Impossible coin sequences?

[Squeegee Beckenheim]

I have no way of knowing.

Do you?

Mathematics is quite a good way of working these things out.

 

[Piggy]

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?

If the system is truly random and unbounded, any finite number of identical results could not be outside the results space.

 

[Piggy]

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.

It's that last bit that introduces the uncertainty.

When it comes to human hands and air, can you describe the results in terms of the physical system?

 

[Piggy]

Mathematics is quite a good way of working these things out.

What things?

By that I mean, math is very good at describing the probabilities of a system whose operations are exactly known -- e.g., a physical system that's bound to exhibit every possible configuration of its parts in a truly random manner -- but where there are unknowns, math can't reach.

 

[Alan]

It will be approximately 50/50, as I have explained before.

How do human hands and air and metal and so on change the likelihood of it getting any other sequences?

 

[Piggy]

Yes, achieving the feat in question would not be cheap.

Here follows an attempt to estimate the cost of getting a run of 100 heads.

I'm going to assume we don't have to use actual coins. Quantum processes are much fairer than metal coins, and we can generate random bits much faster using quantum processes than by flipping metal coins.

For a few billion dollars, we could probably design a machine that flips on the order of 1 billion random bits per second. With mass production, we might be able to get the unit cost down to something like a thousand dollars per machine. For a mere trillion dollars, we could build a billion of those machines. If each machine consumes 100 watts, then the cost of running each machine for ten years would be comparable to the initial cost of the machine. Let's say each machine wears out after ten years and has to be replaced, so the annual cost for operating a fleet of one billion machines is about two hundred billion dollars.

If each machine were to generate 100 random bits before examining the results, we'd expect to operate that fleet of machines for about 4 million years. We can do a lot better by programming the machines to give up on a sequence as soon as the first tail comes up. Then we'd expect to complete our project in less than 80 thousand years, at a total cost of only 16 thousand trillion dollars.

No one said it would be cheap.

It's not impossible, just unlikely. If we're lucky, the project might finish within its first year of operation, much sooner than scheduled and well under budget. If we're unlucky, it might take millions of years, but what's the point of worrying about that? We might as well be optimistic.

Oh, thank you! I love this.

My only beef is that I would require actual coins, to be sure we got it right.

 

[Piggy]

It will be approximately 50/50, as I have explained before.

How do human hands and air and metal and so on change the likelihood of it getting any other sequences?

How on God's green earth should I know?

 

[Alan]

The problem with that is the same could be said about every other single combination.

 

[Alan]

How on God's green earth should I know?

I like that response. It's about the same as all heads.

 

[Piggy]

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.

True, but it doesn't stop it from being indistinguishable from randomness from the perspective of the people using it.

Set up a Diaconis device with a computerized switch that excludes runs of, say, over 25, but randomizes the results otherwise, and nobody in our world will be able to detect that it's not truly random.

In other words, it will exhibit behavior that cannot be distinguished -- within the given timeframe -- from the behavior of a similar system which does not place any such limits on streaks.

That being the case, if the results-spaces are identical from the perspective of the people using the system, I don't see any way to argue that it's not effectively random for the purposes they're using it for.

 

[Piggy]

The problem with that is the same could be said about every other single combination.

Ok, but why are you looking at the problem that way?

 

[Piggy]

I like that response. It's about the same as all heads.

Fine. Do you have a different response to the same question? Do you know how to figure out what that system is doing?

 

[plumjam]

The chance of getting one head in a row is one half, two heads a quarter and so on.

The chance of getting 100 heads in a row is one in 2^100 (that's 2 x 2 x 2 ... with 100 twos)

That's a big number - it's approximately 126765 followed by twenty five zeros.

The number of atoms in a coin is only about 5 followed by 22 zeros.

Now lets assume that every time you flipped a coin you caused one atom to be worn away from it. Let's also assume that once 20% of the coin has worn away, you can no longer tell heads from tails, so you have to swap to a new coin.

If the coins were nickles, you'd (on average) have to wear out about six million dollars worth for each run of 100 heads to occur.

Now you can already see that it's practically impossible to get a run of 100 heads. And, of course, the assumption about only wearing one atom away per flip was ludicrous. No matter how careful you are, you would cause much more wear than that so you'd actually need a whole lot more than $6 million...

So it's not a 6 million dollar question then.

 

[69dodge]

... math is very good at describing the probabilities of a system whose operations are exactly known -- e.g., a physical system that's bound to exhibit every possible configuration of its parts in a truly random manner -- but where there are unknowns, math can't reach.

Math is good at deducing which consequences absolutely must follow from given definitions and assumptions. Definitions are of course true by definition, and whether the assumptions hold in the real world is up to experiment to decide. But if the assumptions do hold, you can be sure that the consequences deduced by math also hold.

Getting back to coins, if on a single flip, a coin might come up heads and it might come up tails, and if also there is nothing in the coin (or the flipping mechanism) that remembers the results of previous coin flips and alters the next flip based on them, then it follows inexorably that 100 heads in a row are possible.

So there are two choices: 1) 100 heads in a row are possible, or 2) If you've already flipped 99 heads (or possibly some smaller number), the coin somehow knows this, and will definitely come up tails on the next flip.

Now, I think it's reasonable to say that a coin is not being flipped fairly if it will definitely come up tails.

So, by definition of "fair", there's a chance that a fair coin which is flipped 100 times will come up heads every time.

Perhaps there are no fair coins in the world. You can consistently maintain that position, I suppose. But just be aware of what it implies.

 

[sol invictus]

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

It illustrates the absurdity of the claim that 100 head runs are impossible. I was hoping it would hence add something to your understanding of this system - but days later you still haven't responded to my questions about it, so I guess not.

By the way there's another argument that's rather clear. You assert 100 head runs are impossible, p(100)=0. You also agree that 5 head runs are possible, p(5)>0. At some point someone asked you how many heads in a row are needed before it becomes impossible, and you said you doubted there is "a bright line".

But there must be a bright line if you are correct: if p(5)>0 and p(100)=0, there must be an N such that p(N)=0 but p(N-1)>0. Perhaps N=100, or perhaps N<100, but there must exist such an N.

That means that if we flip N-1 coins often enough, eventually we are certain to get a run of N-1 heads (since p(N-1)>0). And then, if we flip once more, we are absolutely certain to get tails. With probability 1, we must get tails. Our coin is magic - it refuses to be fair, because it cannot allow a run of N heads.

And that, of course, is utterly absurd (not to mention in contradiction with the adjective "fair").

 

[AdMan]

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.

You are the one saying that a particular result (i.e., 100 heads in a row) may not be possible.

What everyone else here is saying is that is not correct--all possible unique combinations of 100 random throws is equally possible.

Despite all the verbosity, you still have not demonstrated that throwing 100 heads in a row is not possible.

 

[Piggy]

Math is good at deducing which consequences absolutely must follow from given definitions and assumptions. Definitions are of course true by definition, and whether the assumptions hold in the real world is up to experiment to decide. But if the assumptions do hold, you can be sure that the consequences deduced by math also hold.

Getting back to coins, if on a single flip, a coin might come up heads and it might come up tails, and if also there is nothing in the coin (or the flipping mechanism) that remembers the results of previous coin flips and alters the next flip based on them, then it follows inexorably that 100 heads in a row are possible.

The Diaconis machine remembers 100%.

What do your brain, arm, and hand remember?

I can't answer that question, and I don't know anyone who can.

 

[Piggy]

It illustrates the absurdity of the claim that 100 head runs are impossible. I was hoping it would hence add something to your understanding of this system - but days later you still haven't responded to my questions about it, so I guess not.

By the way there's another argument that's rather clear. You assert 100 head runs are impossible, p(100)=0.

Did but don't.

In any case, when you ask, effectively, "What can be deduced from a system if I randomly misassign your data about that system?", you're asking a tangential question, to say the least.

The issue was never whether a series of 100 values of "T" or "H" was impossible for any system you could imagine.

It was whether or not you'd ever actually encounter a run of 100 heads or tails in a fair coin-flipping system, or a human one, on earth.

 

[Piggy]

You are the one saying that a particular result (i.e., 100 heads in a row) may not be possible.

What everyone else here is saying is that is not correct--all possible unique combinations of 100 random throws is equally possible.

Despite all the verbosity, you still have not demonstrated that throwing 100 heads in a row is not possible.

I've already said I was wrong about that.

But the issue of whether or not a given flipping sysem -- whether a machine or someone's thumb -- will or won't ever give you a streak of 100 is a different question.

 

[Mister Earl]

Seems to me a fairly easy program to do just that could be written in qbasic, with the randomizer re-randomized via system time for as near random as you can get easily.

Something along the lines of
3 T = 0
7 Y = 0
10 X = rnd(1),0
20 if X = 1 then Y = Y + 1
30 if X = 0 then T = t + 1: print "Only "; Y;" heads.":goto 7
40 if Y - 100 then print "Got 100 heads! Took "; T;" tries."
50 end

(Haven't done qBasic in aeons, so I may have made an error there)

#EDIT: Couldn't remember how to randomize the timer heh. I could look it up...