Impossible coin sequences?

[Roboramma]

So if we're dealing with a brain controlling a hand, what sort of system are we looking at, precisely? What will its results space look like when it comes to coin-tossing? Is there anything in that system that will limit streaks, such as unconscious sabotage, for example? Is it truly random, or is there actually a very few number of states that don't vary perfectly randomly?

I don't believe answers to those questions are yet available.

There are a few problems here: one is that to limit streaks requires a memory: now certainly a human brain has a memory, but in order to be effective it has to actually be able to affect the outcome of the toss. So the human brain has to have a tendency to get the hand to flip the coin in a way that it will come up tails when there are too many heads in a row. I can't control a coin toss when I want to. You are suggesting that people just naturally do this unconsciously.
Furthermore, they have to be able to do so flawlessly. If their control is less than 100% there will still be a chance of getting heads, and that means that while it's less likely because of this mechanism, it's still possible, and will happen with enough iterations.

What measurement? You've never made any measurements of people flipping coins that would allow us to draw conclusions about whether it actually does run through all possible combinations or not. Neither has anyone else.

Actually, I have, I just haven't done the specific measurement you're referring to. But I've done measurements that show that it is a random process, which is enough.

 

[Piggy]

Sure, but you seem to be suggesting that the we should assume the existence of such a switch when there is no known mechanism that could cause it and no evidence that it exists.

Oh, no. I've already said, the results-space appears to be infinite.

So I can't say that 100 heads would never come up.

But by the same token, I can't assert that they're possible, because I don't know whether or not there's something in the system that could limit them.

It's possible, for instance, that any given human's arms and hands will exhibit only a small number of actual configurations when flipping coins, so the pool of results is much smaller than we might think. And it's possible that the person's understanding of the previous results might somehow influence which one of those configurations will be executed.

I'm not saying that this actually occurs. All I'm saying is that I don't know enough information about the system to conclude that it will in fact run through every mathematically possible combination of states.

 

[Piggy]

Now perform 100 fair flips and note the sequence. That sequence had the exact same likelihood of appearing as 100 heads. Yet, it happened, so it can't be impossible.

You don't know enough about the system to say that.

You don't know that it will actually, in practice, produce an equal distribution of patterns over very large sequences.

Or if you do, you should at least explain why.

 

[Piggy]

I'm not understanding why you keep bringing up whether or not the "coin-flipping system" is "fair".

Because it makes a difference in what you're talking about.

If you assume a truly fair coin-flipping system, which is what Ivor asked about, that's one thing. If you're talking about real human beings flipping real coins, that's not necessarily the same thing.

 

[Piggy]

This is the point I was making with the monkeys as well. As long as there is a finite chance for any key to be pressed each time, you are guaranteed to eventually get the works of Shakespeare.

No, you're not guaranteed that at all.

Because the question is actually one of predictability.

It may be impossible to predict, by looking at one keypress, which key will be pressed next, but still to be looking at a system which will never produce a single work of Shakespeare.

 

[Piggy]

Of course not. The point is that the same probability and the same laws of physics apply on Mars as on Earth.

You can get 100 heads in a row on Mars, and you can on Earth as well.

The same laws of physics apply, but the set-up is not identical.

I don't think that the difference will affect coin-tossing, but it can certainly affect other systems we might want to measure, and I can't absolutely swear that it won't affect the results you get from people flipping coins.

It would certainly be possible to rig a 2-state Diaconis machine to produce results that are indistinguishable from random by humans, while limiting the absolute length of streaks.

And unless you claim to know all the relevant influences in a system comprised of a human and a coin, you simply cannot assert that the results-space of this system will include all mathematically describable combinations.

 

[Piggy]

Sure, but we have no more reason to believe that human coin flippers are such a system than that the earth will suddenly pause in it's orbit tomorrow. You can invest in nails are rope if you like, but I'm taking my chances.

Equally, we have no more reason to believe that human coin flippers aren't such a system than that the earth will suddenly pause in its orbit tomorrow.

 

[Piggy]

There are a few problems here: one is that to limit streaks requires a memory: now certainly a human brain has a memory, but in order to be effective it has to actually be able to affect the outcome of the toss. So the human brain has to have a tendency to get the hand to flip the coin in a way that it will come up tails when there are too many heads in a row. I can't control a coin toss when I want to. You are suggesting that people just naturally do this unconsciously.

Let's keep some perspective here.

I conceded that my notions about streaks of 100 not being possible were indefensible. Not because I thought a streak of 100 likely, but because I had to concede that I can't find any absolute barrier to it.

By the same token, if you really do want to assert that streaks of 100 actually are possible in a particular system then you have to demonstrate why this is so.

I cannot say that streaks of 100 are impossible, but I also cannot say that they must be possible, given any particular system of flipping coins.

I cannot claim to know that human coin-flipping is in fact unbiased -- no matter how unlikely that might turn out to be, short of zero -- for the same reason that I cannot claim to know that a streak of 100 is impossible.

 

[Piggy]

Actually, I have, I just haven't done the specific measurement you're referring to. But I've done measurements that show that it is a random process, which is enough.

Are you saying that there's a point at which you can determine that the process is actually random, and doesn't just appear to be random given the sample size?

If so, then I'd love to know where that point is.

 

[69dodge]

The machine has a perfect memory, because its construction gives it one. Feed coins into it in the same configuration -- heads or tails up -- as long as you like, you'll see how well the Diaconis machine remembers.

You can set the machine up so that whichever way the coin is facing before it gets flipped, it will face the same way after it gets flipped. So set up, the machine, in a sense, remembers which way you put the coin in. But it doesn't remember the results of previous flips. Those results have no effect on the current flip. If the current flip results in heads, that's because you put the coin in heads up this time; it has nothing to do with what happened last time. That's what I meant.

 

[69dodge]

I cannot claim to know that human coin-flipping is in fact unbiased ...

It's easy to believe that it might be slightly biased---e.g., perhaps the probability of heads is 53% instead of 50%---but it's very hard to believe, at least for me it is, that after getting 99 heads in row, the next flip is absolutely certain to come up tails. I mean, there's nothing special about that particular flip. Presumably, whoever's flipping the coin flips it more or less the same every time. If all the other times, the probability is roughly 50:50, why not this time too?

 

[Piggy]

You can set the machine up so that whichever way the coin is facing before it gets flipped, it will face the same way after it gets flipped. So set up, the machine, in a sense, remembers which way you put the coin in. But it doesn't remember the results of previous flips. Those results have no effect on the current flip. If the current flip results in heads, that's because you put the coin in heads up this time; it has nothing to do with what happened last time. That's what I meant.

Yeah, I know, but in practice it amounts to the same thing.

The Diaconis machine has a memory in practice.

As an origami expert once pointed out, folding a piece of paper changes its memory... and physics takes care of all the rest.

 

[Piggy]

It's easy to believe that it might be slightly biased---e.g., perhaps the probability of heads is 53% instead of 50%---but it's very hard to believe, at least for me it is, that after getting 99 heads in row, the next flip is absolutely certain to come up tails. I mean, there's nothing special about that particular flip. Presumably, whoever's flipping the coin flips it more or less the same every time. If all the other times, the probability is roughly 50:50, why not this time too?

Is it any more difficult to believe that a brain-hand system might actually limit streaks than it is to believe that 100 consecutive heads might actually turn up?

In other words, if we're talking about what can't be proven impossible, then certainly 100 consecutive heads shares the same greenroom with a streak-limiting human brain-body.

 

[Roboramma]

Let's keep some perspective here.

I conceded that my notions about streaks of 100 not being possible were indefensible. Not because I thought a streak of 100 likely, but because I had to concede that I can't find any absolute barrier to it.

Cool.

By the same token, if you really do want to assert that streaks of 100 actually are possible in a particular system then you have to demonstrate why this is so.

Which has been done over and over again in this thread. It really is simple Piggy: as long as there is some non-zero chance that each flip will come up heads, then a streak of all heads is possible.

So, for a streak (however long) to be impossible, at some point it has to be impossible for a particular flip to result in heads. That is counter to the reality of the system: it is never impossible for a human coin flipper to flip a coin and get heads. If you don't disagree with that then the issue seems closed.

I cannot claim to know that human coin-flipping is in fact unbiased -- no matter how unlikely that might turn out to be, short of zero -- for the same reason that I cannot claim to know that a streak of 100 is impossible.

There is a very big difference between biased and deterministic: I'm happy to accept the (rather far off) possibility that as streaks get longer people unconciously start affecting the flips in such a way that continuing the streak becomes less likely than predicted. That's very very different from saying that it becomes impossible. As I said, that requires 100% control of the flipping.

 

[Roboramma]

Equally, we have no more reason to believe that human coin flippers aren't such a system than that the earth will suddenly pause in its orbit tomorrow.

Actually we do, because we know what the components of the system are. The only component of that system that has a memory is the human brain. And we also know that it is not capable of unconscious and perfect control of the flipping. So, even if we make the somewhat odd and un-evidenced assumption that humans tend to unconsciously affect their coin flipping in such a way as to limit streaks, they are still incapable of doing so perfectly, and thus all streaks are still possible.

Furthermore, if this is your argument it seems that you concede that if, for instance, the human flipper were unaware of the outcome of the coin tosses (and thus any possible memory were taken out of the system) that all sequences would be possible, correct?

 

[Roboramma]

Is it any more difficult to believe that a brain-hand system might actually limit streaks than it is to believe that 100 consecutive heads might actually turn up?

In other words, if we're talking about what can't be proven impossible, then certainly 100 consecutive heads shares the same greenroom with a streak-limiting human brain-body.

No, because one is a conclusion based upon what we actually know about the world and the system in question, and another is just something dreamed up.

 

[69dodge]

Is it any more difficult to believe that a brain-hand system might actually limit streaks than it is to believe that 100 consecutive heads might actually turn up?

It's not at all hard for me to believe that the chances of getting 100 heads in a row are about 1 in 2¹⁰⁰. In fact, that's exactly what I believe. 1 / 2¹⁰⁰ is very small, to be sure, but it's not zero.

It's much harder for me to believe that the chances are precisely zero, because that would imply that the 100th flip is qualitatively different from the first 99: each of the first 99 might come up heads or tails, but if they all happen to come up heads, then somehow the 100th flip inexplicably must come up tails. What could possibly be special about that particular flip, compared to all the others that have ever taken place in the history of the world, which makes it certain to come up tails?

 

[MetalPig]

You don't know enough about the system to say that.

You don't know that it will actually, in practice, produce an equal distribution of patterns over very large sequences.

Or if you do, you should at least explain why.

I thought I covered that by mentioning *fair* flips.
What I mean by that is flips where heads and tails both have a 0.5 chance of occurring.

 

[brainmeat]

It's been mentioned a few times, and I am baffled that the thread didn't stop dead...non-zero probability means possible. Why would not having perfect knowledge of a particular system mean anything, unless it somehow affected the probability of a given set of outcomes?

Is a given sequence possible? What's the probability of that? Oh, it's a real number between 0 and 1 that happens to not be zero? Then it is possible.

 

[This is The End]

I'm not understanding why you keep bringing up whether or not the "coin-flipping system" is "fair".

Because it makes a difference in what you're talking about.

If you assume a truly fair coin-flipping system, which is what Ivor asked about, that's one thing. If you're talking about real human beings flipping real coins, that's not necessarily the same thing.

Why did you remove the second half of that quote and then type something that is contrary to the removed part:

I'm not understanding why you keep bringing up whether or not the "coin-flipping system" is "fair".

As Sol pointed out many posts ago, even if it was weighted 90% tails to 10% heads, a run of 100 heads would still be entirely possible.

It's possible, for instance, that any given human's arms and hands will exhibit only a small number of actual configurations when flipping coins, so the pool of results is much smaller than we might think. And it's possible that the person's understanding of the previous results might somehow influence which one of those configurations will be executed.

So even if your "human brain/arm/hand" idea reduced the chance of heads to 1%, any run of heads would still be possible.

The only way your theory would work is if somehow the "human brain/arm/hand" idea reduced the heads possibility for a particular flip to zero.

Anyway, by now I've had to turn tail on the "can't get to 100" idea. That didn't pan out too well.

Good.