Impossible coin sequences?

[Alan]

That predicting a sequence of 100 coin tosses beforehand, even with many repeats, is rather ludicrously unlikely. So is, of course, any state of the world that derives from a series of essentially random earlier events.

It isn't a question of predicting what will happen.

 

[Roboramma]

Piggy is not saying that such sequences are just so unlikely that they might as well be impossible. He's saying that they are literally impossible.

And he beleives that to be the case because the real outcome of fair coin tosses differs from the outcome predicted by probability theory.

Here's a quote:

It means that I do not assume that idealizations trump observation. Mathematical models are always, to some extent, idealized... they've got some of the nubby bits of reality scrubbed off of them.

Trouble is, those nubby bits can sometimes be important.

What I observe is a world that has a level of variability and volatility which limits the range of "streaks" in dice rolling, coin tossing, and card picking if the conditions are not rigged.

Basically he thinks some unknown physics somehow stops streaks from going on for too long.

What I'd like to know is if he thinks short streaks (of say ten heads) are less common than predicted by theory, or if it's only extremely long streaks that are less common, and if the latter how he came to that knowledge.

 

[Alan]

It's a belief in micro-streaks but not macro-streaks, to paraphrase something from discussions about evolution.

Big streaks are just lots of little streaks put together.

 

[TubbaBlubba]

It isn't a question of predicting what will happen.

In a way, it is. In the set containing all possible sequences, only a nearly infinitesimal fraction hold any significance to us, such as HTHTHTHTHT..., TTTTTTTT... and HHHHHHHH...

Landing one of these would be equivalent to writing down an equal number of random sequences, and then by chance flipping it. This is as unlikely, but entirely possible, as you could always have written down whatever sequence DOES pop up. Likewise, any sequence that pop us, could by the grace of culture and evolution, have happened to have special significance to us.

 

[Alan]

I do not consider calling something possible like this to be a prediction, but I don't understand the relevance of the example to that so perhaps I misunderstood and we are talking about different things.

 

[TubbaBlubba]

The only reason we're talking about "100 heads" is because the 100 heads sequence happens to hold a cultural (cognitive, whatever) significance for us, indeed it is the only thing separating it from the most of the set of all possible sequences. A predicted sequence would hold a similar significance, and should thus be equally possible or impossible.

If the sequence HHHHH... held no significance to us, but instead HTHTHTHHHTTHTTTHTHTTHTHTHTHTHHTHTTHHTHTTTT... did, we'd be talking about that instead.

 

[Alan]

I know both of those things but I still don't know why that means that thinking something is possible is a prediction. Something doesn't need to ever happen in order to be possible. It is late now and I am tired so maybe it will make sense in the morning.

I am saying that all of the combinations are possible.

 

[TubbaBlubba]

I know both of those things but I still don't know why that means that thinking something is possible is a prediction.

That's not what I'm saying, I'm saying that the HHHHHHHHH... sequence is only removed from others in that it has a special significance to us. By writing down a sequence ("predicting it") before tossing a coin, you give that sequence a similar significiance.

I am saying that all of the combinations are possible.

Not only that, equally likely.

 

[Alan]

Yes.

I have basically been saying that in the other thread for days, or perhaps wrote it some times but never submitted it.

I am not specifically referring to getting 100 heads any particular time but it being possible like any other. That is why I don't like it being characterised as a prediction even though that is possible too.

 

[TubbaBlubba]

I'm not characterising it as a prediction, I'm just saying that applying significance to a sequence does not alter its likelyhood or possibility of appearing.

 

[Alan]

Okay. I understand what you meant now.

 

[godless dave]

The math for this is beyond me, but I just wanted to mention this scenario was on "Fringe" last night. John Noble's character flipped a coin ten times in a row and it came up heads every time. This indicated to him that something fishy was going on with the laws of physics in the area.

 

[TubbaBlubba]

Another analogy, if anyone else misunderstands me, would be to interpret the tosses as bits (say H = 1, T = 0) and look for something like a sequence of the first primes.

The importance being, of course, that you determine a significant coin toss before investigating the sequences, rather than looking for significance in already existing sequences (this would be the Texas Sharpshooter Fallacy, drawing the target around the bullet hole)

 

[TubbaBlubba]

The math for this is beyond me, but I just wanted to mention this scenario was on "Fringe" last night. John Noble's character flipped a coin ten times in a row and it came up heads every time. This indicated to him that something fishy was going on with the laws of physics in the area.

That would be a very reasonable conclusion, indeed, it probably makes a good reality check for those who enjoy lucid dreaming.

 

[Elaedith]

I have a problem conceptualising this but I don't think it is simply the 'gambler's fallacy'. I take the gambler's fallacy to refer to the belief that a coin has some type of memory, and is less likely to lead heads if it has the last 99 times. I have no problem realising that this it a fallacy, or that any sequences of letters and numbers is just as likely as the same set of letters or numbers in any other sequence regardless of the length of the sequence.

Where I get into trouble is with seeing the 100 coin tosses as a 'sample' of coin tosses. A small sample of anything is more likely to depart from chance expectations than a large sample. If the coin has 50/50 chance of landing heads and tails then you would expect that a small sample of tosses is more likely to depart from 50/50 heads/tails overall than a large sample. So surely the probability of getting 70% heads or tails rather than 50% is more likely with 10 tosses than with 100. If 50/50 is predicted by chance alone (the null hypothesis), then any departure from chance expectations becomes less likely to occur when the null is true as the sample size gets bigger. So how is it that a sequence containing 100 heads can be just as probable as a sequence containing 50/50 heads and tails? Perhaps I am confusing the concepts of sequence and sample here but I have trouble distinguishing them in my mind.

I would appreciate it if anyone can explain this is a way that makes sense, as I have to teach statistics (depsite not really having any maths background) and this issue keeps nagging at me.

 

[sol invictus]

A small sample of anything is more likely to depart from chance expectations than a large sample. If the coin has 50/50 chance of landing heads and tails then you would expect that a small sample of tosses is more likely to depart from 50/50 heads/tails overall than a large sample. So surely the probability of getting 70% heads or tails rather than 50% is more likely with 10 tosses than with 100.

Yes - much more likely.

So how is it that a sequence containing 100 heads can be just as probable as a sequence containing 50/50 heads and tails?

It's as probable as any other specific sequence. But there is only one sequence with 100 heads, while there are a very large number with a total of 50 heads and 50 tails. So 100 heads is as probable as any other sequence, but it is much less probable than obtaining a total of 50 heads and 50 tails (i.e., of obtaining any one of the many sequences that contain 50 and 50).

Does that help? If not, here's an example with 4 coin tosses. There are 16 possible sequences (you can write them out if you like). Of those, one has 4 heads, and six have 2 heads and 2 tails. So while the 4 head sequence is just as likely (1/16) as any other, you are six times as likely to flip a total of 2 heads as you are a total of 4 heads.

Since we are often concerned not with the sequence itself, but merely the total, it's easy to lose sight of that distinction.

 

[Elaedith]

Yes - much more likely.



It's as probable as any other specific sequence. But there is only one sequence with 100 heads, while there are a very large number with a total of 50 heads and 50 tails. So 100 heads is as probable as any other sequence, but it is much less probable than obtaining a total of 50 heads and 50 tails (i.e., of obtaining any one of the many sequences that contain 50 and 50).

Does that help? If not, here's an example with 4 coin tosses. There are 16 possible sequences (you can write them out if you like). Of those, one has 4 heads, and six have 2 heads and 2 tails. So while the 4 head sequence is just as likely (1/16) as any other, you are six times as likely to flip a total of 2 heads as you are a total of 4 heads.

Since we are often concerned not with the sequence itself, but merely the total, it's easy to lose sight of that distinction.

Yes it does make sense, but I think that focusing on content rather than sequence probably reflects the way the question is often worded.

 

[ddt]

Yes it does make sense, but I think that focusing on content rather than sequence probably reflects the way the question is often worded.

Indeed, and then it's easy to lose sight that "100 heads" and "50 heads, 50 tails" are not equally probable events. Most of these probability problems can simply be solved by counting - whether it's coin flips or die throws or the birthday problem - but you first must make sure you count indeed equally probable events.

 

[Thabiguy]

I have six yes-or-no questions for Piggy.

1) You claim that the sequence
HHHHHHHHHHHHHHHHHHHHHHHHH
HHHHHHHHHHHHHHHHHHHHHHHHH
HHHHHHHHHHHHHHHHHHHHHHHHH
HHHHHHHHHHHHHHHHHHHHHHHHH
(100 heads) can never be produced in the real world by tossing a fair coin. When tossing a fair coin 100 times in a row, this outcome is impossible, and its probability is zero. Is that correct?

2) Is the sequence
TTTTTTTTTTTTTTTTTTTTTTTTT
TTTTTTTTTTTTTTTTTTTTTTTTT
TTTTTTTTTTTTTTTTTTTTTTTTT
TTTTTTTTTTTTTTTTTTTTTTTTT
(100 tails) also an impossible outcome?

3) Is the sequence
HTHTHTHTHTHTHTHTHTHTHTHTH
THTHTHTHTHTHTHTHTHTHTHTHT
HTHTHTHTHTHTHTHTHTHTHTHTH
THTHTHTHTHTHTHTHTHTHTHTHT
(100 alternating heads and tails) also an impossible outcome?

4) Is the sequence
TTHTTTHHTTTHTTTTHTHHHHHHT
THTHTTTHHHHHTTTTHTTTTTTTT
HTHHHTTHTHHTTHTHTHTHHHHHH
HHHHTTHHHHHTHHHTTHHTTTHHT
(from first 100 decimal digits of pi, odd digit=T, even digit=H) also an impossible outcome?

5) Is the sequence
TTHHTTTTTTHTTTTHTHHTHHTTT
TTTTTTHHHTHHTHHTHHTHTTTTH
HHTHTTHTHTTHHHHHTTHHTHHHT
HTHTTTHTTTHHTTHTTHHHHTHTH
(from the following text, odd word length=T, even word length=H) also an impossible outcome?

"Tossing a fair coin can never produce one hundred heads in a row. It's not possible. The laws of our universe prohibit tossing one hundred consecutive heads. One hundred tails seems impossible by that logic as well. But apparently, we can toss some sequences of heads and tails, therefore they can't be all impossible. The obvious question therefore is: why are some of them impossible? What makes a sequence so special that it cannot occur in the real world? How can we determine whether a particular sequence can occur or not? Maybe we'll find an answer somewhere in the thread."

6) When one tosses a fair coin 100 times in a row and gets a particular sequence, is it impossible for that sequence to ever occur again?

 

[W.D.Clinger]

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.

1267650600228229401496703205376

which is approximately 1.2676506002282294e30

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.

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

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, and a correspondingly longer amount of time spent flipping

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.