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Impossible coin sequences?

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Do you think that it is less likely to come up than any other single combination?
 
Piggy said:
But I can't say it can't.
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Consider this simple thought experiment.

You have a fair coin in an ideal, random tossing system. You can toss it 100 times. Each time, it can come up heads.


If you toss it 100 times, what is the maximum amount of heads that can come up?
 
Piggy said:
No, you're right, it can't be the one and do the other at the same time.

Realized that while replying.

I was thinking that if we had a system that showed human-scale random behavior -- that is, if we agreed it was random, or "fair" -- then by definition it would show roughly an even number of each of the two possible end-states at a sufficiently large scale of results (otherwise it's biased).

So if we were to record each incident of an end-state (say, heads gets a red dot and tails gets a blue dot) then as long as we have a large enough pool of results, we'll see about an equal mix of red and blue dots in an arrangement on a string.

We're essentially using the coin as a measuring device (since we're not deciding anything based on the toss) that we dip into the system, reset, and dip in again, like taking the pressure on a tire with a tire gauge.

If you're using something like Diaconis's device, your results-space will be a solid color if all coins go in with the same side up, and it will be an irregular mix if the coins' starting orientations are randomized.

You could put a switch on the Diaconis machine to change the result of the flip, and use that with all-heads or -tails starting positions to create regular patterns of results.

But for a system that makes fair tosses, once randomness became evident, you'd have an irregular pattern with roughly equal portions of red and blue.

(Which is what makes all-reds or -blues different from the entire large set of patterns that are irregular and roughly equal, not just different from any one of them.)

So how large do clusters of red and blue get in this system? How large could they possibly get?

Well, the largest cluster of red or blue dots must contain less than half the number of measurements (flips) because there's as much red as there is blue on our chart. And it can't be half, because that would be the results-space from a system that is regular (2 red and blue hemistrings) or biased (nearly equal hemistrings, maybe with some foam in between), not random.

The largest cluster also can't be 1, because that would make the system perfectly regular like a metronome, which we know it's not.

So the largest cluster in a FCTS should be somewhere above 1 and somewhere below half the number of flips required for randomness to become sustainably evident.

If randomness always emerges by 200 flips or less -- if you can always detect randomness by that point -- then I'd have to be right that runs of 100 are impossible in a FCTS.

Well, maybe that's ok if you have some way of knowing the range of randomness your system can produce, but coin-flipping is open-ended, a potentially infinite string.

It could be that the results-space of human coin flipping is like a 2-D version of our universe, with vast expanses of weak noise and spots of enormous clumping which are gigantic relative to the bits of stuff that make them up but extremely small and slender relative to the space they inhabit.

Entering the current universe at a random point, even billions of times, you're not likely to land on one of the truly big clumps.

So we can agree that our system's fair, but if the results-space really is infinite, then you can't say anything about the size of the biggest clump, since your small-scale randomness, as representative of the entire system as it may be, could be right up next to a Jupiter of results that you haven't quite gotten to yet.

Which brings it down to the question of whether your physical set-up produces that kind of results-space or not.

For example, if you put a direction detector in a turbulent chamber and used it to print a ribbon by coordinating its direction with a color wheel, you'd get a random variation of colors streaming from the machine.

At the same time, there'd be a limit to the possible length of time you could get one color, if you've created a situation of continual turbulence, and that limit would be determined by how you set the mechanism up.

Is it possible to build a coin-flipper that does something like that? Randomizes the results but limits streaks?

I don't know. I guess you could do it, because even though you set it up to be trending one way or another at any given time, if you did it right there might be no way for the flippers to be sure they were in a trend, or to know which direction it was going, or when it would change, since there's no large-scale regular pattern, and local turbulence could conceal weak larger-scale trends.

In other words, you could make it "fair to the participants" within a certain number of flips even though it was always biased in one way or other and does have limits which you could figure out if you played it a very long time (let's say, a billion years).

Maybe someday someone will discover some necessary "turbulence" in the physical setup of fair human coin tossing that does keep it from trending too long in one direction, but if anybody ever does, it certainly won't be me, so I have to accept the possibility of the Jupiter of streaks looming out there somewhere on our infinite string.
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You're making this vastly more complex than it really is.

If you would ever get around the answering my question, you might understand why. It's got to do with your red/blue thing above - if you (randomly or otherwise) re-color heads and tails on each flip, 100 heads could be an even mix of red and blue, while some more random sequence of heads and tails would be all red or all blue.

100 heads is no more or less special than any other sequence of 100. The only way that can be wrong is if the coin has a memory.
 
It is not possible to make a machine that randomises it and limits streaks, by definition. Anything that limits streaks stops it from being random.
 
The size of the "results space" doesn't matter. It's 50/50 whether a coin comes up heads every time you flip it. The largest cluster can be 100% of all coins flipped (and by that I mean all coins flipped ever, in the entire history of the universe). The largest cluster of heads can be 0. It can be exactly 50/50 split between heads and tails.

None of these results is any more or less likely than any other specific result.
 
So Piggy finally admitted that 100 heads in a row is possible? He just used 918 words to say it so we'd forget he was wrong....
 
Sceptic Tank said:
<snip> The largest cluster can be 100% of all coins flipped (and by that I mean all coins flipped ever, in the entire history of the universe). <snip>
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All heads for the entire history of flipping.... now THAT is some steep odds!!

ETA: Then again, if there were infinite universes, it would be a guarantee. The people in that universe would probably be pretty confused about that.
 
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OnlyTellsTruths said:
All heads for the entire history of flipping.... now THAT is some steep odds!!

ETA: Then again, if there were infinite universes, it would be a guarantee. The people in that universe would probably be pretty confused about that.
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There's probably an infinite something in this universe too, in that case. Einstein had a theory about what that was. :)
 
alex04 said:
And with that; where do you draw the line at impossible? H(90)? H(70)? H(40)? H(20)? H(10)?
.
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I like this response, from a testing standpoint. Impossible indicates a prohibition of some kind, as opposed to finite, non-zero probability. If some given number of flips results in some prohibited state, you have to establish why that number is different. Gambler's fallacy, etc and so on.

by the way, enjoying the forum immensely.
 
alex04 said:
Piggy's point is quite clear. The fallacy, is saying that there's something special about H(100).

And with that; where do you draw the line at impossible? H(90)? H(70)? H(40)? H(20)? H(10)?

My understanding, is that putting a limit anywhere is simply incorrect, and meaningless.
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I would put the limit this way: It is impossible to get 5 heads in a row by flipping a fair coin only 4 times.
 
Its not impossible, just improbable, it is so improbable that is has seldom happened and will seldom happen.

The average number of tosses to get 100 heads in a row is (2^(100+1))-2 = 2,535,301,200,456,458,802,993,406,410,750

Feel free to try and beat that number.

Your chance of getting 100 in a row will be 1 in 2^100 = 1 in 1,267,650,600,228,229,401,496,703,205,376
 
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I have a math query...

Was thinking about this as I was in bed last night-

So; I decided I would not continue flipping coins if I hit a tails- instead I would start over again. How would this effect the speed in which I flipped 100 heads?

Going by the asumption I can flip and write down the result in 5 seconds.

Also would be grateful if you could work out the MAX time taken to flip 10/25/50/75 in a row. I can halve this to find the average I guess.

Thank you fellow mathematicians -
 
Wezznco said:
I have a math query...

Was thinking about this as I was in bed last night-

So; I decided I would not continue flipping coins if I hit a tails- instead I would start over again. How would this effect the speed in which I flipped 100 heads?
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This thread deals with that question. I think the short answer is that it speeds things up by a factor of 50.

Wezznco said:
Also would be grateful if you could work out the MAX time taken to flip 10/25/50/75 in a row.
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There's no maximum. It could take arbitrarily long. But sufficiently long times are very improbable.

Wezznco said:
I can halve this to find the average I guess.
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No, it's not that simple.
 
AdMan said:
Not sure what system you're talking about.

Good. It can, by the way.
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I'm talking about the physical system. After all, what other system is there? When we're talking about coin flips, we're necessarily talking about some physical system, whether it's a machine or my right hand.

If you want to say that any particular system has a particular results-space, and not some other, you're going to have to show why that is.

It's not enough to demonstrate what all the possible combinations are. You also have to demonstrate that the system will achieve them all.
 
Yes. I was referring to a fair coin.

Is a heads more likely to come up than a tails? No.
Any result is as likely as any other, whether you have just one flip or any number.
 
sol invictus said:
You're making this vastly more complex than it really is.

If you would ever get around the answering my question, you might understand why. It's got to do with your red/blue thing above - if you (randomly or otherwise) re-color heads and tails on each flip, 100 heads could be an even mix of red and blue, while some more random sequence of heads and tails would be all red or all blue.

100 heads is no more or less special than any other sequence of 100. The only way that can be wrong is if the coin has a memory.
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In this context, your scenario creates an entirely different question.

The original question concerned how many times a (very crude) measuring device could be applied to a system and end up with a particular result, if the system itself were truly random.

If you randomly recolor the results of those measurements, you get a results-space that's the interaction of the two systems.

So if you want to answer questions about the first system (the coin flips) you have to reverse the re-coloring you did in the second step.

In other words, the re-coloring does not (because it cannot) add anything to our understanding of the first system.

If it were true that a run of 100 were impossible in the first system, and it could be deduced from the results-space, the random recoloring would mask that fact.

If it were true that the results-space must be monochrome, the random recoloring would mask that, too.

As far as I know, we're not asking about the intersection of systems.

But back to the issue of a series of coin flips... we should be just as cautious about proclaiming what a physical system can do as we are about proclaiming what it can't do.
 
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