather don’t have any data to evaluate for so called “loosers” and simply assume that money made and lost in coolers and heaters should wash itself out and converge the same way flopped sets do. Classic survivorship bias.

Look at your results. You played hundreds of thousands of hands in the game you know and love. Your strategy is very sound. You know how to deal with ranges, betting patterns, deception. You know how to exploit and what to do not to be exploited.
Above all, you game select to leverage your knowledge to the max.
And yet, your results, as you posted them a few blogs back, are so mediocre.
Moreover, according to your beloved Bell Curve, you have a pretty decent chance to be nothing more than a lucky looser.

Think about it.


bellatrix posted on February 04, 2011 at 23:25 PM


"And yet, your results, as you posted them a few blogs back, are so mediocre. "

Hey now, no personal attacks!

I made it pretty clear that my strategy wasn't sound when playing in a 10/20 game. I have never claimed to be a winner at stakes 10/20+. I know of my leaks there!
I have played tilted and I have played against better opponents than I and I have done both.

You attack my mediocre results when in fact they confirm what I've been saying, my winrate is higher at the lower stakes games than at higher stakes, because this IS a skill game, not a luck game!

And who are you? You go on attacking me here as if I was some girl you need to educate?
I attended the International Math Olympiad as a teenager, I browsed to physics studies in Germany, publishing papers 4 years after my first year of study. I have a PhD in Physics and was a Lawrence Fellow at LLNL. I work at Caltech/JPL as a postdoc. I have amassed hundreds of thousands of $ grants from NSF and NASA using my knowledge of math and science (thanks for your tax $, by the way, this foreigner is very thankful). I would be happy to take on Taleb on fat tails in poker anytime!

You are running your own personal scheme of inverse gambler's fallacy here. If I tell you the chance of flipping heads was 0.5 you'd tell me that it wasn't valid, because there is a chance of flipping x times tails in a row and that it would take forever to regress to the mean. But the fact is still the same and in a simple coin toss (assuming a fair coin), there are no fat tails in the distribution, the central limit theorem is valid! The same goes for a simple deck of cards, wtf would make it change from a Gaussian to something else? You have provided me with no evidence.

copoka posted on February 05, 2011 at 04:26 AM


“Hey now, no personal attacks!”

That’s right!
Please, don’t make it personal. I know, I never did.

I’ve never ATTACKED your results. I just QOUTED them.
All I said, that according to CLT your results HAVE A CHANCE to be a results of a losing player.

Do I think you played a losing poker in games you played?
Absolutely not!!! Not even in those 10/20s.
But I do believe, that if you’d have a chance to have all the information about your hands (obv. Impossible in such an incomplete information environment as poker) you would most likely see, that you were rubbed of what was rightfully yours in a process by factors not only outside of your control, but also factors that are not governed by predictable and reasonable laws of probability.

You keep on accusing me in not understanding probabilities of a coin toss and not believing that deck of cards behaves in accordance with CLT.
Please stop! We have no disagreement there!
I do understand how it works. There is no gambler’s fallacy on my part.

What I think you are missing is that you are making statistical arguments about poker results assuming that the Central Limit Theorem applies to them simply, and that there is One True Probability Distribution, with its One True Mean and One True Variance, governing a player's results, and that these absolute and true numbers are unveiled with a sufficiently large sample of hand histories.
In reality there is a separate distribution for each position relative to the blinds and button, and each of those distributions depend in turn on who is seated in the other seats, and their internal moods, and the player's own internal mood, and whether or not the player in question or the other players has been drinking, how they each feel about how each of them is running, whether or not it's the beginning of the month and people have just cashed their pay- or social-security checks, etc.
What's more, those probability distributions have a big, fat tail, with surprising outliers mucking up the statistics -- and that fat tail is just as volatile as everything else about these ever-changing probability distributions governing a player's results.
Simply put, there is just too much of a human factor involved. Not in what deck of cards can produce, but in giving and getting action. And human factor, as shown so convincingly in behavioral finance for eg, is anything but predictable, orderly and reasonable.
That’s the reason I genuinely question whether the overall distribution is remotely well-behaved enough for the CLT to apply at all.

This is precisely the reason why so many otherwise brilliant minds, with credentials that probably can compete even with your impressive ones failed so miserably when it comes to calculating probabilities in a speculative fields when human nature runs wild. Poker is undoubtedly one of those.
That’s the reason why we see so many “Texas sharpshooters” around.

One more thing.

It’s is very strange and surprising to me, that you’ve chosen to jump on me with dreamed up “personal attack” charges and naïve coin flip arguments but never actually touched my points.
If you can’t deal with me, how do you plan to take on Taleb?

Peace out.

bellatrix posted on February 05, 2011 at 19:08 PM


All right, cool.
I do believe that I played those 10/20s bad at the end of my downswing - tilted, not optimally. I don't think I was "robbed".

I guess we'll just have to disagree on the CLT applying to poker. I believe it does, you believe it doesn't. In the end it doesn't really matter that much. I have never experienced a fat tail. All of my downswings and upswings were within the normal realm of probability. The whole discussion started with tournaments, something where the fat tails have no discernable influence in the negative direction, since you are expected to lose.

I did feel like you were attacking me, because you write these long novels on my blog. While I really appreciate the comments and I am very thankful for you reading my blog, you should know that everybody is self-conscious about their results. Calling mine mediocre is not gonna win you any cookie points with me :).

I did choose to deal with coinflips - one of the most simplest probability distributions, because you were talking about getting AA in x number of hands and "not getting your fair share" of something. That has nothing to do with the players at the table, the psychology, etc. I am very open to the idea that WINRATE of a certain player in an ever changing field of opponents even at one certain limit does not follow a set distribution. However, I have not seen any evidence to the contrary. If you could please show me a graph or some proof of that it doesn't, I would be very thankful. I compiled a dataset of about 1 million years a couple of years ago of >1BB/100 winners at 6max LHE (low limits), I did not see any evidence at the margins of something funky going on. The Gaussian was self evident taking random set of hands at a time (so random players to account for the fact that I had different players in my dataset). However, you well know that 1 million hands is not enough, surely something might happen within the errors. I would be super thankful, if you could show me evidence (besides anecdotal) where you think there is an underlying function on top of the Gaussian.

You do have a valid point that there is survivor bias in poker. But that doesn't mean that everybody that is playing poker is just some random donk that got lucky. ;-) There is enough evidence to show that skill becomes self evident in poker over just tens of thousands of hands (see for example the NC trial case, you can lose on purpose in poker).

Anyway, glad it wasn't a personal attack. Thanks again for reading the blog!

copoka posted on February 07, 2011 at 04:32 AM


Well, agree to disagree is very civilized approach and I’m all for it.

Just a couple of points still.

“you were talking about getting AA in x number of hands and "not getting your fair share" of something.”

That “something” is mighty important. That “something” is an action I did not get in situations where I have way the best of it.

Deck actually did its job according to CLT. No problems there.
But if you want to tell me that action in poker is being distributed according to Gaussian too, you need to tell me what would be the chance of getting action 22 times out of 100 when probability of getting action is at minimum 80%. Ahhh, what the hell! Make it 50%!!!!

“I would be super thankful, if you could show me evidence (besides anecdotal) where you think there is an underlying function on top of the Gaussian.”

Thankfully, you don’t need to go any farther than 2+2 to have plenty of what you ask for.
It’s filled by examples (accompanied with graphs) of 2.85BB/100 over 250k hands winners (or something very similar) experiencing horrible 500 BB downswings in the same games where they had such an impressive results for so long.

Simple application of CLT tells us that fact of having 500BB downswing makes practically impossible for 2.85BB/100 WR to be a correct figure.

When I point out apparent impossibility to reconcile both numbers, I usually hear:
“Well, you just cant use Gaussian alone. You need to use Bayesian theory to find an explanation.”
Ok, so why you guys don’t need BT when you like results and just can’t do without it when you don’t?
Reasonable question, is not it? Never got an answer. LOL

But that’s OK. Please, apply your BT as they do it every in science and engendering and tell me, what is the chance for those two numbers to work together? If It’s not 1 in 100000 (CLT only result) – what is it?
1 in 50000? 1 in 5?

No can do – the answer is.
No way to get data needed, to many assumption to be made etc.
So it’s pretty much useless. That’s the word. Right?

With you Bella, I don’t have this problem. You are not even getting BT involved.
You simply ask to provide you with evidence that bell curve alone is not enough.

That’s easy. Just go through 2+2 and you’ll find boatload of downswing vs established WR examples.
Amazing outliers are all over the place. I’m sure you’ll have a lot of fun.

Oh, yeh. One more thing.
Poker is the game of skill. No arguments there.
It is just some skilled players are luckier than others.

All the best.

bellatrix posted on February 07, 2011 at 08:14 AM


The key here is not pitting the downswing against the long period at 2.85BB/100. The key is grabbing random 100 hand samples from that 250k distribution and seeing if the BB won over those random hands follows a Gaussian.

There could be two factors at these huge downswings.

a) you were never that good to begin with, still a winner, but nowhere near 2.85BB/100. Even after 250k hands in a 20BB/100 SD, you are within 2 sigmas of having only a <2BB/100 winrate.

b) you are playing worse during the downswing. Now that should still count towards your lifetime winrate, but now it will average out to be much worse than it actually is under your A-game. However, even the underlying distribution during that downswing is normal.

If you send me over the database of the 250k hands, I will gladly go over it and try to find the deviation from the Gaussian in random samples from the 250k hands. Not in one drastic event, in the sample as a whole.

"It is just some skilled players are luckier than others."

Amen to that! :)

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