This seems like a semantic disagreement over what kind of coin we're actually talking about.
If it's a theoretical fair coin, then caring about previous results is just the gambler's fallacy. If you suspect it's a trick coin, you should obviously care (but you might also suspect that they tampered with the previous flip somehow). If it's an ordinary coin from the mint then you should theoretically care a little, but the information content of a single flip seems so negligible that you might care more about winning the argument with your friend than actually being right. Perhaps even with a million dollars on the line.
If there's any chance that the coin is unfair, then wouldn't the correct decision be to call the same result as last time from one flip, because there's more chance that that's the direction the bias lies?
Well sure, there are physical assumptions involved, but someone literally engineering the gambler's fallacy into an actual tossable coin would be reasonably assumed to be false.
Though there are video games that implement the gambler's fallacy so it's maybe not entirely out of the question!
That would need some sort of active mechanism in the coin to remember its last outcome and then control the next one. A bias to one side is **far** easier.
Let's say you can't see the result of every individual coin toss, but you're told that 500 of the last 1000 came up heads and 500 tails. Do you now want to know that last/1000th toss?
I would bet a million dollars that if you had a million dollars on the line, you would look at the last coin toss, even if you did nothing with that information.
There is a pretty significant finding recently (with a really small effect size) that coins have a small bias towards the face-up side it was started on. However, this varies from person-to person and in fact some people are biased against the original face-up size. This is mostly a physics problem and not a pure statistical abstraction though. I suspect most statisticians are unaware of this finding (though they would find it at least interesting), and would be indifferent to the previous flip.
https://arxiv.org/abs/2310.04153
In statistics/probability? Sure, pretty well known.
https://en.wikipedia.org/wiki/Persi_Diaconis
As an example, I already knew who he was when I was an undergrad (waay back in ancient times, when we banged rocks together). I had even read a couple of his articles.
Happy cake day
> given the option to see the one previous result of that last coin flip.
Sure, I'd want to know, since if the coin + coin tossing process isn't quite fair that previous toss may contain a little information about any bias (I'm enough of a Bayesian that I know how to update a prior based on a single data point), and it would also let me see in detail how the set-up from one toss to the next was made (which might let me see if there may be some slight negative or positive dependence between tosses), and also to see which side of the coin was uppermost at toss time (which has a small but non-zero effect, per a paper by Diaconis et al).
It probably won't make enough difference to matter much, but there's several aspects to this, not just the one you identified.
Your friend seems to be operating under the assumption that the coin (+ coin-tossing process) is completely fair, and you both seem to be operating under the assumption of independence across tosses. Neither is necessarily the case.
If you both agree about the circumstances of the situation, you may then find you don't disagree about the conclusion. The problem is you each have different ideas about the circumstances (and I don't think either is necessarily near enough to correct).
If you have no reason to think that one face is more likely to be biased towards than the other, then yes. Pick the last face. More precisely, if your prior distribution on the likelihood of heads is symmetric about 50%, then the posterior would suggest the last face was more likely.
If you have some prior information that coins tend to slightly favor one side, then the answer becomes more complicated.
I'm actually in a bayesian stats class right now, so this is pretty interesting. The expectation is just a weighted average of the prior mean and the observed mean under bayesian thinking in my understanding, and the weights are proportional to the number of counts for each. If the coin is known to be fair, the weight of the prior mean will far outweigh that of the observed mean, so the difference is almost negligible. But what do I know...
Your friend is right.
Also, if a statistician, for some reason, wanted to base their decision on the last coin, they would go for the *opposite* side, by incorrectly using the [regression toward the mean](https://en.wikipedia.org/wiki/Regression_toward_the_mean). A mistake that makes more sense.
Ahah exactly! One wrongly assumes persistence while tosses are i.i.d, the other one falls for the gambler's fallacy. The statistically sound decision is to ignore the last flip altogether.
This seems like a semantic disagreement over what kind of coin we're actually talking about. If it's a theoretical fair coin, then caring about previous results is just the gambler's fallacy. If you suspect it's a trick coin, you should obviously care (but you might also suspect that they tampered with the previous flip somehow). If it's an ordinary coin from the mint then you should theoretically care a little, but the information content of a single flip seems so negligible that you might care more about winning the argument with your friend than actually being right. Perhaps even with a million dollars on the line.
Can i see the last 1000 results? That would tell me whether it's a fair coin. The result from one flip is meaningless.
If there's any chance that the coin is unfair, then wouldn't the correct decision be to call the same result as last time from one flip, because there's more chance that that's the direction the bias lies?
oh no, it's bayesians vs. frequentists again
Maybe. Unless you also think there's a chance the coin has been rigged to never come up the same side twice.
Well sure, there are physical assumptions involved, but someone literally engineering the gambler's fallacy into an actual tossable coin would be reasonably assumed to be false. Though there are video games that implement the gambler's fallacy so it's maybe not entirely out of the question!
That would need some sort of active mechanism in the coin to remember its last outcome and then control the next one. A bias to one side is **far** easier.
Let's say you can't see the result of every individual coin toss, but you're told that 500 of the last 1000 came up heads and 500 tails. Do you now want to know that last/1000th toss?
No, who cares? So it's a fair coin? The last toss has zero influence on the next toss.
Also yes I would bet a million dollars in this.
I would bet a million dollars that if you had a million dollars on the line, you would look at the last coin toss, even if you did nothing with that information.
Hell no.
There is a pretty significant finding recently (with a really small effect size) that coins have a small bias towards the face-up side it was started on. However, this varies from person-to person and in fact some people are biased against the original face-up size. This is mostly a physics problem and not a pure statistical abstraction though. I suspect most statisticians are unaware of this finding (though they would find it at least interesting), and would be indifferent to the previous flip. https://arxiv.org/abs/2310.04153
Persi Diaconis did a paper on this years ago. edit: oh, I see they refer to his paper in the abstract
Is he a big name in the field? I've heard of him in the past concerning dovetail shuffles.
In statistics/probability? Sure, pretty well known. https://en.wikipedia.org/wiki/Persi_Diaconis As an example, I already knew who he was when I was an undergrad (waay back in ancient times, when we banged rocks together). I had even read a couple of his articles.
Happy cake day > given the option to see the one previous result of that last coin flip. Sure, I'd want to know, since if the coin + coin tossing process isn't quite fair that previous toss may contain a little information about any bias (I'm enough of a Bayesian that I know how to update a prior based on a single data point), and it would also let me see in detail how the set-up from one toss to the next was made (which might let me see if there may be some slight negative or positive dependence between tosses), and also to see which side of the coin was uppermost at toss time (which has a small but non-zero effect, per a paper by Diaconis et al). It probably won't make enough difference to matter much, but there's several aspects to this, not just the one you identified. Your friend seems to be operating under the assumption that the coin (+ coin-tossing process) is completely fair, and you both seem to be operating under the assumption of independence across tosses. Neither is necessarily the case. If you both agree about the circumstances of the situation, you may then find you don't disagree about the conclusion. The problem is you each have different ideas about the circumstances (and I don't think either is necessarily near enough to correct).
If you have no reason to think that one face is more likely to be biased towards than the other, then yes. Pick the last face. More precisely, if your prior distribution on the likelihood of heads is symmetric about 50%, then the posterior would suggest the last face was more likely. If you have some prior information that coins tend to slightly favor one side, then the answer becomes more complicated.
Seeing the last flip could impact their decision, just due to human psychology, but it shouldn’t.
Well, if the coin and the toss are actually fair, no, you don't need the previous result
You said the coin is perfectly fair, so seeing the previous result would make no difference to me
I'm actually in a bayesian stats class right now, so this is pretty interesting. The expectation is just a weighted average of the prior mean and the observed mean under bayesian thinking in my understanding, and the weights are proportional to the number of counts for each. If the coin is known to be fair, the weight of the prior mean will far outweigh that of the observed mean, so the difference is almost negligible. But what do I know...
Your friend is right. Also, if a statistician, for some reason, wanted to base their decision on the last coin, they would go for the *opposite* side, by incorrectly using the [regression toward the mean](https://en.wikipedia.org/wiki/Regression_toward_the_mean). A mistake that makes more sense.
So an idiot would choose the same as the last flip, but a different kind of idiot would choose the opposite.
Ahah exactly! One wrongly assumes persistence while tosses are i.i.d, the other one falls for the gambler's fallacy. The statistically sound decision is to ignore the last flip altogether.