Showdown statistics - strange problem

Yes, but 7 isn't just one particular result (or category), but three different results instead: 1+6, 2+5, 3+4. I'm not sure if you can use the total sum category here.

When you physically visualize heads up poker showdown statistics - you can compare this to flipping a coin. And it's a fair comparison, it gives exactly the same type of string of zeros and ones, assuming there will be no folds.

But with more than 2 players it's difficult to use a real life comparison for visualizing, even though it's the same game after all.

This is a super esoteric question.

So you are recording the winner of each hand (e.g. 6,4,1,2,2,5...), but the term "non-repeatable" indicates you discard the result if the same configuration+winner repeats? Why?

And you want to know what math governs the expected ordering of that list?

Only the configuration of active players in showdown isn't repeatable. Winner repetition doesn't matter. By this, I'm simulating a certain environment which theoretically could exist. I'm just wondering how is it possible that when there are no folds - heads up showdowns are 100% random, but if more players added, it suddenly all gets twisted.

If it's not explainable using math or poker terminology, then maybe at least a real life comparison would be possible here - like coin flips for heads up.

Ok so you're determining winners randomly. But then how are you choosing the configuration of active players at showdown? Is that also random?

The players are playing, so it's the cards and them who creates the whole scenario. It's like a description of a rare situation at a table which has already happened. But I suppose both factors you mentioned are determined randomly (for the purpose of the experiment).

According to your own rules:

• In HU there's only one possible showdown configuration, seat 1 and seat 2. So you'll get one random entry and the string terminates.
  
• In 6-max if you don't allow folding there's still only one possible showdown configuration - all players. So you'll get one random entry and the string terminates.
  

Pretty boring. Ok so allow folds.

If you choose showdown configs randomly (i.e. just pulling a ball from a hat of all possible configurations) then it's slightly more interesting. There are 57 possible configurations of at least two players in a 6-max game. But it's still just random noise. There's no signal - just a random string of integers between 1-6.

Still boring.

Ok so measure real poker data. Now it gets... well not quite interesting, but at least not completely trivial. You start getting a noisy signal in the string. You could either encode players (e.g. Bob) or positions (e.g. whoever has the BTN is seat 1).

The string will then encode:

• Earlier entries more likely to come from common formations (e.g. BTN vs BB).
  
• Later entries at the end of the string represent extremely rare multiway spots (e.g. UTG vs HJ vs CO vs SB) - and these are far more likely to be random. This is because there are only 57 possible configs so you need to find weird super rare family pots to reach 50 unique configurations of showdown.
  
• The likelyhood of any seat is strongly correlated to their VPIP, e.g. you'll likely get BB most often if measuring seats because they play the most hands and are more likely to see showdown.
  
• You weakly encode the win% of each seat. How likely they are to see showdown is gonna have a far bigger impact on their likelihood to appear anywhere in your string tho.
  

What about real HU?

Still nothing, string ends after one entry according to your arbitrary rules.

By "there are no folds" I meant only the folds that would cancel the showdown.

Thanks for a detailed explanation, I can see you've attached the GTO to your answer. I need a while to get a closer analysis of it.

And I asked this question for the purpose of getting more overall perspective on poker. Like a wider look, less strategic, but which sees only what's the most important thing. Sometimes I think it should be called "multi-way randomness" or "public randomness with multiple personal access"

Yes that is true in this particular example. But the main point is that random does not imply that all results are of equal frequency. This was just a simple example. You could make a single die, for instance, that had a 1 on three sides, a 2 on two sides and a 3 on the remaining side. If you rolled that die 10000 times you would generate a random sequence of 1s, 2s and 3s, but you would have more three times as many ones as threes. ItÂ’s still random, just not a uniform distribution.

I still maintain that youÂ’d get similar in your example. Players 1-6 would each have a certain probability of winning - p1, p2, etc. These would not necessarily be equal. What you would get would simply be a random sequence with ones having frequency p1, twos with frequency p2, etc.

I still can't understand though how does it happen in heads up poker, that a coin being literally constantly flipped is sliced into 4 phases making the odds percentages miraculously irrelevant from the showdown result perspective. I think this is the most fascinating point in NLHE.

I just don’t know how you come up with these sorts of ideas

This particular idea comes from my will to understand what is impossible to explain and in general, my ideas come from several years of experience in designing poker variants and just gaining knowledge.

Good news is that I gave up with new poker variants. My haters can celebrate now the end of my trolling phase.

I shifted gear to a free pro - advice mode 😃

I remember that a few years ago I managed to guess the color of about 15 cards in a row (RED or BLACK) after shuffling the deck.

I must have had some sort of clue on the rhythm of that sequence. The probability is too low to be this lucky.

If it was possible to transfer this to a heads up game and mix it with some skills, then the result would be highly profitable.

Exactly how improbable must an event be to conclude that you couldnÂ’t have just been lucky? The odds of winning a Powerball jackpot are 1 in 292 million. The odds of correctly predicting 15 straight card colors are no greater than 1 in 2^15 or 1 in 32768. ItÂ’s actually less improbable than that since you can predict that red will come out with probability greater than 1/2 if youÂ’ve already seen more black than red cards dealt. When someone hits a Powerball jackpot, does that imply that they knew something about the pattern of numbers drawn in that lottery? ItÂ’s much less probable than your example, so it couldnÂ’t just happen by luck, right?

Ok, but notice how many people tried to hit that Powerball jackpot. I don't know precisely, but probably there were millions of them. And guessing red / black with real cards isn't very popular, so this should be also taken into the probability measure.

What do you mean by the bolded.

When a particular event takes place, from general perspective it does matter how many source points (with their own perspectives) there were that were trying to make it happen. Each source point multiplies the number of attempts made. I think this should be taken into account when measuring how rare this event is.

Cool breakdown. Feels less like a strict formula and more like weighted probabilities based on VPIP/aggression.

So, the more attempts the more likely something is to happen. This is not groundbreaking. Again, how does this change the probability?

It changes the probability from general perspective (more likely to happen), but it doesn't change the probability from a single source point (percentage that you will achieve this).

You can attach perspective type (general / subjective) to probability.

So no direct answer to my question then. Exactly how improbable must an event be to say it could not have occurred just by chance? The very fact that a particular set of numbers was drawn counters your argument. That event had a probability of 1 in 292 million of occurring, and there was only one set of numbers drawn. Does that mean Powerball is fixed since an event with probability of 1 in 32000 indicates non-randomness?

1) It's hard to tell the precise percentage value, the same way as it is to tell the exact bet size for a certain situation where betting is the optimal move. I can only suspect that 1 / 32000 indicates non - randomness, but only for a well known and natural category of event, like the guessing red / black card is. Strange, fantasy - like and weird categories don't count.

2) If Powerball had 1 in 292 million chance that anybody will win, then yes it would mean that it's fixed. But this number refers only to a single person perspective. The chance that somebody will win is much bigger, but still impossible to measure until the number of lottery tickets bought is counted.

What I mean is that guessing 15 red / black cards is from my own perspective nothing strange, but as an event happening in general for all people, considering the very small popularity of this activity, it's very unusual in fact.

Forget how many people played Powerball. 9,12,22,41,65 with 25 as the power ball. That was the result of the most recent Powerball drawing. That particular result had a 1 in 292 million chance of occurring by random chance. ThatÂ’s much less probable than your lucky streak of card prediction. If your streak must be non-random then why would you not suspect that the wildly improbable result of the last drawing is similarly not random?

Well, itÂ’s likely because there had to be SOME combination of numbers drawn; the one actually drawn seems not particularly noteworthy. Suppose the actual drawing had given 1,2,3,4,5 with a PB of 6. Would you conclude that itÂ’s not random? Maybe, but thatÂ’s because we perceive a pattern that makes it noteworthy. There is only a tiny fraction of all possible results that we would perceive as noteworthy. Thus we think that it MUST be nonrandom when we see one. That isnÂ’t true, however. 1-2-3-4-5 PB6 is no less likely than the actual result.

In like fashion, suppose you repeat your experiment and predict the color of 15 random cards. For simplicity assume you shuffle after each (so we donÂ’t have to account for cards already dealt and can assume a probability of 1/2 of making the correct prediction). Record your results as a string of Rs and Ws (Right or Wrong). You happened to get RRRRRRRRRRRRRRR when you did this. Had you gotten something like WRRRWWWRWRRWRRW, you probably wouldnÂ’t have posted anything. Yet this particular string of Rs and Ws is no more or less probable than your actual result. It just is less noteworthy.

If you really do have the ability to see a pattern in how the color of cards in a random deck goes, then repeat the experiment. If your hypothesis is true, you should be able to correctly predict 15x in a row with probability greater than 1 in 32000. More testable, your probability to correctly predict should be greater than 1/2, so if you did this 1000 times, you would make 15000 predictions. If you got Sat 9000+ right, then maybe your hypothesis has some merit.