Impossible coin sequences?

[Alan]

And add just one to that "random-looking" sequence and that 101-long sequence is less likely to occur than 100 heads. Woahhhhhhhhhhhhhh.

 

[Piggy]

And add just one to that "random-looking" sequence and that 101-long sequence is less likely to occur than 100 heads. Woahhhhhhhhhhhhhh.

Here's an interesting question: How long a streak would allow us to judge the fairness of the coin-flips?

Theoretically, you could never say it wasn't an honest coin, as long as the string of results is potentially infinite.

By the same token, a given Diaconis machine might actually be producing random results, over an extremely long timespan, and we'd just never be able to tell.

 

[joller]

So I can't say that there is literally a zero chance, not even 1/1-nonillionth of a chance, that real coin-flippers would somehow sabotage extremely long streaks, were they to occur.

Are the coin flippers aware of the series?
i.e. it has to be the same coin flipper for all 100 flips right?
Or would you accept 10 coin flippers doing a single flip each in order, not aware of the others results (only the score keeper is).
I.e. is this a blind test, doble blind test, or not a blind test at all>

It seems you'd only accept a single coin flipper, aware of the whole series.

How I see the bias would work: the real-life coin flips are non-random, to a small extent.
Toss a coin 200 times and see what stats you get.
Even if you get close to 50-50, not everyone tosses the coin the same way, so you'd need a 'toin coss' protocol.
The 50-50 result, which i think was partially your point with the smooth spaces earlier, is quite unlikely, probabilistically, even though it's the most likely result.
Throw 2 d-6 dice. 7 is the most likely result, but it's less likely than ie. '6 or 8' together.

How do I place the coin before the toss? Where is it placed, its it always heads-up before the toss? How high do I toss it? Do I catch it or let if fall? do we have a minimum amount of revolutions to qualify for the test?

It would be interesting to ask some experienced illusionists how much they can bias a coin toss with a random, not-pre-selected and not wighed coin.

If they can acheive a good result (meaning strong bias), then I think it is fair to assume that a flipper aware of the series might subconciously be able to influence the toss to break a long series.

How big is his chance to influence it, and if he would influence it reliably enough to break the series reliably is difficult to answer due to the very small probability he would have a chance to break a 99 long series in the first place.

 

[hodgy]

Just a side note - noting Piggy's error that we would by definition a need a zillion years of coin flips to get the 100 heads. The 100 heads sequence is just as likely to occur on the first 100 flips as on the zillionth zillionth.

 

[joller]

Just a side note - noting Piggy's error that we would by definition a need a zillion years of coin flips to get the 100 heads. The 100 heads sequence is just as likely to occur on the first 100 flips as on the zillionth zillionth.

You are correct, but on average we'll only hit our pre-determined sequence in half a zillion years.

A typical scenario, that deals with data more random than your average coin toss - finding encryption keys through a brute force attack.
Say you have an encrypted message and you know the plaintext (the message before encryption). Therefore you can test all 2^64 individual key in the following manner.
Select a key.
Encrypt the message with your key.
Check if the encrypted message is equal to the original encrypted text you were given.
Repeat till you find the key.

It turns out you're very lucky if you get the key in say around the 25% of the tested keyspace (and usually around the 50% mark, and of course, unlucky around 75% mark) even though as you correctly stated, the chances of you selecting the correct key (randomly) for the first attempt are the same as for every other attempt (assuming perfect randomness - and results of encryption are anb extremely good approximation of pure randomness, though we can do even better than that)

EDIT: if I remember correctly, we'd be dealing with a normal distribution here, and 68% of results fall within a standard deviation of the mean. The mean here will be 50% of the exhaustive keyspace search.

The standard deviation for a series of 100 coin tosses is 5, so if the coin is fair and the toss truly random, 68% of results will be 45-55, a probability of hitting 50, the most likely result, is below 8 percent.

 

[Piggy]

Are the coin flippers aware of the series?
i.e. it has to be the same coin flipper for all 100 flips right?
Or would you accept 10 coin flippers doing a single flip each in order, not aware of the others results (only the score keeper is).
I.e. is this a blind test, doble blind test, or not a blind test at all>

It seems you'd only accept a single coin flipper, aware of the whole series.

Yeah, I was thinking about that this morning. You'd have to posit ESP, I suppose, if the flipper couldn't see the results, and I'm not willing to go that far, although I've been on threads with skeptics who will.

Or I don't know, maybe there is some strange-but-not-provably-physically-impossible scenario out there. When you're talking about "no chance" it gets real dicey to say.

But for my money, no, I can't see a way around that.

 

[Piggy]

It would be interesting to ask some experienced illusionists how much they can bias a coin toss with a random, not-pre-selected and not wighed coin.

If they can acheive a good result (meaning strong bias), then I think it is fair to assume that a flipper aware of the series might subconciously be able to influence the toss to break a long series.

I believe the article on the Diaconis machine talked about that, but I believe the illusion was to "flip" the coin so it spun and wobbled while remaining face up or down, and the magician made the choice of heads or tails between grabbing the coin and laying it on his arm.

I don't know how good a person could get at real flips, but I'd bet it could be mastered with sufficient practice.

 

[Piggy]

Just a side note - noting Piggy's error that we would by definition a need a zillion years of coin flips to get the 100 heads. The 100 heads sequence is just as likely to occur on the first 100 flips as on the zillionth zillionth.

That's true, but it's not just as likely to occur on the first 100 flips as it is somewhere between the zillionth and the zillion zillionth.

Anyway, no, that's not what I'm saying -- in fact, if the system is potentially infinite, then if you look at it theoretically there's literally no limit (as long as you have a finite number of flips) to the streak from flip 1 forward that would be allowed in a random results space which, at some scale, evenly balanced heads and tails without forming regular patterns.

No matter how large your "blob" got, you could just say it's a tiny detail in a much larger arrangement.

 

[ehcks]

Let's imagine flipping a coin some extremely high number of times. A number that no one is likely to ever do, but this thread seems hypothetical anyway.

If a coin were flipped a billion times, what's the largest consecutive run of heads you'd statistically expect? And how would you make that calculation?

Alternately, if I wanted to know how likely a run of consecutive heads was in certain numbers of throws, I'd divide the number of possible sequences with that many heads by the total number of possible sequences.

Let's say a coin is flipped a billion times. There should be 2^billion possible sequences. How many would have 100 consecutive heads?



Or am I missing the entire point of the thread? I'm new here, and all.

 

[TubbaBlubba]

Some of the discussion in this topic reminded me of this Dilbert strip.

http://www.dilbert.com/dyn/str_stri...000/000000/00000/2000/300/2318/2318.strip.gif

 

[DevilsAdvocate]

It's not that, really. It's just that I can't tell you that there is not even 1/1-nonillionth of a chance that real human beings flipping real coins would not somehow unconsciously kill the streak if it lasted long enough.

I can tell you that I find the notion pretty far-fetched.

But I can't tell you that there's not some hyper-astronomical chance that it might actually turn out to be true.

Why do you think the real human being flipping a real coin would somehow unconsciously kill or sabotage a streak (that would have gotten heads) rather than continue or perpetuate a streak (that would have gotten tails)?

If someone can influence the coin flip somehow unconsciously, and they have a long sting of heads going, what makes you think they would somehow unconsciously influence the flip to come up tails and end the streak rather than somehow unconsciously influence the flip to come up heads and continue the streak?

 

[Thabiguy]

Here's an interesting question: How long a streak would allow us to judge the fairness of the coin-flips?

Theoretically, you could never say it wasn't an honest coin, as long as the string of results is potentially infinite.

Of course you could say that. You couldn't say that with zero chance of being wrong, but you could say that with arbitrarily low chance of being wrong.

Which happens to be the very best degree of confidence available. If people only said things with zero chance of being wrong, it would be a silent world.

 

[Mobyseven]

It's not that, really. It's just that I can't tell you that there is not even 1/1-nonillionth of a chance that real human beings flipping real coins would not somehow unconsciously kill the streak if it lasted long enough.

I'm sorry, but this really is just silly.

Even if, even if there was some unconscious kill switch in our brains that implores us to sabotage the streak -- even in that case, we've still got a non-zero probability of getting a hundred heads. Because at each step of the way the coin still needs to be flipped.

The psychological state of the person is, for all intents and purposes, irrelevant as it pertains to the question of possibility. The system we are dealing with in reality is much larger than just the person; despite the best of intentions, the person can still succeed.

Of course, it's possible that the person might start to cheat...but provided they're still actually flipping the coin (as is required) they have a non-zero probability that their cheat will fail (I.e. the coin will keep landing heads).

So no: not even in this bizarre fantasy world of yours is there ever zero probability of getting a run of a hundred heads.

 

[Piggy]

Why do you think the real human being flipping a real coin would somehow unconsciously kill or sabotage a streak (that would have gotten heads) rather than continue or perpetuate a streak (that would have gotten tails)?

I have no clue. How in the world would I know the ins and outs of something that has such a small chance of being true?

 

[Piggy]

Of course you could say that. You couldn't say that with zero chance of being wrong, but you could say that with arbitrarily low chance of being wrong.

Which happens to be the very best degree of confidence available. If people only said things with zero chance of being wrong, it would be a silent world.

Perhaps, but regardless, this thread has pretty consistently been about zero-chances, not absurdly small chances.

 

[Piggy]

Let's imagine flipping a coin some extremely high number of times. A number that no one is likely to ever do, but this thread seems hypothetical anyway.

If a coin were flipped a billion times, what's the largest consecutive run of heads you'd statistically expect? And how would you make that calculation?

Alternately, if I wanted to know how likely a run of consecutive heads was in certain numbers of throws, I'd divide the number of possible sequences with that many heads by the total number of possible sequences.

Let's say a coin is flipped a billion times. There should be 2^billion possible sequences. How many would have 100 consecutive heads?



Or am I missing the entire point of the thread? I'm new here, and all.

There's a paper linked on the parent thread, btw, that discusses the number of flips needed to make certain streaks likely. I believe it's somewhere between 2,000 and 3,000 flips to make a streak of 10 likely.

I suppose that would mean figuring out the point where more than half of all such runs would contain a streak of that length.

But then you get back to the point someone made just upthread, that if you were to do a sufficient number of very long series of coin flips, some of them would show streaks at the beginning, others at the middle, others at the end.

There was a Radio Lab episode very recently that discussed coin tosses. As part of their experiment, the RL guys flipped a streak of 7 very early in (or perhaps at the start of, it's hard to tell) a run of 100.

Later, they talked to a statistician and said they flipped 7 heads, and he gave them the very long odds. But when they mentioned they'd made a total of 100 flips, he said, oh, that means it's only about 1 in 6.

So it goes from a long shot to about like throwing a dart at a calendar and hitting a Tuesday, based on what they did after making the streak.

Which makes me think of a scenario like, suppose me and my brother-in-law forgot who owed who 20 bucks, so we agree to flip a coin 10 times and bet $2 on each flip.

Well, we decide that he flips, he takes heads, and he gets 10 out of 10 heads.

I say, "Dude, that's gotta be a rigged coin", and he says, "No, you see, I always intended to flip this coin another 2,990 times, which is what I'm about to do, so this isn't unusual at all."

So I say, well, ok, and take off, but then I realize I left my smokes inside and I go back in and the guy's popped open a beer and is sitting with his feet up watching TV.

I say, "You lying SOB, you never intended to make those 2,900 flips... you were cheating me, after all!"

 

[Piggy]

I'm sorry, but this really is just silly.

Even if, even if there was some unconscious kill switch in our brains that implores us to sabotage the streak -- even in that case, we've still got a non-zero probability of getting a hundred heads. Because at each step of the way the coin still needs to be flipped.

The psychological state of the person is, for all intents and purposes, irrelevant as it pertains to the question of possibility. The system we are dealing with in reality is much larger than just the person; despite the best of intentions, the person can still succeed.

Of course, it's possible that the person might start to cheat...but provided they're still actually flipping the coin (as is required) they have a non-zero probability that their cheat will fail (I.e. the coin will keep landing heads).

So no: not even in this bizarre fantasy world of yours is there ever zero probability of getting a run of a hundred heads.

Very nice. Can't argue with that.

To get the zero we've got to go supernatural, and I just can't go there.

 

[Rodney]

To get the zero we've got to go supernatural, and I just can't go there.

So if someone defies odds of better than 1 in a million (or perhaps even 1 in a billion) and wins the JREF Challenge, would you conclude that something paranormal happened, or that s/he just got lucky?

 

[Perpetual Student]

So if someone defies odds of better than 1 in a million (or perhaps even 1 in a billion) and wins the JREF Challenge, would you conclude that something paranormal happened, or that s/he just got lucky?

Since there is no scientific basis or demonstration of any such thing as "paranormal," I would look for other scientific explanations -- including the occurrence of an unlikely probability.

 

[Mobyseven]

So if someone defies odds of better than 1 in a million (or perhaps even 1 in a billion) and wins the JREF Challenge, would you conclude that something paranormal happened, or that s/he just got lucky?

Personally? I'd sit up and take notice, but I'd wait for reproducibility before changing my mind on anything.