Double Headed Coins and skepticism

[Alan]

Something does not have to happened in the past in order to be possible.

 

[Roboramma]

I seriously doubt it. Unless I moved into mathworld, which I don't plan to do.

If the bet were based on a series of randomly generated numbers by a computer, and not actual coins, would you still take it?

(understand you've gone to bed, maybe a response tomorrow. )

 

[This is The End]

I'm going to try and clarify the question to Sol so we can get some solid numbers from a few people who feel like confirming the math.

We will have 2 scenarios.

Scenario 1: The people start over after every miss (after every tails).

Scenario 2: The people start over after every 100 flips.

There are 7 billion people flipping.

Let's say it takes 1 second per flip, just to keep the math easy.

How long should a run of 100 heads take in each scenario?

 

[Guybrush Threepwood]

I used to teach test prep for exams like the SAT, GRE, LSAT, and such, and I can tell you that the human brain is indeed very bad at statistics when it comes to abstract problems, and situations -- such as the shared birthday question -- that are simply of no importance in everyday real life.

Vegas, of course, intentionally manipulates the casino set-up to exploit that fact.

I don't think that prayer and homeopathy are believed in because of a failure to comprehend the stats intuitively... seems to me there are other psychological blind spots at work there.

But consider that even babies who have not yet learned to talk are cracker-jacks at the kind of stats that are important to us in everyday life.

Babies will stare measurably longer at unexpected events than they will at expected ones, and we can use that fact to judge when they sense that something is fishy.

If a baby is looking at a clear plastic box filled with red and white balls, and the white ones greatly outnumber the red ones, for instance, and an adult removes several red balls in a row from the box, the baby shows perplexity at this situation. S/he knows something is wrong with that.

Also, we know that babies will attend longer to novel sounds. But only recently has it been discovered that when babies are exposed to various pairings of sounds, it doesn't take long for them to learn which pairings are common and which pairings are rare, and to respond with increased attention when a rare pairing occurs with an unexpectedly high frequency.

Turns out, our brains are naturally built for statistics, but only within the realm of what biology has determined is particularly useful.

This is why we're flummoxed by the birthday problem, but we will refuse to bet on a coin that comes up heads too often in a row, because we know the real universe does not work that way.

If you've got a fair coin and a fair set-up -- normal atmosphere, a human hand doing the flipping, etc. -- then the actual randomization of the world we live in sees to it that 100 heads in a row never actually happens. Yes, it's possible on paper, but not in our actual universe.

Fair enough, we are using entirely different interpretations of the phrase 'good at statistics'.
I don't regard the ability to recognise that an outcome that is 8 or 10 standard deviations from the mean is unusual as deserving of the description 'good at statistics'. You apparently do.

 

[sol invictus]

I'm going to try and clarify the question to Sol so we can get some solid numbers from a few people who feel like confirming the math.

We will have 2 scenarios.

Scenario 1: The people start over after every miss (after every tails).

Scenario 2: The people start over after every 100 flips.

There are 7 billion people flipping.

Let's say it takes 1 second per flip, just to keep the math easy.

How long should a run of 100 heads take in each scenario?

It's not very different considering the numbers involved. In 1, you save the time that would have been devoted to flipping out the remaining 100 after the first tails. In other words, the difference between the two are the flips-that-weren't.

How many flips-that-weren't are there? Well, on average you'll have flipped tails after 2 seconds. Therefore flippers in scenario 1 will save (on average) 98/100 flips - in other words they will go 50x faster towards their goal. But as we saw, in scenario 1 it would take 10^20 seconds for 10 billion flippers to achieve 100 heads in a row - so in scenario 2 it will take more than 10^18 s (still more than the age of the universe, but now only by a factor of 2 or 3).

 

[Cuddles]

I don't know why the physics of this world prohibits very long streaks, but that's the way it is in practice.

What a truly bizarre claim. Are you seriously arguing that because you have never seen an event that is so unlikely you should not expect ever to see it, that event must therefore be impossible?

 

[blutoski]

I think we're still just arguing varying levels of possibility here. Piggy just seems to have considered 'same person wins the one in a billion weekly lotto every week of their life' just as unlikely as flipping 100 heads. Not even close. It's more along the lines of 'same person wins the one in a billion weekly lotto once or twice in their life.

Someone else can do the math, but if all 7 billion people on the planet started flipping tomorrow, a lucky 'winner' would flip 100 heads fairly quickly*.

(I was going to suggest using a computer to simulate the above, but Piggy likes real world scenarios.)

*Also, as I said in a previous post, it is way more likely, and would happen way sooner, if the people get to start over after every miss (tails) instead of after every 100 flips (which would be a true 100 in a row test).

I've actually run simulations to see how quickly a sequence would appear on average. That may be the best way to approach this question.

I think piggy's overfocused on 100 heads as if it's somehow unique. Every sequence of 100 reslts is equally likely or unlikely. I think another poster made this point already, but it's worth repeating.

It's just human nature to be totally unimpressed with this sequence: TFTTFTFFFTFTTTTFFFTF, but the odds of that are 1/(2^20) which is just as miraculous a result at TTTTTTTTTTTTTTTTTTTT. The only reason it's 'less noticeable' is our overall cultural numerical illiteracy.

 

[Alan]

About the odds given for sadhatter getting ten 100s in a row, I am dissatisfied with those. The odds of some (not one specific) 'notable' combination are worth calculating, since that's not the only combination people would accuse him of being mistaken about. Ten 1s, ten 2s, ten 3s, ten 4s, a 1-2-3-4-5-6-7-8-9-10 sequence, a 49-50-51-52-etc sequence or so on. It should be divided by however many reasonably suspicious notable sets there can be, then by how many times he's rolled the die or seen somebody do it (since I think he just said he saw it).

 

[Simon Bridge]

Can we please come back to the article? Thank you. It is not the intent of this thread to discuss what counts as "possible" or not.

Part of the article demonstrates a way of assessing how confident one can be of a claim given the evidence, as the evidence mounts.

For example: skeptics are often criticized for being closed-minded about paranormal claims ... but we can show that we do not need very many honestly determined disproofs for any particular type of claim to be very confident that no such claim can be true. When testing a particular claim, we are wise to be very generous in our prior assumption of "innocence" on the part of the claimant. However, we are aware of a history of such testing of thousands of such claimants, none of which have demonstrated anything paranormal. Thus, we would be very silly, on the basis of that result, to go about our everyday life as if paranormal abilities exist.

I don't think anyone here will disagree about this - the difference here, I hoped, was that having a quantifiable example could be illustrative of many aspects of our interactions with believers. I had hoped that it would be that aspect which would interest members here.

Still, at east there is some discussion.

 

[Simon Bridge]

Because it disconfirms, to an extremely high degree, the null hypothesis of random chance.

No it doesn't. Any particular sequence of numbers has the same odds.

To put it another way, there are many, many combinations that are consistent with random chance.

to put it another way: how do you sort out those sequences of numbers that are consistent with random chance from those that are not consistent?

However, the set of combinations that is consistent with cheating is much smaller. So when we see a result that belongs to the set that is consistent with cheating, a bunch of red flags go off.

You are saying, for example, that there is only one way of getting HHHHH, and many ways of getting any other combination - so the odds against are pretty long (in this case you'd reject chance at 95%).

However: there is only one way of getting HTHHHT too. Many ways of not getting it. Would you consider this result more or less likely to occur by chance alone?

Have a look at: The Longest Run of Heads ... when people are asked to simulate a number of random tosses, they always produce one with too few long runs in it.

If you are comparing the probabilities of getting 5 heads on 5 tosses as opposed to any other combination you are right. However, the way the example is set up is properly thought of as a sequence not a combination ... the math for any other sequence is identical. The same calculation will "prove" that no possible sequence can occur by chance ... you need to revise the calculation.

The key word is "cheating". Some particular result is attributed a special value - perhaps you win $1 each time you roll H? In which case, there are many ways of losing and only a few of winning. However, on the strength of the argument posed in my article, I will maintain that you have rejected the hypothesis too soon.

If we are talking about 5 successfully called tosses in a row then your analysis would be correct in that it is as unlikely as getting any particular sequence out of all possible sequences. Of course we note that rejecting chance as an explanation of events is not the same as accepting any other particular explanation - which is where other peoples discussion of controls and so on comes in.

It's quite difficult to talk clearly about probabilities. Part of the idea of discussing the article here is to discover where I am confusing people (and where I'm getting confused) and rewrite to compensate.

 

[Simon Bridge]

A Fair Coin Revisited
... the first paper: "A Fair Coin", basically started with an empirical probability estimate before launching into the bayesian analysis. It has been pointed out in this forum that regular hypothesis testing is good enough ... in the link (above) I have had a formal go at the more usual math.

Summary: revising the experiment to the case that you have a randomly pick a coin with a known 99% chance of the coin being fair and 1% chance of being double-headed, using only forward probabilities you would reject the hypothesis (that the coin is fair) to 95% confidence after 5 heads in a row ... compared with bayesian statistics for a 0.99 prior which suggests 11 heads to be more reasonable.

The two statistical approaches disagree strongly for low numbers of runs - the rejection level for a single toss being above 0.5 for forward probabilities and close to 0 for the bayesian.

This suggests that relying on forward probabilities in hypothesis testing will lead us to reject chance as an explanation for events significantly too soon. But check my math, I was a tad rushed.

 

[Simon Bridge]

I live in the real world of weight and heft and air currents and muscles and neurons.

And in this world, a fair coin will not come up heads or tails on 100 consecutive fair flips.

We may not understand exactly why our world works that way, but it does.

Its a red herring.

Piggy is saying that the math used in this thread involves idealized circumstances - however, in nature, with long runs of the experiment, small effects can accumulate to disturb the classical statistics ... so we leave regular stats and enter chaos math and fractals.

And it works both ways - chaos effects can reinforce runs as well as disrupt them.

However, the effects are very very small ... the run needed to show chaos effects in this example is much longer than that needed in classical statistics to conclude that, to high confidence, that the coin is double-headed.

So I doubt this is really something I need to include in the article ... unless Piggy can come up with a study?

 

[Piggy]

If the bet were based on a series of randomly generated numbers by a computer, and not actual coins, would you still take it?

Of course not. Unless I had reason to believe that the method of randomization were a monkeys-and-typewriters kind of setup.

You know the old saying... an infinite number of monkeys with an infinite number of typewriters would eventually type all the works of Shakespeare.

Or, as PatMacDonald once sang....

They say that a monkey, in the right frame of mind,
given enough paper and given enough time,
is bound to type Shakespeare eventually.
Oh, baby, don't give up on me.

Well, Barbara did give up on Pat, and it also turns out that the saying is wrong.

Someone bothered to test the idea, and debunked it. In reality, monkeys and keyboards don't exhibit the behavior of running through all the possible configurations. Instead, they have a rather limited repertoire with a lot of repetition.

Take all the monkeys and keyboards you can find, give them all the time in the world, and they'll never type Shakespeare, or even "The quick brown fox jumped over the lazy dog".

So the question is, when it comes to coin flips, what sort of situation are we looking at?

Is it in fact a situation that's actually described by the kind of math that folks here are using?

I don't believe it is.

As I've mentioned before, the math that's being used here describes a highly idealized system. And there's a danger in asserting that such a model accurately describes the world we live in. There's the danger that it ignores important factors, that it is not sufficiently robust.

For example, the math describing black holes -- which someone mentioned earlier -- at first seemed to imply necessarily that they should continue to expand.

But the math was too clean, not sufficiently robust. It ignored some important features of our messy universe. It took Hawking to demonstrate that, in fact, black holes should evaporate due to the behavior of particle-antiparticle pairs on the event horizon.

Same for our expanding universe. It was assumed for quite some time that the rate of expansion is decelerating. Turns out, it's speeding up. Again, the model was not sufficiently robust to describe the real world.

So the question is, does that math actually describe real coin flips here on earth?

There's no reason to assume that it does.

There's no reason to assume that fair coin flips are the kind of physical system that will eventually run the gamut of all possible configurations. Perhaps, instead, coin flips are a kind of system in which a limited number of small-range configurations tend to repeat themselves, so that we would never see all of the mathematically possible sequences no matter how long we ran the experiment, even infinitely.

And I believe that's precisely the kind of system it is. In fact, I would literally bet my life against a Lotto jackpot that 10 billion fair coin-flippers running for, say, 10 years would never produce a streak of 100 heads or tails.

The physical randomness caused by quirks in the flipping device (which must be present in a fair system) combined with the physical vagaries of air and the surface on which the coin must land will cause a shifting back and forth between heads and tails on a much smaller scale than 100.

That's been my experience, and as far as I know it's been everyone's experience throughout all of history.

Seems to me, the small scale patterns we observe will persist, no matter how long we run the game. The universe will continue to operate the way it always has, and streaks will be limited to much less than 100 flips ad infinitum.

We can certainly do the math, but I don't think anyone here has any basis for asserting that this math actually does describe the physical reality of fair coin flips. The best you can say is maybe, maybe not.

So why do I opt for an absolute verdict of "impossible"?

Well, it's simply because I'd be lying if I told you any different.

My brain firmly believes that runs of 100 are impossible, and in this case I agree with my brain, given the evidence at hand.

Sometimes I disagree with things my brain firmly believes. For example, the McGurk effect.



My brain absolutely believes that two different sounds are hitting my ears. But I know that's not true.

Same for the rotating snakes illusion. My brain is absolutely convinced that those things are spinning and there's nothing I can do about it, even though I know it's not true.

But in this case, I agree with my brain.

Because of the battering of air and the inevitable unevenness of surfaces and the general randomness built into our real world, I conclude that the math discussed on this thread does not in fact describe the real-world system and that this system will not run through all possible configurations, but will, on a large scale, simply repeat the types of runs we see on smaller scales, so that we will never achieve a run of 100 on a fair coin with fair flips no matter how long we play the game.

 

[Piggy]

What a truly bizarre claim. Are you seriously arguing that because you have never seen an event that is so unlikely you should not expect ever to see it, that event must therefore be impossible?

See above.

 

[Piggy]

Fair enough, we are using entirely different interpretations of the phrase 'good at statistics'.
I don't regard the ability to recognise that an outcome that is 8 or 10 standard deviations from the mean is unusual as deserving of the description 'good at statistics'. You apparently do.

Not quite. What I'm saying is that we're good at certain types of stats, not at others. I think that our BS detectors are accurate when it comes to a streak of 100 coin flips. The red flags would go up for good reason, and we'd conclude that something was wrong with the set-up.

 

[Piggy]

So I doubt this is really something I need to include in the article ... unless Piggy can come up with a study?

No, I don't think one exists, although it would be fascinating to do one.

 

[Piggy]

It's just human nature to be totally unimpressed with this sequence: TFTTFTFFFTFTTTTFFFTF, but the odds of that are 1/(2^20) which is just as miraculous a result at TTTTTTTTTTTTTTTTTTTT. The only reason it's 'less noticeable' is our overall cultural numerical illiteracy.

Ah, but that's an improper comparison.

The proper comparison would be TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT..... versus everything else, not versus a single alternative chosen at random.

 

[Alan]

At what approximate point would it become impossible? After 50? Or just 100?

 

[Piggy]

At what approximate point would it become impossible? After 50? Or just 100?

I'm not sure how one would determine that, and I doubt it's a bright line.

Like the Earth's atmosphere, there are some things we can say are clearly inside it, other things we can say are clearly outside it, but no dividing line where we can draw the boundary.

I was trying to figure out if the paper cited by Simon Bridge above provided any indication of a possible answer, but I'm way too rusty on my math (as in, decades rusty, and never did much of it in the first place, just what I had to).

If we've got some math that's grounded in real-world experimentation which indicates that runs of 100 are no big deal, given a sufficient number of permutations, then I'll change my mind, of course.

 

[bokonon]

I don't know why the physics of this world prohibits very long streaks, but that's the way it is in practice.

It doesn't, and it isn't. Your experience just hasn't been expansive enough to realize it.