## Anti-Gambler's Fallacy

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Vytron
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### Re: Anti-Gambler's Fallacy

douglasm wrote:...That's missing the point. No one in this thread is saying "after X losses, the next must be a win". What we're saying is "after X losses, we're pretty sure the game is rigged".

You are pretty sure that the game is rigged BECAUSE you'd have expected at least one win by now, even though, no matter how many flips you do, you're NEVER guaranteed to get a win. The Gambler's fallacy is that, thinking that if you go long enough a win MUST HAVE happened, just because it was very likely to begin with. But every time you lose the chance of losing remains the same, no matter how many flips you do, the next flip will be as likely to lose, there's no n big enough that will ensure at least one win. You could keep getting loses forever because the next time you're flipping the coin it'll be as likely for you to get a loss as it was on your first flip.

They say if you put enough monkeys hitting random keys on a keyboard long enough they'll start writing the bible, the works of Shakespeare, books that will be written in 20 and 100 years in the future, a very detailed description of your life including events yet to come, and a description of how to simulate an universe that includes themselves. What they don't tell you is that, it's also possible that all those monkeys, by pure chance, just keep hitting the space bar forever and you get nothing written. This is also a possibility, because no key that was already hit will make less likely that the space bar will be hit or that any other key will eventually be hit.

That's how pure chance and randomness works, you can't predict the future, and the least likely outcome can still happen because it was still possible.

Cauchy wrote:Yeah, those both sound more plausible. I think you underestimate the unlikeliness of this event.

The actual unlikelihood of the event doesn't matter. I think the discussion was very hurt by having something so extraordinary happening, and I'm sure all the thread would have looked different if we spoke about 8 loses in a row instead of 8000. Let's stop discussing 8000 loses in a row and start discussing a much lower number, one that could make you doubt your prior knowledge, but doesn't make you go "this couldn't be happening."

--------

And now, I'll provide some variation:

In this one, you get amnesia after every flip. That is, you get a result and are told if you won or you lost, and then decide to continue or not. All you know is how much money left do you have, but you don't know if you had more, or less, because of the amnesia.

So, you started with 1000000, and are down to 200000, but you don't know it. For all you know, you could have started with 500 and made a fortune by now.

What I will claim is, if you'd continue playing in this variation, then you should continue playing in the original one.

The thing with the gambler's fallacy is adjusting your beliefs and strategy based on recent events, even though those events have no effect whatsoever on future ones. If you know the coin is 70-30, then you can make a flip and forget it. This time you're not shocked after 8000 loses in a row and suddenly start believing in Santa.
Last edited by Vytron on Sat Sep 12, 2015 7:16 am UTC, edited 1 time in total.

quintopia
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### Re: Anti-Gambler's Fallacy

It's not the number of losses in a row that I can't comprehend. That the coin comes up tails that many times is easy to imagine. No harder than omniscient gods that always tell the truth anyway. What is literally incomprehensible is the idea of knowing something with absolute certainty. It is literally asking me to assume the existence of a real infinity in the universe, and how am I supposed to be able to imagine that?

Vytron
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### Re: Anti-Gambler's Fallacy

Can you imagine a perfect coin that always lands on either head or tails, half the time?

douglasm
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### Re: Anti-Gambler's Fallacy

Vytron wrote:The Gambler's fallacy is that, thinking that if you go long enough a win MUST HAVE happened, just because it was very likely to begin with.

That's not the Gambler's Fallacy. The Gambler's Fallacy is "I lost 10 times, I must be due for a win". It's a prediction that a future event will be opposite past events in order to balance them out. The Gambler's Fallacy's "logic" would say you most definitely should continue playing because after 8000 losses you're practically guaranteed the next flip will be a win. This is exactly the opposite, in multiple ways, of the reasoning going on in this thread.

Your complaints are about predicting the result of the next coin flip. Our complaints are about questioning the validity of the game. These are related, but very different, issues.

Vytron wrote:The actual unlikelihood of the event doesn't matter. I think the discussion was very hurt by having something so extraordinary happening, and I'm sure all the thread would have looked different if we spoke about 8 loses in a row instead of 8000. Let's stop discussing 8000 loses in a row and start discussing a much lower number, one that could make you doubt your prior knowledge, but doesn't make you go "this couldn't be happening."

--------

And now, I'll provide some variation:

In this one, you get amnesia after every flip. That is, you get a result and are told if you won or you lost, and then decide to continue or not. All you know is how much money left do you have, but you don't know if you had more, or less, because of the amnesia.

So, you started with 1000000, and are down to 200000, but you don't know it. For all you know, you could have started with 500 and made a fortune by now.

What I will claim is, if you'd continue playing in this variation, then you should continue playing in the original one.

The thing with the gambler's fallacy is adjusting your beliefs and strategy based on recent events, even though those events have no effect whatsoever on future ones. If you know the coin is 70-30, then you can make a flip and forget it. This time you're not shocked after 8000 loses in a row and suddenly start believing in Santa.

This negates the entire point of the OP, and is again confusing the issue. The fallacy you are trying to refute is about independence of sequential random events. The actual problem is about certainty about trends.

Vytron wrote:Can you imagine a perfect coin that always lands on either head or tails, half the time?

Exactly half the time? Only in a physics teachers' world where cows are spherical, pulleys are massless and frictionless, and planes are infinite.

PeteP
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### Re: Anti-Gambler's Fallacy

Vytron maybe what you miss is that when you determine probability via repeated trials you aren't actually determining the exact true probability distribution you are showing that a certain probability distribution is likely. When you repeatedly throw a coin and after 10k trials it's about 50% heads, 50% tails then it's likely that it's a fair coin but that is a probability. It could just as well be a coin with a 90% chance for head that just happened to fall like that. And with more trials you only increase your confidence but it could always be just coincidence.

When looking at the 8000 losses in a row everyone knows that that could theoretically happen with a 70%/30% coin, but the more likely explanation is that either it isn't a 70/30 coin or there are extra factors influencing the trial. We know that it being such a coin and the sequence just randomly happening is possible, we just consider other explanations vastly more likely. That is why it doesn't matter much that it could happen, we are dealing with probabilities and can only look for the most likely explanation which with 8000 losses in a row usually isn't that it happened by chance. That it can be wrong and is wrong in this hypothetical is an unavoidable consequence of dealing with probabilities.

Cauchy
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### Re: Anti-Gambler's Fallacy

I feel like you're looking at a different problem from the rest of us, Vytron. While I'm solving the problem as written, you've made a mental substitution, switching out "You know for certain that the coin is biased in your favor: 70 to 30." for "the coin, when flipped, will always come up heads with a probability of 70%.". These are not the same statement, but the latter one is an obvious simplifying assumption to make. However, this simplification misses the crux of our argument, which is why you're always talking past us and bringing up examples like "what if it was only 8 tails in a row" or "what if you had amnesia" that miss the point we're making.
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elasto
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### Re: Anti-Gambler's Fallacy

Cauchy wrote:I feel like you're looking at a different problem from the rest of us, Vytron. While I'm solving the problem as written, you've made a mental substitution, switching out "You know for certain that the coin is biased in your favor: 70 to 30." for "the coin, when flipped, will always come up heads with a probability of 70%.". These are not the same statement, but the latter one is an obvious simplifying assumption to make. However, this simplification misses the crux of our argument, which is why you're always talking past us and bringing up examples like "what if it was only 8 tails in a row" or "what if you had amnesia" that miss the point we're making.

Yeah. The problem with your interpretation, Vytron, is that it is mixing a Platonic, mathematical object with the real world. It'd be like if the problem said 'imagine a line segment AB, 1 meter long, and imagine you balance it on its end and stand on it...'

Probabilities are a mathematical thing: They represent knowledge about the future. The question in the OP is asking us to claim perfect knowledge of the future, then asks us how we'd react in the real world. Well, in the real world we'd doubt our own sanity and/or the reality of the world long before losing 8000 times in a row. Or at least we should.

The trolley dilemma and similar scenarios also do the same thing, but at least they have an ethical dimension to them and so are worth exploring for that reason. This puzzle doesn't even have that, so it devolves into a rather benign conclusion:
- If we are taken to be perfect mathematical beings in a Platonic universe, since our perfect knowledge of the future is correct, we keep on playing
- But if this is anything remotely close to the real world - say, for example, we have examined the coin at an atomic level and know for certain from physical laws that it will fall 70/30, then if we lose 8000 times the most likely conclusion is that our knowledge of physics is wrong: Maybe physical laws evolve over time (perhaps starting right now), maybe we're in a simulation with a broken RNG, or whatever. Either way we definitely shouldn't continue betting on the 70 side!

Vytron
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### Re: Anti-Gambler's Fallacy

I don't see the "real world" mentioned in Cradarc's post. What I see is, you're presented to some world, in where you can gain certainty about some things, and in this world, you got 8000 loses in a row by pure chance. The puzzle isn't dragged out of the thread to reach your location and ask you what you would do, you're dragged into the situation of the puzzle, and it seems some people just don't find such a world plausible, so they wouldn't move into it, even though one moves to the worlds presented in other puzzles in a regular basis.

In the real world, a coin that is perfectly 70-30 may not exist, or it may change slightly with each flip, or be damaged and change its bias entirely. After witnessing 8000 loses in a row people would assume something like that happened, and if after them the coin went back to normal, you'd assume whatever was making the coin always lose is now gone. Call it an unknowable mystery and move on. I find that dull.

In the puzzle's world, no outcome from the coin should affect your certainty, because if it did then you weren't certain to begin with. People reject "You know for certain that the coin is biased in your favor: 70 to 30" from the puzzle, and go "no, I don't know for certain".

SirGabriel
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### Re: Anti-Gambler's Fallacy

Vytron wrote:They say if you put enough monkeys hitting random keys on a keyboard long enough they'll start writing the bible, the works of Shakespeare, books that will be written in 20 and 100 years in the future, a very detailed description of your life including events yet to come, and a description of how to simulate an universe that includes themselves. What they don't tell you is that, it's also possible that all those monkeys, by pure chance, just keep hitting the space bar forever and you get nothing written. This is also a possibility, because no key that was already hit will make less likely that the space bar will be hit or that any other key will eventually be hit.

They've actually done experiments on this. The monkeys pick a key they like (usually S) and keep hitting it until in breaks, and then they poop on the keyboard.

Not sure that has any relevance to the puzzle, but it's an interesting fact.

quintopia
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### Re: Anti-Gambler's Fallacy

Vytron wrote:In the puzzle's world, no outcome from the coin should affect your certainty, because if it did then you weren't certain to begin with. People reject "You know for certain that the coin is biased in your favor: 70 to 30" from the puzzle, and go "no, I don't know for certain".

Exactly. Furthermore, there is little value to accepting such a hypothetical (by, for instance, imagining one has some disease that makes it possible to live as though the odds ratio for some event is infinite, amnesia or mind control of what have you), since there is nothing to learn from it. Other logic puzzles let us explore facets of understanding we can actually apply in the real world for the small cost of accepting something impossible. However, no aspect of this problem has a place in the real world. It quite literally asks us to mentally become some complete other person that we could never possibly be, and then asks what we would do as that person. What value is there in that?

Vytron
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### Re: Anti-Gambler's Fallacy

Are you saying the puzzle as posed is bad? I think it's very interesting, specially to know that if people recorded noise, stopped at a random frame, and the noise looked just like themselves, they couldn't accept it could happen by chance and would go to find more likely explanations.

What I found weird is that people claim I'm imposing my own interpretation over the puzzle, and my interpretation has problems, and it's wrong. Yet I'm on the side that can claim there's stuff you can now with absolute certainty. Descartes for example "I think, therefore, I exist" ("I am", whatever). I'm absolutely certain that I exist and people doubting their own existence are fooling themselves.

Linking it again with blue eyes, in that puzzle people leave when they're absolutely certain about the color of their own eyes. And they do. They're absolutely certain that the Guru is telling the truth and that if there was a single person with blue eyes, they're absolutely certain that it would leave the island the first day.

Nobody questions the answer to the puzzle. Nobody rejects "everyone is a perfect logician". Absolute certainty can be comprehended there, but not here.

Sizik
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### Re: Anti-Gambler's Fallacy

Which brings us back to this.

SPACKlick wrote:Vytron I think the issue is once you make this puzzle pure abstraction there's no point of interest in it. "If you are playing a game where your odds of winning are 7:3 and you can keep playing should you" is dull, the answer is obviously yes.
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King Author wrote:If space (rather, distance) is an illusion, it'd be possible for one meta-me to experience both body's sensory inputs.
Yes. And if wishes were horses, wishing wells would fill up very quickly with drowned horses.

quintopia
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### Re: Anti-Gambler's Fallacy

Vytron wrote:They're absolutely certain that the Guru is telling the truth and that if there was a single person with blue eyes, they're absolutely certain that it would leave the island the first day.

Absolute certainty is not a prerequisite for blue eyes. In fact, at no point does it refer to absolute certainty. It just says they leave when they "figure it out" . Nor do they need absolute trust in the Guru. It would be enough to state that they must know their eye color beyond a reasonable doubt and that they have no reason to distrust Guru, either her perceptions or her descriptions of them. This problem depends critically on the notion of absolute certainty. I guess I would say that there is one sense in which it is not flawed: it illustrates the extent to which the concept and state of absolute certainty contrasts with the real world.

Vytron wrote:I think it's very interesting, specially to know that if people recorded noise, stopped at a random frame, and the noise looked just like themselves, they couldn't accept it could happen by chance and would go to find more likely explanations.

Yeah? There are other threads on these fora that illustrate this point just as clearly. I guess I get some value out of knowing you would accept such a thing happening by chance without seeking further explanation. It's very odd.

elasto
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### Re: Anti-Gambler's Fallacy

Vytron wrote:I don't see the "real world" mentioned in Cradarc's post.

I was specifically asked what I would do were I in this situation. I exist in the real world, where there is a non-zero probability of insanity, unknown physical laws, being in a simulation etc.

Contrast this with the blue eyes puzzle, where I am not asked what I would do were I one of the inhabitants, but what an inhabitant who is a known perfect logician, surrounded by other known perfect logicians would do...

As stated many times now, if instead of asking what I would do were this to happen to me, it asked what a known perfect logician in a known perfect universe should do, the answer is boring, and has been stated repeatedly:

SPACKlick wrote:Vytron I think the issue is once you make this puzzle pure abstraction there's no point of interest in it. "If you are playing a game where your odds of winning are 7:3 and you can keep playing should you" is dull, the answer is obviously yes.

----

Vytron wrote:Are you saying the puzzle as posed is bad? I think it's very interesting, specially to know that if people recorded noise, stopped at a random frame, and the noise looked just like themselves, they couldn't accept it could happen by chance and would go to find more likely explanations.

Are you saying you wouldn't look into other explanations if that happened IRL - that someone had manipulated the source or recording of the noise? You'd shrug your shoulders and say 'it happens'?

As quintopia says, that's pretty odd.

Vytron
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### Re: Anti-Gambler's Fallacy

quintopia wrote:Absolute certainty is not a prerequisite for blue eyes.

I disagree. In blue eyes, if the Guru says they see someone with blue eyes, and you don't see anybody else with blue eyes, you should be absolutely certain that your eyes are blue and that it's the reason the Guru said that, to know your eyes are blue.

Otherwise, you just scratch your head, and think maybe my eyes are blue and the Guru saw me, or maybe they're green, but the Guru has optical problems and perceives green as blue, or the Guru is hallucinating that they see someone with blue eyes that isn't me, or he speaks a rare tongue and said "hello everyone!" but it sounds like "I see someone with blue eyes". Or I could have hearing problems and misheard what the guru said, or I could be hallucinating myself, or maybe it's me who perceives blue as brown and all these guys have blue eyes and the Guru is talking about them...

Or the Guru is a liar.

In any case, the Guru saying "I see someone with blue eyes", in the real world, does NOT make you conclude your eyes are blue just because you don't see another blue eyed people. In fact, if the Guru said "I see someone with eyes that shine in the darkness" or something you'd find it unlikely yourself, you'd not believe it and look for a different explanation.

But Blue Eyes happens in the puzzle's world, and people have no problem moving there.

quintopia wrote:Yeah? There are other threads on these fora that illustrate this point just as clearly.

I had no idea about it. Things should be able to happen by pure chance. In my experience, humans tend to have unrealistic expectations about randomness and think things are much less unlikely to happen than they are. They're often surprised, but when they are, they refuse to believe in pure chance being at work here, they'll assume their model of randomness was wrong, even if it wasn't (the multiuniverse theory that always has at least one universe where the most unlikely thing happened by pure chance).

It's not that I don't get surprised, it actually surprises me a LOT that people refuse to believe something very unlikely happened by pure chance. We still don't have an explanation of why consciousness exists at all in our universe and we have failed to simulate it (or don't have a way to check if we have succeeded at it), so perhaps consciousness appearing in the universe was more unlikely than the OP's premise. But since it already happened, then we assume it had a chance of happening close to 1...

elasto wrote:Are you saying you wouldn't look into other explanations if that happened IRL - that someone had manipulated the source or recording of the noise? You'd shrug your shoulders and say 'it happens'?

Well, I have anecdotal evidence: A few years ago, overnight, a huge tree appeared in my garden. My mother saw it, my sister saw it, my neighbors saw it... but nobody remembers seeing it before, or seeing it grow up, and we don't know how it got there. And there wasn't and ground removed, or noises of people having gotten there on the night to plant it, it seemed just as if the tree had been there for years. As if, the tree didn't appear overnight, but instead, everyone that had seen the tree forgot about its existence, overnight, and were surprised to see it the next morning.

Once it was noticed the neighbors were very angry about it and ended poisoning the tree and killing it so that it wouldn't damage their property, or whatever, so the tree isn't there anymore.

So, what was up with that? I had no idea, it just happened.

quintopia
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### Re: Anti-Gambler's Fallacy

Vytron wrote:I disagree. In blue eyes, if the Guru says they see someone with blue eyes, and you don't see anybody else with blue eyes, you should be absolutely certain that your eyes are blue and that it's the reason the Guru said that, to know your eyes are blue.

Otherwise, you just scratch your head, and think maybe my eyes are blue and the Guru saw me, or maybe they're green, but the Guru has optical problems and perceives green as blue, or the Guru is hallucinating that they see someone with blue eyes that isn't me, or he speaks a rare tongue and said "hello everyone!" but it sounds like "I see someone with blue eyes". Or I could have hearing problems and misheard what the guru said, or I could be hallucinating myself, or maybe it's me who perceives blue as brown and all these guys have blue eyes and the Guru is talking about them...

Or the Guru is a liar.

In any case, the Guru saying "I see someone with blue eyes", in the real world, does NOT make you conclude your eyes are blue just because you don't see another blue eyed people. In fact, if the Guru said "I see someone with eyes that shine in the darkness" or something you'd find it unlikely yourself, you'd not believe it and look for a different explanation.

Then you are wrong. It is sufficient that:
• Everyone is 99% confident the the Guru speaks truly and perceives correctly (as I mentioned in the last post).
• Everyone is 99% confident everyone else heard the same words from the Guru he did.
• Everyone is 99% confident that he and everyone else sees everyone else's eyes clearly and perceives them correctly.
• Everyone is 99.99% confident of their own and everyone else's sanity and perfect-logicianhood.

With such high confidence levels, everyone will behave exactly as the do in the accepted solution, with high confidence that they are behaving correctly and high confidence that the outcome will be the desired one. None of this interferes with the logic used by the perfect logicians to determine their behavior. There are many ways such confidence levels could have been established inductively prior to the described scenario taking place. Yes, such confidence would never be achieved in the real world, but we all agree that all of these puzzles do not occur in the real world. However, we can imagine a world in which it takes place, but a world in which perfect certainty is achievable is not even imaginable.

Vytron wrote:In my experience, humans tend to have unrealistic expectations about randomness and think things are much less unlikely to happen than they are.

Then we agree that 8000 tails in a row or an image of my face appearing in white noise on a TV screen are both far more likely to occur for reasons other than random chance. Now that we see we are both sane, we can get to convincing those that still play the lottery that winning is far more unlikely than they think it is.

Or did you mean to write "think things are much less likely to happen"? The rest of the paragraph would imply you did...

Vytron wrote:Well, I have anecdotal evidence: A few years ago, overnight, a huge tree appeared in my garden. My mother saw it, my sister saw it, my neighbors saw it... but nobody remembers seeing it before, or seeing it grow up, and we don't know how it got there. And there wasn't and ground removed, or noises of people having gotten there on the night to plant it, it seemed just as if the tree had been there for years. As if, the tree didn't appear overnight, but instead, everyone that had seen the tree forgot about its existence, overnight, and were surprised to see it the next morning.

Once it was noticed the neighbors were very angry about it and ended poisoning the tree and killing it so that it wouldn't damage their property, or whatever, so the tree isn't there anymore.

So, what was up with that? I had no idea, it just happened.

This whole story is extremely unbelievable. You should make a mini-documentary and get it played on Syfy.

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### Re: Anti-Gambler's Fallacy

quintopia wrote:Then you are wrong. It is sufficient that:
• Everyone is 99% confident the the Guru speaks truly and perceives correctly (as I mentioned in the last post).
• Everyone is 99% confident everyone else heard the same words from the Guru he did.
• Everyone is 99% confident that he and everyone else sees everyone else's eyes clearly and perceives them correctly.
• Everyone is 99.99% confident of their own and everyone else's sanity and perfect-logicianhood.

I definitively wouldn't be confident enough for that. My eyes could be brown even though it's extremely likely they're blue due to the evidence presented. I wouldn't be surprised if I looked in a mirror and they turned out to be brown, because it was plausible that the chain of logic events failed at some point. After all, there's 100 people in the island, 1 could be insane, and it could be the guru.

I know that it's extremely likely my eyes are blue, but I wouldn't know if my eyes were blue.

quintopia wrote:Or did you mean to write "think things are much less likely to happen"? The rest of the paragraph would imply you did...

Yes, I meant to say "likely."

There's factual evidence about some woman that won the lottery much more times than it would have been expected to be possible, and she did. Things just happen.

quintopia wrote:This whole story is extremely unbelievable. You should make a mini-documentary and get it played on Syfy.

Yet, it happened.

The documentary would be dull without any evidence. And, anyway, if there was evidence it'd look like a video taken at date X where the tree wasn't there, and then at date X later where it was, but you wouldn't be able to tell if the earlier video was actually later (after the tree was removed), and it'd just have a bunch of people talking about how they hadn't noticed the big tree before (because apparently that's the most logical explanation... there was a huge tree there for years, and nobody saw it until that day...) - how do you even investigate what happened?

quintopia
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### Re: Anti-Gambler's Fallacy

Vytron wrote:I definitively wouldn't be confident enough for that.

I wouldn't be either.

Vytron wrote:Yes, I meant to say "likely."

In which case, I disagree. From what I've seen, people are more likely to overestimate small probabilities than underestimate large ones. People who play the lottery for instance.

Vytron wrote:The documentary would be dull without any evidence.

That's what dramatization is for. And lack of evidence is not any impediment to Syfy. This is the channel that airs "ancient aliens" you know.

lordatog
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### Re: Anti-Gambler's Fallacy

We can get around the certainty issue. Suppose the game is being run by a sufficiently advanced alien, wielding godlike power you can't hope to understand. At the start of the game he tells you that the odds are 70:30 in your favor. He then pulls you aside and whispers to you "We both know I can't make you totally certain of those odds. However, you had better ACT as though you were certain, because I'll be reading your mind and if you allow doubt to influence any of your choices in this game I'll kill you on the spot."

What would you do in that case? Keep playing, obviously. The odds are in your favor, after all.

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### Re: Anti-Gambler's Fallacy

Vytron wrote:Can you imagine a perfect coin that always lands on either head or tails, half the time?

As an abstraction, I can imagine a perfect coin that lands on heads or tails exactly 50/50.

However, I can't imagine an actor in that scenario who is holding that coin and knows that the coin is exactly 50/50, with absolute certainty.

Or, rather... I can, but then if I do, the question in the OP is trivial and uninteresting, because it boils down to simply "Due to an unexplained circumstance, there is literally nothing that could convince you that X is false. Will this particular situation convince you that X is false?" There's nothing to actually discuss.

I'm reminded of the infamous Aeroplane on a Treadmill puzzle from however many years back... the exact details are unimportant, but the relevant part is that it was a puzzle that, when poorly worded, could be read in one of two ways. The first way lead to an interesting puzzle, where your immediate intuition about the puzzle was incorrect, and deeper understanding of the situation revealed a different answer. Classic puzzle stuff. The second way devolved into technicalities and I saw described as boiling down to "Due to an unexplained circumstance, this plane cannot move. Can the plane move?"

One of those is an interesting puzzle. One is not. But most of the threads ended up being about the people reading the puzzle the two different ways yelling at each other cross-purposes rather than anything actually useful.

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Vytron
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### Re: Anti-Gambler's Fallacy

Because, the people reading the puzzle in the "interesting" way didn't really discuss with each other about their stopping points or why are theirs better than the others, because they accept that it's a personal choice. It turns out to not be very interesting after all, but only in the sense that people provide different answers.

What I don't like it's that the answers are clearly cut, either:

1. It's obvious you should keep playing. If you have certainty that the coin is 70-30, you have certainty that you will win in the long run, and can play without even looking at the outcomes you get.

Or

2. It's obvious you should NOT keep playing. There's a point at which it's more likely your assumptions are wrong than the coin is 70-30, and at this point (which happens much earlier than 8000 flips, so we don't really even get there, so the puzzle doesn't happen in the first place) you stop playing.

Why not a third option? Why not something in-between?

Like, use the coin to decide what to do.

Decide what to do over the next 10 flips:

Flip 1:

If it's a loss, jump to flip 2. If it's a win, decide to keep playing.

Flip 2:

If it's a loss, jump to flip 3. If it's a win, decide to keep playing.

...

Flip 6:

If it's a loss, jump to flip 6. If it's a win, Jump to flip 5.

Flip 7:

If it's a loss, jump to flip 7. If it's a win, Jump to flip 4.

Flip 8:

If it's a loss, jump to flip 8. If it's a win, Jump to flip 3.

Flip 9:

If it's a loss, decide to stop playing. If it's a win, Jump to flip 2.

Flip 10:

If it's a loss, jump to flip 10. If it's a win, Jump to flip 1.

So you have this program, which is very unlikely to run. And you run this program whenever you doubt your prior knowledge, instead of just deciding to quit.

At least, for the OP given, this program would make you continue of the coin normalized, and make you quit if the loses keep coming on a row (i.e. you win whatever the mod decides would happen.)

brenok
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### Re: Anti-Gambler's Fallacy

I can't make sense of this program. The way it's written, it would get stuck on flip 6, and there is no way to get to flip 10 (where it would get stuck again)

phlip
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### Re: Anti-Gambler's Fallacy

Vytron wrote:Because, the people reading the puzzle in the "interesting" way didn't really discuss with each other about their stopping points or why are theirs better than the others, because they accept that it's a personal choice.

It's not (just) a personal choice. It's a choice that depends wildly on information that isn't given in the OP... specifically, the source of the information that the coin is weighted 70/30. Is the source of that our own experiments? Well, we should be able to calculate a reasonable estimation at the probability our conclusion is correct from the data. Did someone else do the experiments? The same, but multiply by our confidence that the other person is an honest scientist. Have we just been Told by an Authority that it is true? Then we have to consider how much we trust the authority, taking into account how likely it is the authority is lying, or mistaken, and our estimation of how much confidence the authority has in the fact, even as they state it as an absolute.

And that's before we get to the human elements of how to weight the various factors, or the initial prior from before we did our experiments (or whatever) that those experiments are modifying. And various other things that are hard to quantify, like "how likely is it that all my senses are false and I'm actually a brain in a jar", which most people agree is a very small probability, but you can't put an actual hard number on.

Even given all of that information, very few people will be able to go through all of that and put down an actual number for how much confidence they have in the 70/30 statement, more sort of vague fuzzy ranges.

And that's just how you get to your starting point of what you think the probability is before the first of your 8000 flips. The question then of a "stopping point" is a further fuzzy factor, even if all of the above got completely resolved, and you had a hard nailed-down prior. A hard nailed-down prior would let you give specific answers to "after n flips, all of which lose, what is your new confidence that the coin is 70/30"... but to go from there to "after n flips, all of which lose, what is your new confidence that continuing to play is beneficial" brings in questions of utility curves and the like, which are all personal based on way too many parameters (some objective, some subjective) to list here. And even then, to move on to "how many all-losing flips should you take before you stop", adds even more fuzzy layers of risk-averseness. It's not a binary of on flip 37 you're still almost sure it's 70/30, but then on flip 38 you're now almost sure it's rigged, and you give up. It's a fuzzy function of a fuzzy function of a fuzzy function, that you are asking when it crosses a fuzzy threshold, and then being surprised that people can't give you a specific answer.

In practise, science tends to cut through the Gordian knot with things like "p<0.05"... an arbitrary declaration of a threshold on a proxy for confidence level. Because that's something that can be actually calculated, and serves essentially the role that we need. And, in practise, that's mostly good enough.

Vytron wrote:Why not a third option? Why not something in-between?

Like, use the coin to decide what to do.

This much, at least to an extent, is a reasonable scientific practise... To reduce the possibility of seeing spurious patterns by chance, it's a good thing to (a) collect data, (b) see patterns in data, (c) collect new data to see if the patterns continue. In common detailing of the scientific method, this is "make predictions; test predictions". The idea being that if you only look for patterns in existing data, then you will find things in the noise that are only there due to chance... and then if you check if the patterns really exist in the data you already have, then obviously it will fit. But if you collect new data after you spot the pattern, then you can more directly test that pattern... if it's not there, it should regress to the mean. So if you, say, flip a coin 10 times, the probability that there's some sort of pattern there just by chance is actually reasonably high. But if you spot a pattern, then flip the coin again 10 more times, looking for that specific pattern, and then the probability that the same pattern is there just by chance is much lower.

Of course, you can do all of this redundancy with your allegedly-70/30-but-actually-always-loses coin and still come to the conclusion that you should stop playing long before you get to 8000 flips.

Code: Select all

`enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};void ┻━┻︵​╰(ಠ_ಠ ⚠) {exit((int)⚠);}`
[he/him/his]

MOJr
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### Re: Anti-Gambler's Fallacy

The problem with that puzzle is that the probabilities are really small here, so people who understand those numbers say that they don't make sense.

For example do you know how many tosses on average do you need for 50/50 coins to produce 8000 heads in row in that sequence? Try guessing...

Spoiler:
Its around 10^2408 tosses. Give or take few hundred folds. Not trillions, trillion trillions...

For example, if you made a trillion tosses (1 000 000 000 000) the largest number of heads in a row in that sequence would be around 39...
(wolfram alpha: "chance of getting 39 heads in a row on 1000000000000 coin flips"

So that means that you need around 10^2408 number of tosses to counter the 8000 tails in a row to know the coin is OK...

My answer to the puzzle:
Spoiler:
In some hypothetical universe where I would know with certainty of at least 10^(2408+ (don't know the exact math for 70/30 coin)) flips before that the coin is OK, then I would continue to flip.

In a normal universe I would stop after 30 tails if I was PRETTY sure that the coin is 70/30 before starting to flip. (30 is that this would happen to 1 person in the world in around 100 000 trials)
Then I would say the game is rigged and called it a day.

Vytron
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### Re: Anti-Gambler's Fallacy

Spoiler:
MOJr wrote:My answer to the puzzle:
In a normal universe I would stop after 30 tails if I was PRETTY sure that the coin is 70/30 before starting to flip. (30 is that this would happen to 1 person in the world in around 100 000 trials)
Then I would say the game is rigged and called it a day.

What I argue is, once you're in the normal universe and it happens by chance, you're quitting the game illogically. Though I guess the point of all this is that quitting illogically would happen extremely rarely and so, it's not illogical to have a trigger that sets at some point and makes you quit for no good reason, as long as that trigger happens in extraordinary circumstance.

I guess the discussion is basically over, is my summary correct?:

Such an outcome wouldn't be expected to happen on a normal universe, because the chances that something else is going on are much higher than it happening by chance, so you should quit even if it happened by chance, because you should assume something else is going on.

Znirk
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### Re: Anti-Gambler's Fallacy

Vytron wrote:Such an outcome wouldn't be expected to happen on a normal universe, because the chances that something else is going on are much higher than it happening by chance, so you should quit even if it happened by chance, because you should assume something else is going on.

It's a bit of a leap to compress a discussion filled with "I would ..." statements into a "You should ..." statement; but other than that, I agree that's the general gist.

PeteP
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### Re: Anti-Gambler's Fallacy

Vytron wrote:
Spoiler:
MOJr wrote:My answer to the puzzle:
In a normal universe I would stop after 30 tails if I was PRETTY sure that the coin is 70/30 before starting to flip. (30 is that this would happen to 1 person in the world in around 100 000 trials)
Then I would say the game is rigged and called it a day.

What I argue is, once you're in the normal universe and it happens by chance, you're quitting the game illogically. Though I guess the point of all this is that quitting illogically would happen extremely rarely and so, it's not illogical to have a trigger that sets at some point and makes you quit for no good reason, as long as that trigger happens in extraordinary circumstance.

No in a normal Universe your conclusions are wrong, not irrational. Acting on the most likely possibility is rational it just isn't guaranteed to be correct.

Vytron
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### Re: Anti-Gambler's Fallacy

Well, if you get 10 heads in a row after flipping a coin I find it irrational to think a higher being was present in the room, or that you're dreaming, just because you find it unlikely. This doesn't change as the event get harder and harder, just the chance of being wrong diminishes (i.e. you're more unlikely to be in the universe where you witnessed it by chance).

I use "you should" instead of "I would", because otherwise, people doing it would be doing something that they would not recommend other people doing. Leading by example, if you're taking the optimal course of action, then other people should, too.

Unfortunately, I guess we're just speaking about words at this points.

PeteP
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### Re: Anti-Gambler's Fallacy

Vytron wrote:Well, if you get 10 heads in a row after flipping a coin I find it irrational to think a higher being was present in the room, or that you're dreaming, just because you find it unlikely. This doesn't change as the event get harder and harder, just the chance of being wrong diminishes (i.e. you're more unlikely to be in the universe where you witnessed it by chance).

I use "you should" instead of "I would", because otherwise, people doing it would be doing something that they would not recommend other people doing. Leading by example, if you're taking the optimal course of action, then other people should, too.

Unfortunately, I guess we're just speaking about words at this points.

That reveals your own irrationality. You are categorically giving alternative explanations a probability of zero so that they will never be the more likely explanations. If something isn't logically impossible I would for many things consider giving it a probability of 0 (declaring it as definitely impossible) as irrational.

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### Re: Anti-Gambler's Fallacy

PeteP wrote:That reveals your own irrationality. You are categorically giving alternative explanations a probability of zero so that they will never be the more likely explanations.

No, I accept that there's more likely explanations after witnessing the events, what I'm saying is that it's irrational to think about them on the universe where it happened by chance. That is, either it happened by chance with probability 1, or it didn't.

Suppose I have 5 cards of different colors, red, blue, black, white and yellow. I shuffle them and pick one. Now, you could say the card could be of any of the 5 colors, and give them a likelihood of 1/5 (20%), or whatever. But suppose before the experiment start, I have decided, that no matter what, I'll pick the red card. I.e. the card is red no matter what.

Well, I'll say that the card is red, and that it's irrational for you to consider that the card could be blue, black, white or yellow, because it's as if I showed you the card, and you saw it was red, but still thought it could be of the other 4 colors. That you're seeing it doesn't change anything, either the card is red, or it isn't.

In the same way, either the 8000 flips happened by chance, or they didn't, and if they happened by chance, it's irrational to try to find a different explanation (indeed, you could exhaust all your resources trying to find a different explanation, and find none. How can you, scientifically run tests that allows you to know if it happened by chance or not?)

My point is this string isn't special.

I mean, where do you draw the line? Does a fairy exist that decides how a coin falls when you flip it? And most of the time she decides for the coin to fall as you expect, but on a whim, she may decide suddenly the coin will keep showing heads for a long while?

I like doing the experiments for real, and so, I'll flip an actual coin and report the results...

Okay... whoa! I actually did this, and didn't expect to get so many heads in a row, and only 2 Tails... Maybe the coin is broken? Let's try it again...

Tails, Heads, Tails, Heads, Tails, Tails, Tails, Tails, Heads, Tails.

Okay, so 4 Tails in a row... I actually have no idea how likely is this. I didn't have anything only 3 times in a row, if I did then the fourth time in a row also happened.

But, anyway, what I was going to say is, that, what is special about this string?

11011111010101000010
(Actual string after 20 coin flips)

Because, on the long list of possible strings after 20 coin flips, having this extra string is as hard as having the 00000000000000000000 string, or the 11111111111111111111 happen. But this isn't surprising (unless the 11111 and 0000 were surprising) so why should the others be?

As you keep getting the same result, does the existence of the fairy becomes more likely? What I'd like to know is, at what points the existence of the fairy has 0.00000001%, 0.0000001%, 0.000001%, 0.00001%, 0.0001%, 0.001%, 0.01%, 0.1%, 1%... of happening, and at what point the existence of the fairy is more likely than the string happening by chance? Because, if I do 20 flips, and send them to one person, and keep doing it and sending a string to different persons, I'll get either the 00000000000000000000 string or the 11111111111111111111 string eventually (And I'll get either of them before I get the 11011111010101000010 one again!) and send it, and then this person, irrationally, will think I lied, or did something different, or that something went wrong. They won't believe I got it by chance, even though I did. I find the nonbelief of things happening by chance irrational, because the 11011111010101000010 string I just got was just as unlikely to be gotten, but your brain poses different standards, because 11011111010101000010 looks about right, while one that doesn't change didn't. But both were as likely.

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### Re: Anti-Gambler's Fallacy

Vytron wrote:In the same way, either the 8000 flips happened by chance, or they didn't, and if they happened by chance, it's irrational to try to find a different explanation

That would be true if and only if you actually know that they happened by chance. The whole point of this entire discussion is that, in a realistic world, you don't know that. You're trying to reason from the position of an omniscient god, when all you would really have is the perceptions of an ignorant mortal.

PeteP
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### Re: Anti-Gambler's Fallacy

Vytron you clearly have heard the stuff about "every specific random order" being unlikely but that one will occur. But from how you use the argument in this thread I am not sure you understand why that argument matters because otherwise you would understand when it isn't relevant. So before you use it the next time please describe for what situations that matters.

phlip
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### Re: Anti-Gambler's Fallacy

douglasm wrote:
Vytron wrote:In the same way, either the 8000 flips happened by chance, or they didn't, and if they happened by chance, it's irrational to try to find a different explanation

That would be true if and only if you actually know that they happened by chance. The whole point of this entire discussion is that, in a realistic world, you don't know that.

Exactly. A decision can't be "rational" or "irrational" depending on information you don't have when you make the decision. In the world where you're given a coin, truthfully told it's 70/30, but lose the flip 10 times in a row by chance, and the world where you're given a coin, falsely told it's 70/30, and lose the flip 10 times in a row because the coin's rigged, your information is the same in both cases. It doesn't make sense for a decision to be rational in one case but irrational in the other.

Code: Select all

`enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};void ┻━┻︵​╰(ಠ_ಠ ⚠) {exit((int)⚠);}`
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MOJr
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### Re: Anti-Gambler's Fallacy

Vytron wrote:But, anyway, what I was going to say is, that, what is special about this string?

11011111010101000010
(Actual string after 20 coin flips)

Because, on the long list of possible strings after 20 coin flips, having this extra string is as hard as having the 00000000000000000000 string, or the 11111111111111111111 happen. But this isn't surprising (unless the 11111 and 0000 were surprising) so why should the others be?

You are mixing apples and oranges here. If you have a "string" (in other words a series of events occurring in sequence where you care about the order) then yes strings 11011111010101000010 and 00000000000000000000 and 11111111111111111111 have the same probability of occurring if using 50/50 coin.

But if you care about the number of heads in that sequence, not the order of them then you get a really different results. Then the result 20 tails will happen only once in ~1000 000 throws, while results with some number of ones will happen much more often.

You can see the propabilities of number of heads for the original throw here here. See how the propability of ending up with less than 7 heads is really around zero.

So what is the difference, between the probability of having from 20 flips string with all tails, and having a string with lets say with 14 heads.
And the answer is:
string of all tails has a probability of 0.000000003486784401%.
string of all heads has a probability of 0.079792266297612001%
String with 14 heads has a probability of 19.16389827534425796%
Do you see the the MASSIVE difference in propabilities?

Vytron wrote:As you keep getting the same result, does the existence of the fairy becomes more likely? What I'd like to know is, at what points the existence of the fairy has 0.00000001%, 0.0000001%, 0.000001%, 0.00001%, 0.0001%, 0.001%, 0.01%, 0.1%, 1%... of happening, and at what point the existence of the fairy is more likely than the string happening by chance? Because, if I do 20 flips, and send them to one person, and keep doing it and sending a string to different persons, I'll get either the 00000000000000000000 string or the 11111111111111111111 string eventually (And I'll get either of them before I get the 11011111010101000010 one again!) and send it, and then this person, irrationally, will think I lied, or did something different, or that something went wrong. They won't believe I got it by chance, even though I did. I find the nonbelief of things happening by chance irrational, because the 11011111010101000010 string I just got was just as unlikely to be gotten, but your brain poses different standards, because 11011111010101000010 looks about right, while one that doesn't change didn't. But both were as likely.

This exact thinking is how i came to the answer 30. You could send a results of throws to every person on the planet, so some should be all zeroes (19 throws). So lets add some security margin against that so lets use 30 flips. Of course I would be wrong, but the question for me personally the dilemma is, either I am the unluckiest guy from all the people on 500 000 planets, or something fishy is going on.
And I think the fairy starts to appear around the 400 heads, where every proton in the universe could be flipping the coin since the beginning of time and still don't flip 400 tails in a row

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### Re: Anti-Gambler's Fallacy

This problem seems to be related to Stopped Brownian Motion. Even assuming I am convinced at all times of the validity of the coin's bias, if no sequence of results would convince me to quit, eventually I will reach the only stable state of the system. Therefore, as a logical player I must choose some condition to quit. Personally, I would quit whenever my current worth reached, say, 30% of my all-time maximum worth (generally, I quit when current/max < k, where k is the probability of a single loss). In this scenario, I would quit after 7000 consecutive losses.

ybot968
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### Re: Anti-Gambler's Fallacy

There can not ocur 7000 misses in a 70-30 biased coin

The standard deviation number rules fluctuation here from +/3 sd

ybot968
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### Re: Anti-Gambler's Fallacy

Cradarc wrote:You have \$1 million. You are playing a coin flipping game in which you have to bet \$100 every round, but you can quit any time.

If the flip is heads, you win back twice the money. If the flip is tails, you lose your money. You know for certain that the coin is biased in your favor: 70 to 30.
You play 8000 rounds in a row and lose all of them. Do you keep playing?

Suppose you lose every one of your first N rounds. For what value of N would you stop playing?

A coin can be altered to hit 70-30 or whatever

Nobody will face 8000 misses, even in 50-50

With this such 70-30 ratio 40% edge!!, in a binomial experiement, fluctuation is not as wild a other ranges

1 standard deviation every 1000 trials= square root of 1000x(7/10)x(3/10)=14,4913

mean of 1000= 700

655(-3sd rounded) 670 (-2sd) 685(-1sd) 700 mean 715(1sd) 729(2sd) 743,47(3sd)

range 655 to 744 every 1000 coin flip at 70-30

The worst scenario should be losing 45 units(655)

Supose -4 standard deviation(1 in 10k samples) you lose 60 units every 1000 played

So far from this case

PeteP
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### Re: Anti-Gambler's Fallacy

Not "can't" it's just unlikely. (And I don't think it's necessary to bring standard deviation into this, for this purpose it's just a more complicated way of saying that it's unlikely to deviate that much.)

ybot968
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### Re: Anti-Gambler's Fallacy

PeteP wrote:Not "can't" it's just unlikely. (And I don't think it's necessary to bring standard deviation into this, for this purpose it's just a more complicated way of saying that it's unlikely to deviate that much.)

The chance to ocur is astronomic!

You could wonder when you stop playing in case of many misses.

When you have got an actual 70-30 , any result out of +/-3 sd should start an alert, a hidden condition might be working.

Measure said he would quit after 7000 misses!, You would be convinced of something fishing much more earlier.

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### Re: Anti-Gambler's Fallacy

Vytron wrote:No, I accept that there's more likely explanations after witnessing the events, what I'm saying is that it's irrational to think about them on the universe where it happened by chance. That is, either it happened by chance with probability 1, or it didn't.

Suppose I have 5 cards of different colors, red, blue, black, white and yellow. I shuffle them and pick one. Now, you could say the card could be of any of the 5 colors, and give them a likelihood of 1/5 (20%), or whatever. But suppose before the experiment start, I have decided, that no matter what, I'll pick the red card. I.e. the card is red no matter what.

Well, I'll say that the card is red, and that it's irrational for you to consider that the card could be blue, black, white or yellow, because it's as if I showed you the card, and you saw it was red, but still thought it could be of the other 4 colors. That you're seeing it doesn't change anything, either the card is red, or it isn't.
It took me some time to parse this, but I think I understand what you're saying. There are 5 cards, and you're choosing one (knowingly) but I don't know which one you've chosen. You could choose any color, but you've decided to always choose red and I don't know this. You're asking how I would approach this situation? If I knew nothing else about the situation or about the rules you have chosen to follow, I would assume the 5 cards are equiprobable to begin. If I assumed anything else, it would have to be because of some outside knowledge that I (so far) don't have. At this point, my hypothesis is perfectly rational.

After playing a few rounds, perhaps it was by chance that each time you chose the red card, or perhaps you have a different rule than my initial assumption. After a series of losses (the absolute number isn't that important, let's say 10 since at that point I should have gotten at least one 90% of the time), perhaps I would reconsider my rule. At this point, a new hypothesis might be "you always choose red". Of course, an equally valid hypothesis might be "choose red 10 times, then choose at random" or "choose red 10 times, then choose blue 10 times". It would not be irrational to consider any of these. Even if you told me you were planning to pick the red card, it is still rational to consider the possibility that you are lying or otherwise wrong.

It looks like you're saying that it is irrational to consider the possibility that something didn't happen by chance, when you know you're in the universe in which it happened by chance - which is a perfectly circular argument. We are arguing that we might not actually know that, in which case it may be rational to consider other possibilities, especially in the face of something so astronomically unlikely.