## Anti-Gambler's Fallacy

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Cradarc
Posts: 455
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### Anti-Gambler's Fallacy

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?
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Cauchy
Posts: 602
Joined: Wed Mar 28, 2007 1:43 pm UTC

### Re: Anti-Gambler's Fallacy

Since I have entered a spooky realm where probabilities no longer mean anything, yeah, I'd quit. I'd quit by 100. I'd be weirded out by 10, incredulous by 20, and questioning the veracity of the rules by 30.
(∫|p|2)(∫|q|2) ≥ (∫|pq|)2
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PeteP
What the peck?
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### Re: Anti-Gambler's Fallacy

I would assume that I'm being cheated in some way, 0.3^500=3.636029e-262. If the knowledge about the 70% is absolute than it's happening in some other, maybe they have telekinesis or managed to swap the 70% coin with a coin which has tails on both sides.

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

PeteP wrote:I would assume that I'm being cheated in some way

Why?

Suppose that every time the coin is flipped, 70 universes are created where it landed in your favor, and 30 where it had an unfavorable results.

If after 8000 times all you got was unfavorable results, that just proves those universes where you got them exist. The probability is >0 so actually, if you flip the coin an infinitely high number of times you'll have for certain a stretch of 8000 unfavorable results in a row. Actually, infinity isn't necessary here, a Graham Number of flips would ensure a mindblowingly number of such strings of unfavorable results.

So the answer is you keep flipping, because for any of those strings, you catch up and recover all your money, and come up ahead in profits. Of course you can still be in the 9000 or 10000 string of lost games, because they exist, and that's the reason of the Gambler's Fallacy in the first place (you always come up ahead, but only if you have infinite money).

Similar answer:

Suppose that you land on a random string of Pi, and you lose every time a 7, a 8 or a 9 comes up, but win otherwise. You lose 8000 times in a row. Do you keep playing?

Cradarc
Posts: 455
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### Re: Anti-Gambler's Fallacy

I logically agree with Vytron (although realistically I don't gamble at all, so N = 0). You can easily reverse the situation and say you won 8000 times in a row. Would you still quit? The fact that a series of events though to be near impossible happened in the past is irrelevant to future events.

I think how a person responds to this question reveals how they view the meaning of probability.
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Vytron
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### Re: Anti-Gambler's Fallacy

Yes. I think the critical part of the puzzle is "certainty."

Suppose you entered the experiment with no prior info, and you started flipping coins to see if it was biased. How many times would you have to flip it to be certain about its bias?

It doesn't matter, because the puzzle tell us that we have a priori knowledge that the chance of getting the favorable result is 70%. For certain.

If you can keep flipping it and then change your opinion about it, then you didn't know "for certain", so people quitting the game aren't as certain as the puzzle requires.

Here's a fun variation:

Suppose that before the experiment starts, I have printed a million photos of my hands, in all possible configurations, from both closed, to showing 10 fingers, and everything in-between. So there's 10 different photos.

Then I grab one of the photos, and put it face down over the table.

So I tell you that you can make 7 guesses about how many fingers I'm showing on the photo, and if one of your guesses is correct, you win.

Or suppose you took the photos, as in, YOU are running the experiment, you shuffle the photos yourself, you put them face down, and you report if your guesses are right, I'm not involved in this other than rewarding you for every time you report a win (you could even cheat me!)

So, when you have the photo in front of you, and you are sure it will not change, and you make a guess, this is an independent event.

If after 8000 times of trying you always fail to guess the number of fingers on the photos that you pick, (say, you always guess 1-8 fingers are shown, but got the photos of my closed fists or 9 fingers shown or hands opened) then this doesn't change the facts you know: the photos include all configurations, the configurations you got were as likely as all others (this is key, a string 999999999999999999 is no more special than 167554428992169877), you could have won 70% of the time with different choices. There's no reason to quit.

The biased coin is just like this, except I'm giving a reason for the certainty. It's always possible to get one of the photos you didn't guess. But all the photos previously gotten have no effect on the next one you'll get.

Gwydion
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Location: Chicago, IL

### Re: Anti-Gambler's Fallacy

Vytron, you said it really well, except in the example given you use incorrect probabilities - using photos of hands including 0-10 fingers would leave us with a probability space of 11 outcomes, not 10. Dropping the option of 0 fingers fixes it...

For me, whether I continue has little to do with the rules, and lots to do with my personal finances. For example, after losing \$800K of my million I need to take stock of what the remaining \$200K is "worth". If I owe a loan shark \$200K and will be killed if I don't pay, then no I definitely don't keep playing (and probably would have quit sooner). If I have millions more in the bank and am just messing around for fun, then I probably keep going. If we take the problem statement as given (namely the belief that the odds for each toss are truly 70-30), the consecutive losses don't mean anything except a run of excruciatingly bad luck.

On the other hand, if I was told by a carnival worker that the coin was 70-30 in my favor (or in some other way can't be certain of the odds), I don't know if I play at all. In an abstract situation we can reason about our behavior in theory, but in practice I can't picture anyone offering me such a bet without there being a catch - things that sound too good to be true, often are. Using Vytron's (modified) example, it looks like I win on a 1-7 and lose on 8-10, but maybe the pictures aren't equally distributed. Maybe there are a lot more 8-10 in the pile than the other numbers. A real-life example of this can be seen at the end of pinball games. Many tables offer a "match" chance for a free game after losing, comparing the last two digits of your score to a random number. Of course, the number may be random but it isn't necessarily a "fair" game. The machine owner can set the likelihood of a match to a number well below the "expected" 10%, and the machine will perform accordingly.

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

Okay, yes, a zero there breaks thing up.

Gwydion wrote:Using Vytron's (modified) example, it looks like I win on a 1-7 and lose on 8-10, but maybe the pictures aren't equally distributed. Maybe there are a lot more 8-10 in the pile than the other numbers. A real-life example of this can be seen at the end of pinball games. Many tables offer a "match" chance for a free game after losing, comparing the last two digits of your score to a random number. Of course, the number may be random but it isn't necessarily a "fair" game. The machine owner can set the likelihood of a match to a number well below the "expected" 10%, and the machine will perform accordingly.

Ah, but you're not forced to go for 1-7. I could give you a big stack of prearranged cards, and you can select any 7 numbers that you wish, and I don't touch the cards afterwards. So if you suspect the deck has more 8-10s after several tries, you could switch and include 8-10s in your guesses. Except, now it's those ones that don't pop up and you still fail at winning, with whatever 3 numbers you don't include failing every time.

If you quit this game then I presume you believe I have supernatural powers or something (I knew the future and knew which numbers you'd guess, so I pre-arranged the cards in a way that would ensure you'd lose), because, remember, you're not gambling your whole 200000 at once, just 100 from it. You can still walk away with 199900, so you should bet one time more. Quitting is fallacious because you assume just because you failed 8000 times in a row that you'll lose your 200000 because you will never win, but the next card is still there with a number from 1 to 10, and you can guess any 7 numbers...

Cauchy
Posts: 602
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### Re: Anti-Gambler's Fallacy

Vytron wrote:If you quit this game then I presume you believe I have supernatural powers or something (I knew the future and knew which numbers you'd guess, so I pre-arranged the cards in a way that would ensure you'd lose), because, remember, you're not gambling your whole 200000 at once, just 100 from it.

It is far more likely in my mind that you have super powers than that the coin comes up tails 8000 times in a row due to natural causes. And if I'm doing the entire experiment by myself, then it's far more likely in my mind that I unknowingly have super powers that cause me to always mess up than that I got the wrong answer 8000 times in a row due to natural causes. For point of reference, (.3)^8000 is on the order of 10^{-4000}. This is a probability at which I question the validity of everything, everywhere, and certainly the validity or every tenet stipulated in the problem.

I don't know anything to a probability of 10^4000:1, so there's no way my certainty in the distribution of the coin holds up against the data presented.
(∫|p|2)(∫|q|2) ≥ (∫|pq|)2
Thanks, skeptical scientist, for knowing symbols and giving them to me.

douglasm
Posts: 630
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### Re: Anti-Gambler's Fallacy

To continue, I would need a reason to believe in the correctness of that probability that is stronger than the evidence of 8000 consecutive failures. With those odds, 8000 consecutive failures is evidence so strong as to be practically unbeatable. It's somewhere in the same realm of incredulity as the idea that a tornado going through a scrapyard would, purely by chance, assemble a complete modern airplane in perfect working order.

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

So basically "the puzzle as presented is impossible to happen unless there's some force behind it that can explain the observed outcomes."

I disagree, because the random distribution of outcomes does include one where you lose every time, so it's always possible that you observe this outcome.

For instance, it's not "1 in 10^4000", that's the chance of always losing, but you have to include ALL the patterns that would make you believe you have superpowers. What if you had a string Lose, Win, Lose, Win... out of the 8000 tries? Well, if that happened and you thought something fishy was going on, then you'd do it in 2 out of 10^4000 cases.

What about Win, Lose, Win, Lose...? There's now 3 out of 10^4000 cases...

And there's the one where you win every single time.

And the one where you lose every third try, and win the rest and vice-versa.

And the one where you win every third try, and lose the rest and vice-versa.

...

Even the times where you get the expected outcome might make you think you have super-powers!

Say, one where you lose 3 times, and win 7, lose 3, and win 7...

One where you win 7, and lose 3, win 7, and lose 3...

The idea here is that the 10^4000 is full of random cases that have some pattern that make you suspect you have superpowers, including:

You won the first one and lost the rest
You won the second one and lost the rest
You won the third one and lost the rest
...
You won the 8000th one and lost the rest

And:

You won the first and second one and lost the rest
You won the first and third one and lost the rest
You won the first and fourth one and lost the rest
...

All of those that would make you believe some higher power was controlling your rolls?

Losing all is just one of them, but they're plenty, so it seems a case of looking for patterns in the noise. You got a Lose all pattern. It could have happened without any intervention just like all these other unexpected patterns. It means nothing for the 8001st try. For 10000 tries of picking 7 out of 10 numbers that started with 8000 Loses in a row, you're expected to win 1400 times out of the next 2000 tries, just like those with 8000 wins in a row, all extraneous strings posted, and a "looks about right" string you had expected to get.

douglasm wrote:With those odds, 8000 consecutive failures is evidence so strong as to be practically unbeatable. It's somewhere in the same realm of incredulity as the idea that a tornado going through a scrapyard would, purely by chance, assemble a complete modern airplane in perfect working order.

That's like saying the chance of getting 8000 consecutive failures is 0. If in your experiments you never get 8000 failures in a row, you haven't ran enough experiments. If you ran experiments forever, there will be an infinite number of strings with 8000 failures in a row. If you ran them for long enough to finally witness the 8000 in a row one, you'd realize there's no magic force that stops it from being the very first try.

Finally, if you're still thinking of not playing the 8001st time, I have a last variation:

You, and other 10^10^10^10^10^10^1.1 people are put on a chamber, and all of you play the game 8000 times.

You get 8000 failures in a row.

What now? Are you as surprised? From the 10^10^10^10^10^10^1.1, how many did you expect to get your result? Do you think you're the only one? There's actually a huge group that did it and you're part of them.

How is it different from the actual case? Because, other people doing things independently from you doesn't change the chances of your own outcomes. Knowing other people are also flipping their own coins should have no effect on your decisions, and if, somehow, there being enough people doing it makes it feel "more right" that you got that outcome then there's a problem with the reasoning and you should probably play the 8001st time.

PeteP
What the peck?
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### Re: Anti-Gambler's Fallacy

Vytron wrote:
PeteP wrote:I would assume that I'm being cheated in some way

Why?

Suppose that every time the coin is flipped, 70 universes are created where it landed in your favor, and 30 where it had an unfavorable results.

If after 8000 times all you got was unfavorable results, that just proves those universes where you got them exist. The probability is >0 so actually, if you flip the coin an infinitely high number of times you'll have for certain a stretch of 8000 unfavorable results in a row. Actually, infinity isn't necessary here, a Graham Number of flips would ensure a mindblowingly number of such strings of unfavorable results.

I know it's a possible event the fact remains that it's unlikely to happen, that with a big enough number it is likely to happen at some point matters little. If the estimated 10^24 stars in the known universe each had 10 planets and if on all these planets there were 10^9 people each playing this game 1000000 times each second for the last 10^10years, the chance of one of them ever getting 8000 loses in a row with a win chance of 70% is still almost as incredibly unlikely. You say run enough experiments but if the whole universe banded together to run this experiment they can only hope that there is no such thing as the heat death of the universe because they will wait for this event a long time. Yes if you get the numbers big enough it has a chance of happening. Say you get 10^10^10^10^10^10^1.1 people and it never happened which is really unlikely, but no matter you add another fifty ^10. And still it didn't happen at what point do you start searching for an explanation even if it never is impossible?

The problem says I'm certain the coin is biased by some magical means, but not that I'm certain the outside factors can't change the probability of the throw. Maybe they developed a technological way to control the outcome of a coin throw precisely perhaps with targetted burst of airs. Or they achieved complete invisibility and someone invisible is very skilled at making it look as if the coin was actually thrown. Maybe magic is real. Maybe I do frequently win but the coin can almost instantly change it's surface from head to tail so I win but never know because I never saw it. Maybe they manipulated my brain in some way and I'm hallucinating everything which also might explain why I'm so certain that it's 70 to 30 for no good reason.

Yes if I'm in hypotheses space and can be sure that the coin has that probability and there is no outside influence I would keep playing. Except if I grew tired of repeating this game thousands of times. We are all well aware that if by some magic we could be sure that we really have a 70% chance of winning, it would be very likely we get our money back and more if we continue.

PS: If you are trying to find reasons it's more likely than it seems to get an result where we would make the same arguments you need to give categories of other results that would also make one think it's magic, single other results don't change the result much.

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

The odds make it unlikely for this to happen however, let's be bayseian about this and assign huge weight to our prior information drowning out the puny 8000 failures.

The value I stop playing at if the \$1,000,000 is all the money I have: I will pick a number that covers certain bills, expected costs and a sink fund. At the moment it's about \$18,000 and stop when I hit there.
If the \$1,000,000 exists as a "here is a free pile of money on top of your existing resources" I play to bust.

Interestingly, in this scenario, I don't know when I stop betting at the other end. That depends on how long a coin toss takes.

taemyr
Posts: 111
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### Re: Anti-Gambler's Fallacy

Vytron wrote:
You, and other 10^10^10^10^10^10^1.1 people are put on a chamber, and all of you play the game 8000 times.

You get 8000 failures in a row.

What now? Are you as surprised? From the 10^10^10^10^10^10^1.1, how many did you expect to get your result?

I am still just as surprised. I expect a huge number of people to get the same result. This does not mean that I should not be surprised that I am one of those.

After all, for every one that got this result there is 10^4182 people who did not.

Or to put it in different words; You are playing a straight poker against a single opponent and you get a 4 of a kind. Should you be surprised if this loses? Does your surprise about this result change if you where playing in a tournament with 500 000 000 other tables of 2 players, where this result would be expected to occur at about 3 or 4 tables?

Cradarc
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Joined: Fri Nov 28, 2014 11:30 pm UTC

### Re: Anti-Gambler's Fallacy

Everyone who's arguing about strong evidence for some other forces at work needs to answer these variations:

Suppose you draw the line at N fails in a roll. That is, the instant you get N fails, you quit playing. In a variation of the story, you are robbed of \$100N prior to playing the game. Do you still play the game?

Suppose you have complete control over the probability system. You can make it based on quantum properties. You can make it so that the only way to predict the results is to break sacred physical laws (like conservation of energy, or information traveling faster than speed of light). Would you still be equally skeptical?

Suppose you win if the coin lands on tails while another person wins if it lands on heads. A robot is conducting the coin flip, and you win 8000 times in a row. The other person wants to quit. Would you try to convince them to stay or let them quit ?

Here's a more philosophical one:
What is the probability that you, as a self-described intelligent system, would be present at that location at that particular instant in time with that particular number assigned to your economic worth by a society of 7 billion self-described intelligent systems?
14 billion years ago, the universe was just a bunch of hot gas and dust. How many probabilistic events, each succeeding with probability less than 30%, are needed to produce the exact state you are in now?
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Vytron
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### Re: Anti-Gambler's Fallacy

PeteP wrote:The problem says I'm certain the coin is biased by some magical means, but not that I'm certain the outside factors can't change the probability of the throw. Maybe they developed a technological way to control the outcome of a coin throw precisely perhaps with targetted burst of airs. Or they achieved complete invisibility and someone invisible is very skilled at making it look as if the coin was actually thrown. Maybe magic is real. Maybe I do frequently win but the coin can almost instantly change it's surface from head to tail so I win but never know because I never saw it. Maybe they manipulated my brain in some way and I'm hallucinating everything which also might explain why I'm so certain that it's 70 to 30 for no good reason.

But the puzzle says you got 8000 loses in a row because it was possible to do it, there's no magic involved here.

I'm actually going to do the experiment, for real, once. Because, so far we've been only theorizing. These, my fellows, is a close as true randomness as one can get, and here we have, how 8000 flips of a coin that is biased to fall 70% of the time on a 1 looks like:

Spoiler:
0 0 1 1 1 1 1 0 1 0 0 1 0 0 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 0 1 0 1 1 1 1 0 1 0 1 0 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 0 1 0 0 1 1 1 1 0 0 1 1 0 1 1 1 1 1 1 0 1 1 1 0 0 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 0 1 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 0 1 1 0 1 1 1 1 1 1 1 0 1 1 1 0 0 0 1 1 1 0 0 1 1 1 1 0 0 0 0 1 0 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 1 0 1 1 0 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 0 0 1 1 0 1 1 1 1 1 0 1 1 1 1 0 1 0 1 1 0 0 0 1 1 1 1 1 1 0 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1 0 1 1 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 0 1 0 0 1 1 0 0 1 0 1 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 1 1 0 1 0 1 1 1 1 1 1 1 1 0 1 1 0 1 0 0 1 1 0 1 0 1 1 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 1 1 1 1 0 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 0 1 0 0 1 0 1 1 0 0 0 1 0 1 0 1 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 1 0 0 0 1 1 1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 0 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 1 1 1 1 1 0 0 1 1 0 1 1 0 0 1 0 0 1 1 1 1 0 1 1 1 1 1 0 0 1 1 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 1 1 1 0 0 1 1 1 0 0 0 1 0 1 1 1 0 1 1 0 1 1 1 1 1 0 1 1 0 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 0 1 1 0 1 1 1 0 1 0 1 0 1 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 0 1 1 1 1 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 1 1 0 0 1 1 1 1 1 1 1 0 1 1 0 0 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 0 1 0 1 0 1 1 0 0 1 1 1 0 1 1 0 0 1 0 1 1 1 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 0 1 0 1 1 1 1 1 1 1 1 0 0 1 0 1 1 1 0 0 1 1 1 1 1 0 1 0 1 1 1 1 1 1 1 1 0 1 1 1 0 1 0 1 0 1 1 1 1 0 1 1 0 1 1 1 0 1 0 1 1 0 0 1 0 0 1 0 1 0 0 1 1 0 1 1 1 1 0 0 1 0 0 1 1 1 1 0 1 0 1 1 1 1 1 0 0 1 1 0 1 1 0 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 1 1 1 0 0 0 1 1 1 0 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1
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Hmmm, so, how did I do? 5601 1s. I was expected to get 5600. Perhaps this wasn't as random as I thought? I was expecting the difference from the result and from what was expected to be larger. Perhaps I just was lucky, but, the most people are expected 5600, anyway... The largest string of loses I got was 7.

Anyway, just pretend that this is what you could have gotten if you did the experiment by hand with an actual coin, the point is:

This string I'm presenting is as unlikely as the one full of 0s

That is, I got this string, and 10^4182 people did not. The chance for anyone to start flipping the biased coin and end with this specific one is like "a tornado going through a scrapyard would, purely by chance, assemble a complete modern airplane in perfect working order". Yet, I did it, on my first time!

So yeah, the whole point is these flips I had done have no effect whatsoever on the extra flip I'd do next. Had I flipped 8000 0s they wouldn't have an effect on the next flip. So I oughta keep playing.

How impressive are the 7 loses I had on a row, anyway? What if the string had 15 loses in a row? I wouldn't have been surprised, these things happen. The chance of so many wasn't 0, so it could have happened on my first try.

taemyr wrote:You are playing a straight poker against a single opponent and you get a 4 of a kind. Should you be surprised if this loses? Does your surprise about this result change if you where playing in a tournament with 500 000 000 other tables of 2 players, where this result would be expected to occur at about 3 or 4 tables

This is like the surprise quiz puzzle, where the professor tells the students there will be a pop quiz next week and it'll take them by surprise. Turns out giving it on a Friday is surprising because by Thursday night you'd expect it by Friday, so you expect no quiz at all, so it surprises you.

No, for surprises to work you actually have to, before the experiment begins, make a list of all the strings that would surprise you. I named many such strings in my other post (all wins, alternating wins and loses, anything that keeps repeating short patterns, etc). If you'd be surprised by a tremendous amounts of different strings that you could have gotten, it's not surprising that a string of only 0s surprises you too. People keep throwing the number 10^4182 around, but the problem is a lot of those people were surprised too, just by different reasons.

I'd stop playing if the string of just loses was the only surprising one. But really, if a given person starts being surprised when their string has n loses in a row, they get easily surprised, as, apparently, strings with 100 loses in a row are already nearly impossible to get. But just count them, there's a bunch of them and they should be counted in "n in 10^4182."

Once you realize any string you get after 8000 throws was as likely as any other for a biased coin, all 0s is just part of the population, and you don't stop playing, because the fallacy is assuming the coin is very, very likely to lose, just because it did it 8000 times in a row. The bell-shaped curve you'd expect to get if you keep graphing results beyond this point will look right, and in fact, after 1000000 tries it'll have about 70% of wins, because the 8000 loses will just be noise at that point.

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

Vytron wrote:This string I'm presenting is as unlikely as the one full of 0s

That would only be true if 1 and 0 were equally likely outcomes. In fact, your string is (7/3)^5601 times as likely to occur as the string of all 0s.

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

How is this a logic puzzle?

Anywho, the whole situation is implausible. Flip-a-coin-to-see-who-wins is an incredibly boring excuse for a game, and no matter if I win or lose, there is no way I'm wasting my time on 8000 rounds of that. And don't say it's about the money. I have a million bucks, I don't need to win a couple of hundreds.

Cradarc wrote:Suppose you draw the line at N fails in a row. That is, the instant you get N fails, you quit playing. In a variation of the story, you are robbed of \$100N prior to playing the game. Do you still play the game?

Even less likely. I'm busy reporting the robbery.

Cradarc wrote:Suppose you have complete control over the probability system. You can make it based on quantum properties. You can make it so that the only way to predict the results is to break sacred physical laws (like conservation of energy, or information traveling faster than speed of light). Would you still be equally skeptical?

Look, the problem isn't that people doubt it is marginally possible to flip tails 8000 times in a row even with a heads-favouring coin. The problem is that you're asking real people how they would react if they were actually in that situation. They're rightly saying that the situation as described is much more likely to be some sort of con game than a genuine extraordinary streak (to begin with, why would anyone choose to play on the tails side if this was an honest game?). Getting the mark to believe that they're in control and have an edge is an essential part of many cons; so I'd be at least equally sceptical.

Cradarc wrote:Suppose you win if the coin lands on tails while another person wins if it lands on heads. A robot is conducting the coin flip, and you win 8000 times in a row. The other person wants to quit. Would you try to convince them to stay or let them quit?

Ask them to quit, offer them their money back, and suggest a more interesting game to play instead.

Cradarc wrote:[...] How many probabilistic events, each succeeding with probability less than 30%, are needed to produce the exact state you are in now?

Forty-two. Duh.

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

Znirk,
The puzzle part is deciding whether or not it is rational to continue playing the game after N fails in a row. I, myself, would not play the game at all in a realistic setting. That is not the issue though. The issue is, would a rational person (who wants more money) continue playing this game?
I admit the question wasn't framed this way, but since it was posted in Logic Puzzles, I assumed people will treat it this way.

Should the starting amount matter? If one only had \$10,000 and each roll only cost \$1, should that affect how many rounds one is willing to play?
The gambler's fallacy highlights people's tendency to be believe in a "balance force" driving randomness. This anti-gambler's fallacy highlights people's tendency to doubt randomness when confronted with particular cases.
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douglasm
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### Re: Anti-Gambler's Fallacy

Vytron wrote:That is, I got this string, and 10^4182 people did not. The chance for anyone to start flipping the biased coin and end with this specific one is like "a tornado going through a scrapyard would, purely by chance, assemble a complete modern airplane in perfect working order". Yet, I did it, on my first time!

Ah, but your specific string is not predesignated. "All 0s" effectively is predesignated.

Vytron wrote:No, for surprises to work you actually have to, before the experiment begins, make a list of all the strings that would surprise you. I named many such strings in my other post (all wins, alternating wins and loses, anything that keeps repeating short patterns, etc). If you'd be surprised by a tremendous amounts of different strings that you could have gotten, it's not surprising that a string of only 0s surprises you too. People keep throwing the number 10^4182 around, but the problem is a lot of those people were surprised too, just by different reasons.

I'd stop playing if the string of just loses was the only surprising one. But really, if a given person starts being surprised when their string has n loses in a row, they get easily surprised, as, apparently, strings with 100 loses in a row are already nearly impossible to get. But just count them, there's a bunch of them and they should be counted in "n in 10^4182."

Once you realize any string you get after 8000 throws was as likely as any other for a biased coin, all 0s is just part of the population, and you don't stop playing, because the fallacy is assuming the coin is very, very likely to lose, just because it did it 8000 times in a row. The bell-shaped curve you'd expect to get if you keep graphing results beyond this point will look right, and in fact, after 1000000 tries it'll have about 70% of wins, because the 8000 loses will just be noise at that point.

Ok then, let me describe (since there are too many to actually list) what sequences would surprise me. There is, in information theory, a concept called entropy (often "Shannon entropy", after the person who first formally defined it). It is, in essence, a measure of how reliably it is possible to predict the next bit (or letter, or whatever unit of encoding you're using) given the preceding bits (or letters, etc.) in a sequence. High entropy sequences are difficult to predict, low entropy sequences are easy to predict. The math behind the concept is used in computer compression algorithms - a low entropy piece of information can be compressed into a much smaller high entropy piece of information.

A sequence generated by a biased 70-30 repeated coin flip has a specific expected entropy. Any sequence whose entropy is at least a certain amount below expected would surprise me. The lower the entropy, the more it would surprise me. A sequence of all 0s has literally the lowest possible entropy, and would thus be extremely surprising. Every "surprise candidate" example you gave would also have extremely low entropy. The category of sequences with low entropy has, collectively for the entire category, a low probability of occurrence, and I would base my surprise on the chance of producing a sequence with the amount of entropy observed or less.

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

Douglasm has a point.
Although any random string is as likely to occur as any other, there are many more strings with high entropy than with low entropy.
For example, in a string of four,there are two strings with a 4:0 distribution of bits:
0000 and 1111
However, there are 4 choose 2 = 6 different strings with 2:2 distribution:
0011, 1010, 1001, 1100, 0101, and 0110

As the string gets longer, the maximum entropy strings more and more outnumber every other entropy string.

That is irrelevant to this problem though.
The fact of the matter is, it is totally possible to obtain 8000 fails in a row. Saying it is impossible simply because you don't want to believe it is not logical. The question is, what are you going to do about the future? You know the system is memoryless, so why is it that past events would influence your opinion about future events?
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douglasm
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### Re: Anti-Gambler's Fallacy

Cradarc wrote:That is irrelevant to this problem though.
The fact of the matter is, it is totally possible to obtain 8000 fails in a row. Saying it is impossible simply because you don't want to believe it is not logical. The question is, what are you going to do about the future? You know the system is memoryless, so why is it that past events would influence your opinion about future events?

No one is saying it's impossible. We're saying it's so improbable that explanations other than pure chance are more likely, and thus it makes sense to believe that a true alternative explanation exists and to either seek that explanation or assume its existence and act accordingly.

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

douglasm wrote:Ah, but your specific string is not predesignated. "All 0s" effectively is predesignated.

No, as Cradarc said, the spoiler you opened to see the string could have been all 0s. It was possible, just like I got 7 in a row, to get 8, or 100, or 8000. Ah, but if I did, even if I had done exactly the same that I did and posted my result, then, somehow, you'd have believed it was fake, or tried to find another explanation, even though I, would have, in fact, rolled 8000 0s in a row for a coin biased to give 1s 70 out of 100 times.

That, just because the string was something you didn't expect, makes it "predesignated" is the fallacy. You assume it's "predesignated" just because of its low likelihood.

It's as if you are tasked with listing all possible results of such a coin being flipped 8000 times, and you go and not include the all 0s result, arbitrarily, just because you find impossible for it to happen.

douglasm wrote:A sequence generated by a biased 70-30 repeated coin flip has a specific expected entropy. Any sequence whose entropy is at least a certain amount below expected would surprise me. The lower the entropy, the more it would surprise me. A sequence of all 0s has literally the lowest possible entropy, and would thus be extremely surprising. Every "surprise candidate" example you gave would also have extremely low entropy. The category of sequences with low entropy has, collectively for the entire category, a low probability of occurrence, and I would base my surprise on the chance of producing a sequence with the amount of entropy observed or less.

I guess it would be very interesting to see the edge cases. Like, there's a string X that has just the right entropy that you'd expect to see, the minimum case, and if there's an identical string but with a 1 instead of a 0 you jump out of your seat, just because it didn't jump over the bar?

This enters Zeno's Paradox territory: You can't define two lists, where one includes all strings that surprise you, and another where they don't, because there are two strings that are indistinguishable from one another yet are in different boxes, just because one has a missing bit of entropy that makes you go "wow! This string has 10^-4182 less entropy than this other one that doesn't surprise me!"

Attempting to define a case where surprising entropy kicks in is like trying to define that a heap of sand has n grains, and if you remove 1 grain it's no longer a heap.

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### Re: Anti-Gambler's Fallacy

In other words:

Suppose you flip 7999 coins and get some string. It turns out its entropy, and whether it goes over the line or not, will depend on the last flip. If it's a 0 you'll say the entropy is high enough to surprise you. If it's a 1 it won't surprise you.

I claim, it makes no sense because 7999 coin flips should be enough to get you surprised or not, and getting a 0 in the 8000th flip shouldn't make you suddenly jump out of your seat.

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

That's akin to nitpicking about how many grains of sand qualify as a "mound" vs a "dune" or "hill" or other such term. The border between such terms is fuzzy, and so is the point at which a person (myself or otherwise) would be surprised by low entropy. Surprise is not a binary state, there are many gradations to it and no single coin flip in a set this large would make the difference between "definitely surprised" and "definitely not surprised".

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

But there's a definitive "If I get this string then I smell something fishy and stop betting" and "If I get this other string then everything seems normal and I continue betting."

Betting or not betting is binary.

Sure, there are many, many surprising cases that are irrelevant, because the coin seems to output even more 1s than expected, and you're surprised that you're winning more money than advertized.

But in the rest you were surprised, and this surprise made you stop betting, so there should be a case where after 7999 flips you don't know if you'll be surprised and stop betting until the last roll, which is binary, and we go back to the heaps paradox.

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

The precise boundaries are impossible to meaningfully assess, but I think my realistic reaction to something like this would have an edge region, not an edge case. There are a lot of outcomes where I would definitely continue playing. There are a lot of outcomes where I would definitely not continue. In between, there are outcomes where I might or might not continue, depending on factors that essentially combine into another random roll. The difference in entropy between two adjacent sequences in the edge region would make a minuscule difference in the probability of choosing to continue.

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

That still sounds more gut feeling than logic...

What I will say last is that after 1000000 flips the entropy will return to normal and one shouldn't have quit, because the first 8000 0s in a row lose relevance.

Also, I inadvertently proved everyone wrong

By the rules of the puzzle, you got 8000 loses in a row with a 70/30 win/lose coin, so what could have followed was the string I posted, which proves you quit early and would have recovered your loses and made a profit. Please note my string starts with 0 0, so it was actually 8002 loses in a row!

It leads to the next variation:

Suppose that after you quit, you can continue flipping the coin, and see the outcomes you would have gotten if you continued. If you continued and saw the string I posted, at what point would you decide to come back into the game and continue playing, because the bizarre aberration that was causing the loses seems to be now gone?

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

So I have a decision making process that will, in a situation I am 99.999999999999999(insert a bunch more 9s)% likely to never get into even if I lived to the heat death of the universe, make the wrong decision.

I can live with that.

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

(the point has always been, that if after enough tries you get some very unlikely outcome, then there's nothing stopping it from happening the very first try. Happening on the very first try doesn't mean there's "predesign" at work. Your own existence* was probably less likely than the premise of the thread, and yet, you're here)

*Okay, I'll cheat and say your own existence includes your memories of this post, that was something nobody would have predicted, ever, including that I'd use the word predesign in a post, it even surprised me.

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

My own existence has a vastly greater entropy measure than getting 8000 0s on the first try. Happening on the first try is not an indication of "predesign", having extremely low entropy is (probably).

Human brains are predisposed to detect low entropy and assign meaning to it, and they do that because it works.

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

Vytron wrote:Suppose you flip 7999 coins and get some string. It turns out its entropy, and whether it goes over the line or not, will depend on the last flip. If it's a 0 you'll say the entropy is high enough to surprise you. If it's a 1 it won't surprise you.

If the entropy of a single final 0 is high enough to surprise you, it will make you go "hmm, that's less entropy than I expected"; if it turned up as a 1 instead, you'd also go "hmmm, that's less entropy than I expected, it just barely missed my pre-determined threshold value!". It won't make you 'jump out of your seat'.

Thanks for explaining the entropy thing - it's a great way to articulate the instinct I couldn't put into words.

To borrow a metaphor, to find a sequence of all 0s is like finding a pocket watch in the middle of the desert. Sure, it's possible that the random machinations of metals and sand could form a working pocket watch using entirely natural processes; but it's more likely that the watch has a watchmaker.

So too with the sequence of 0s. Someone with an invisibility cloak messing with the coin, or a coin with nanobots reforming the faces are both orders of magnitudes more likely than any sequence with a similar entropy to 8000 tails in a row. (8000 heads in a row, too, for that matter!).

So to answer the question, if it happened to me, I'd stop playing because I would consider the odds of someone deliberately messing with the system despite the coin itself being fair to be a higher probability than it happening by chance.
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SPACKlick
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### Re: Anti-Gambler's Fallacy

I think people arguing over the "highly unlikely scenario demanding additional explanation" just aren't putting enough prior probability into their bayesian analysis.

Imagine before you run the bet, to your satisfaction you are left unmolested and toss the coin grahams number times and it falls out with a ratio of 70/30. You don't tell anyone when you start betting just click a button on a self contained device hidden about your person which you have constructed yourself to measure the results and know passes the information the bet is live to other people. etc. etc. etc.

If you have enough prior information you can be confident this is just once in a universe odds, not trickery and therefore you keep playing.

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

Cradarc wrote:The puzzle part is deciding whether or not it is rational to continue playing the game after N fails in a row. I, myself, would not play the game at all in a realistic setting. That is not the issue though. The issue is, would a rational person (who wants more money) continue playing this game?

If the assumptions are correct, then duh, yes; but then the whole thing is akin to begging the question: If it is rational to play, then it is rational to play. There's no hint of a puzzle, nor of anything worth discussing (except possibly how the person within the story can be so sure of that ex cathedra truth imposed by the narrator).

So we move on to puzzling at something else. If I know that I have a 7 out of 10 edge, then I also know that it is irrational for my opponent to keep playing. And yet she/he/it does, successfully, 8000 times. That is a pretty strong hint that a) something other than the coin is influencing the throws, and b) my opponent is in on it. Thus the puzzle becomes, "What other external effect overrides the coin's balance?", or, "how is the opponent cheating?". I think that's where most of the discussion you're seeing comes from.

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

Vytron wrote:(the point has always been, that if after enough tries you get some very unlikely outcome, then there's nothing stopping it from happening the very first try. Happening on the very first try doesn't mean there's "predesign" at work. Your own existence* was probably less likely than the premise of the thread, and yet, you're here)

Let's play troll coin toss game from discworld. Ie. the game where we throw a coin and then bet about wether or not it comes down again.

Every attempt so far have ended with the coin falling down.

Which do you think is the most likely explanation:

• Some force causes the coin to gravitate to the ground.
• It's actually a 50/50 chance, and the previous tosses have just happened to get the fall to the ground output.

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

SPACKlick wrote:I think people arguing over the "highly unlikely scenario demanding additional explanation" just aren't putting enough prior probability into their bayesian analysis.

And I think that you're putting a 100% probability into your bayesian analysis, which is always inaccurate.

If you have enough prior information you can be confident this is just once in a universe odds, not trickery and therefore you keep playing.

The difficulty is that there comes a point where you can't have enough prior information. A point at which it is more likely that your memories of having prior information are false than that that prior information actually existed.

Additionally, in your specific example: It's more likely that there's something reading your mind and deciding to mess with things now you've started betting than that you've got those results by chance.

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### Re: Anti-Gambler's Fallacy

Madge wrote:To borrow a metaphor, to find a sequence of all 0s is like finding a pocket watch in the middle of the desert. Sure, it's possible that the random machinations of metals and sand could form a working pocket watch using entirely natural processes; but it's more likely that the watch has a watchmaker.

That's like proving god exists because of the complexity of the universe (i.e. the watchmaker is even more complex than the watch, and so and so until you find the universe in the middle of nothing and say it's more likely there was a universemaker. In fact, if you get enough sand it'll have more entropy than a watch.)

The difficulty is that there comes a point where you can't have enough prior information. A point at which it is more likely that your memories of having prior information are false than that that prior information actually existed.

Given by the puzzle, the chance that your prior information is false is zero. It's actually funny that you're told an hypothetical example in which you got 8000 0s in a row, by chance, and still can't believe it happened. Again you're removing all 0s by chance of the possible outcomes for no reason whatsoever.

The interesting thing is that from what SPACKlick said, Graham Number is so numbingly big that if you flipped the coin that many times, you'd expect to get about a Graham Number of strings of 8000 0s in a row. They'd just pop up like galaxies in the universes, with long gaps between each string, but still, such strings would appear in any direction you look.

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

Oh! Oh yeah! I actually thought about it on the night and thought about an example to show what I mean.

Suppose you don't flip actual coins, but actually print out sheets of paper with all possible outcomes. Accordingly, there are more sheets with 1s than 0s, with a 70-30 proportion.

Once you're done, you have a big stack of cards, and you shuffle and shuffle them.

Then you pick the top card, and it's the all 0s card.

Are you so amazed? Is it some higher power at work that put that card at the top of the deck after you shuffled it? Are you dreaming? What is going on?

Because, the all 0s card was printed, it had to be somewhere, so why not on top?

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

Vytron, I don't think boiling it down to one deck of 1*10^4183 cards does it justice. in that scenario there is a single event, the shuffle. In the Coin toss scenario there are 8000 events which does effect the odds of alternate explanations.

I'm convinced by the argument that these odds are so astronomical that it's more likely the bias of the coin has changed during the test or my perception of the test is being changed than me being in the unlikely scenario. My argument above about when to stop applies where the statements of the case are beyond doubt but realistically they aren't

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

Vytron wrote:Suppose you don't flip actual coins, but actually print out sheets of paper with all possible outcomes. Accordingly, there are more sheets with 1s than 0s, with a 70-30 proportion.

Unless you repeat outcomes to weight the probabilities correctly*, the proportion of 1s to 0s would be 50-50.

*e.g. Given two different sheets that are identical but for one coin flip, there should be 7 copies of the sheet with that flip being a 1, and 3 copies of the sheet with it being 0.
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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.
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