Impossible coin sequences?

[69dodge]

Regardless, I would still take my bet of my life against a Lotto jackpot that everyone on earth flipping coins for 10 years wouldn't come up with that streak.

1/2^100 is a hyper-astronomical number. To keep with the coin theme, if a coin were flipped every 10 seconds, I figure you'd have to flip it a little over 400 sextillion years before you'd made 2^100 flips. That's some 26 trillion times the age of the universe.

And each person on earth would be facing the same odds.

I agree with all of that.

But I don't see how it makes my position any less likely, because my position is not that someone flipping a coin will get 100 heads in a row, but rather that the probability of his getting 100 heads in a row is 1/2¹⁰⁰. So all of that improbability (26 trillion times the age of the universe, etc.) is built into my position already.

The question is not whether someone flipping a coin 100 times will or won't get 100 heads. He almost certainly won't. The question is whether the probability of getting 100 heads is 1/2¹⁰⁰ or 0. All that's required for the the probability to be 1/2¹⁰⁰ is that the coin be flipped fairly, which anyway everyone generally does assume is the case for normal coin flips. What's required for the probability to be 0 is that the coin, sometime before the hundredth flip, be wildly unfair. The latter seems much less likely to me than the former.

 

[Piggy]

Depends. Is it a dime or a shilling?

Good point.

 

[Perpetual Student]

This discussion reminds me of those statements that quantum theory allows for some object to suddenly pop up on Mars with a non zero probability (meaning some highly unlikely to the extreme probability). It's an interesting concept, that if quantum theory is utterly correct, must be true. Such statements may give us some perspective about quantum theory even though they are of little practical value. I think the same can be said of the probability of "100 consecutive heads."

 

[RecoveringYuppy]

This discussion reminds me of those statements that quantum theory allows for some object to suddenly pop up on Mars with a non zero probability (meaning some highly unlikely to the extreme probability). It's an interesting concept, that if quantum theory is utterly correct, must be true. Such statements may give us some perspective about quantum theory even though they are of little practical value. I think the same can be said of the probability of "100 consecutive heads."

Don't even think you need to invoke QM. With cosmic rays all over the place there is some small probability that Tom Hanks and a Wilson basketball will materialize, if only for an instant, at any point. Can the word impossible never be used?

 

[Piggy]

So what? The odds are exactly the same for each unique combination of 100 throws. And yet, each time we throw a coin 100 times, we get one such unique, equally probable or improbable combination.

Why is it so hard for you to see that?

It's not hard for me to see.

So given that fact, let me just ask you, why is it that there are innumerable combinations of sand grains on a beach which you would consider typical of what sand on a beach looks like, but if you were to walk out one day and find a perfectly regular pattern of sand, exactly smooth and arranged in neat rows and columns, this would catch your attention, to say the least?

The answer is that perfect regularity is not the way this kind of system behaves on that scale, which puts it into a different class of result from the enormous number of other possible patterns which are typical of how it behaves.

Ditto when you're using a radio dial. The space in between signals is characterized by its noisiness, and large-scale regularity and predictability -- whether that's a series of beeps or someone reciting the Declaration of Independence -- isn't characteristic of that system.

Would static ever spontaneously arrange itself into the sound of someone reciting the Declaration of Independence? I seriously doubt it.

 

[Piggy]

Why are you comparing them? Neither I nor anyone else has suggested that flipping 100 heads in a row is likely. We all accept that it's phenomenally unlikely, so what's your point?
I simply said that your excuse for how it becomes impossible is less likely to be true than not, so if we are to sit here and try to figure out if it is actually impossible, even at this first step it's looking more likely to be possible than not.

But, as I said, even assuming that this unlikely thing is true, it still doesn't make streaks of 100 heads impossible.

Why are you still discussing that particular point with me? I've already conceded that.

What I'm discussing now is a different question: Is it impossible (not just extremely unlikely) that a human being flipping a coin is not perfectly random and will not actually produce all possible combinations of, say 100 flips, no matter how long it goes on?

 

[Piggy]

So, you are now suggesting that people getting tired makes coin flips come up tails 100% of the time?

Nope.

But again, the question is not "Is this likely?" or "Would we expect it?" but -- just as with the run of 100 -- does this have a zero probability?

 

[Piggy]

Any evidence of any possible mechanism that could cause that to happen?

Nope. See above.

 

[Piggy]

That bolded part is just wrong. Saying that 1 in 2^100 is non-zero is not 'an assertion'. It's mathematical fact. Having to demonstrate it happening in real life in order to prove that it's true is an arbitrary and unreasonable condition for you to apply.

Sheesh. When I originally chipped in to this thread I thought you were being misunderstood.

You may have missed the part where the topic changed.

The question now is, what are the odds that a human flipping a coin, if it could go on indefinitely, would not in fact produce all possible combinations at any given scale? In other words, what are the odds that it is in some way biased in a way that would prevent extremely long streaks from happening?

I happened to ask, in one post, if the odds were less than 1/2^100 because it's already established that even those long odds do not equal zero.

 

[Piggy]

That's not true -- I recall further up the thread someone explaining where it comes from. To wit: every time the coin is tossed there is a non-zero probability that it will come up heads. That, together with the fact that each coin toss is an independent event, gives you a non-zero probability that a run of 100 heads will occur.

If you could guarantee that each flip were independent, I'd have no problem. I'm sure we could build machines to satisfy those conditions (or reasonably sure). But when dealing with a brain, arm, and hand, I can allow myself at least 1/1-nonillionth of a doubt.

 

[alexi_drago]

I think the problem is that most people perceive one random looking sequence to be more like another random looking sequence than one that looks like it has some sort of order.
Out of the 2^100 possible sequences the vast majority look more random than ordered and therefore it's more likely that a random looking sequence will be thrown.

I see it kind of like this:

Say there are 20 (or 2^100) evenly matched horses in a race, all are black except one which is brown, it's most likely that a black horse will win even though not one of those black horses has a better or worse chance than the brown horse. So do you have a better chance by betting on a black horse?

 

[Mobyseven]

If you could guarantee that each flip were independent, I'd have no problem. I'm sure we could build machines to satisfy those conditions (or reasonably sure). But when dealing with a brain, arm, and hand, I can allow myself at least 1/1-nonillionth of a doubt.

To be honest, I don't even strictly need to guarantee independence of events, just that at each coin toss both sides have a non-zero probability of occurring. Doesn't matter whether it's a machine doing the flips or a human.

If you want to say otherwise, then either you've stopped talking about coins as we know them, or you mean something very different to the rest of us when you talk about a coin toss.

 

[Roboramma]

Why are you still discussing that particular point with me? I've already conceded that.

I appologise, I thought you had conceded that it may be possible, but were continuing to argue that it may be impossible as well, and we simply can't tell either way.
I was making the argument that it cannot be impossible.

If you have accepted that it is impossible, I'm sorry for belaboring that point.

What I'm discussing now is a different question: Is it impossible (not just extremely unlikely) that a human being flipping a coin is not perfectly random and will not actually produce all possible combinations of, say 100 flips, no matter how long it goes on?

Those are two different questions. It isn't necessary that a human being flipping a coin be perfectly random in order for enough iterations to produce all possible combinations. But Mobyseven says that better than I can.

 

[Roboramma]

Nope.

But again, the question is not "Is this likely?" or "Would we expect it?" but -- just as with the run of 100 -- does this have a zero probability?

Does what have a zero probability? That people getting tired causes coin flips to come up tails 100% of the time? That's not a probability question. It's simply not possible. The "messiness" of the world rules it out, necessarily, as the coin is affected by things other than the tiredness of the tosser.

 

[Roboramma]

You may have missed the part where the topic changed.

The question now is, what are the odds that a human flipping a coin, if it could go on indefinitely, would not in fact produce all possible combinations at any given scale? In other words, what are the odds that it is in some way biased in a way that would prevent extremely long streaks from happening?

I happened to ask, in one post, if the odds were less than 1/2^100 because it's already established that even those long odds do not equal zero.

As I said earlier I'm happy to accept that the odds may be less than 1/2¹⁰⁰ and that there could be things influencing the system that we are not aware of.
However, as I and others have pointed out, in order for the odds of any particular combination to equal exactly zero, the odds of some particular flip in some particular circumstance coming up with a particular result have to equal 100%.
With a normal human executing a normal coin flip, I can't see that. Hell, even a trained coin flipper can make mistakes.

 

[Thabiguy]

What I'm discussing now is a different question: Is it impossible (not just extremely unlikely) that a human being flipping a coin is not perfectly random and will not actually produce all possible combinations of, say 100 flips, no matter how long it goes on?

The question now is, what are the odds that a human flipping a coin, if it could go on indefinitely, would not in fact produce all possible combinations at any given scale? In other words, what are the odds that it is in some way biased in a way that would prevent extremely long streaks from happening?

Although the "new question" may look superficially similar to the original thread topic, it is a very different question - here you are using the word "odds" in a way that requires another interpretation of the whole concept of probability.

Q1. Is it impossible (P1=0) for a sequence of 100 heads to come up when a coin is fairly tossed 100 times?

Q2. Is it impossible (P2=0) for "humans flipping coins" to be unfair in such a way that they are prevented from getting a sequence of 100 heads?

The first question asks about a fair, random experiment, and makes it clear what constitutes a trial and what constitutes a success. It implies frequentist interpretation of probability P1, and in that context, it can be easily and clearly answered. (The answer is "no, the probability is very low, but non-zero".)

The second question doesn't define any fair, random experiment. It isn't clear what constitutes a trial: are we somehow supposed to construct humans randomly (and what would still constitute "humans flipping coins"?) and analyze how often we may end up with humans unable to flip 100 heads in a row? That doesn't make any sense, and the question doesn't ask that anyway. It refers to a single scenario that's already been set up - existing humans either being prevented from flipping 100 heads in a row or not - and asks about the "probability" of the answer (which is already established but unknown to us) being this or that.

The second question therefore intrinsicially implies Bayesian interpretation of probability P2 and it is a question of our belief or confidence in something that we can't know.

Considering that results of real humans flipping coins are in principle affected by the entire observable universe (it's possible for a meteorite to fall nearby, shake the ground and alter the result; it's possible for a cosmic ray to strike a neuron controlling the arm and alter the result, etc.), the hypothesis that humans are prevented from flipping 100 heads is actually a statement about the entire observable universe being set up in a particular way that rules out humans flipping 100 heads. It would therefore seem that the upper bound of probability P2 (our confidence in this hypothesis) might be extremely low, dramatically lower than 2^-100 (but let's not forget that probability P1=2^-100 is a different kind of probability and the two can't be directly compared).

Could we rule out the hypothesis altogether, though (P2=0)? It seems that we can't, because, after all, we could always be living in a virtual reality which was for some weird reasons set up to prevent flipping 100 heads.

The answer to Q2 might therefore ultimately be the same as the answer to Q1 ("no, the probability is very low, but non-zero"), but it is important to note that the two probabilities P1 and P2 represent different concepts that cannot be directly mixed.

 

[Piggy]

To be honest, I don't even strictly need to guarantee independence of events, just that at each coin toss both sides have a non-zero probability of occurring. Doesn't matter whether it's a machine doing the flips or a human.

If you want to say otherwise, then either you've stopped talking about coins as we know them, or you mean something very different to the rest of us when you talk about a coin toss.

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.

 

[Piggy]

Does what have a zero probability? That people getting tired causes coin flips to come up tails 100% of the time? That's not a probability question. It's simply not possible. The "messiness" of the world rules it out, necessarily, as the coin is affected by things other than the tiredness of the tosser.

Can you describe the system perfectly?

I can't. Don't know anyone else who can.

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.

 

[Piggy]

The answer to Q2 might therefore ultimately be the same as the answer to Q1 ("no, the probability is very low, but non-zero"), but it is important to note that the two probabilities P1 and P2 represent different concepts that cannot be directly mixed.

This is, indeed, all that I was trying to say.

 

[alexi_drago]

Write down any random looking sequence of 100 heads and tails, the exact same discussion as has gone on here would apply equally to that sequence as well as the 100 heads.