Double Headed Coins and skepticism

[paiute]

A False Coin - hopefully the link works - I wrote considering the oft-repeated argument (by theists) that the atheist cannot know there is no God because the atheist has not been everywhere. This questions how one can be certain of a negative result only through a finite amount of negative data. eg. How hard do I look for something, and fail, before I can confidently conclude the thing is not there to find?

The idea is that I am tossing a coin and it keeps coming up heads - how many tosses before you conclude, reasonably, I'm cheating? How do your initial prejudices affect the answer to this question?

You don't have to cheat if your opponent has no grasp of probability:
http://www.scribd.com/doc/13366448/The-Wisdom-of-Coinage-A-Play-in-One-Act

 

[Malerin]

I'm not saying that it is impossible that a person can cheat. I am saying that it is possible that the person is not cheating.

I, for one, was talking about any combinations.

What's the chance of any other combination after 100 coin tosses? It's low, therefore that combination probably wouldn't happen either.

Any particular combination will be a statistical long shot. Some combinations are statistically significant: they are better explained given cheating or bias than chance alone.

As Piggy stated, you'd be a fool to bet a coin that's landed heads 100 times in a row will come up tails. This is because you intuitively know that that many heads in a row is far more likely given a two-headed coin (or other form of cheating) than by chance.

 

[Beerina]

i mean hell i have seen a 100 sided dice roll 100 10 times in a row.

No you haven't. If you have, it's prima facie evidence the die is badly weighted.

That's a 1 in 100,000,000,000,000,000,000 event, or one in one hundred quintillion.


To answer the OP, after 5 in a row, I'd say, "Hold on. I want you to do 10 test flips, please." If your opponent seems a touch unsavory, demand it before you start. Of course, a decent prestidigitator could easily swap the fraud one in and out.

Quite frankly, I don't even know if it's possible to weight a coin so it lands on one side more than the other more than a fractional increase. Even the best would be pushing it to do 10 in a row, I'm guesstimating.

I'll also add that it will probably bounce in ways that look funny.

 

[Guybrush Threepwood]

And btw, in real-world terms, our brains turn out to be not bad statisticians. Even infants somehow understand that something's wrong when events occur that are way outside expected probabilities.

We're not very good statisticians when it comes to artificial situations or unimportant business such as the number of folks in a room sharing a birthday. But for everyday work, our brains are Johnny on the spot.

Could you clarify this a bit, as far as I know, and based on my experience, unless we are trained, our brains are awful statisticians.

If you want me to cite evidence: Las Vegas
If that's too artificial for you: Prayer, Homeopathy, Lucky Charms (not the cereal).

 

[Simon Bridge]

I probably wrote my response in a rambling way, sorry. I was trying to address the paper.

The main point I was trying to make is that prior probabilities aside, we will still quibble about the confidence interval thresholds. There is no cultural standard that builds common ground between advocates and skeptics.

That was one of the things I had hoped to bring out in the paper - the difference between the skeptic and the believer is represented by differences in the prior. The trouble with irrational belief is that it does not change with evidence. In the coin-toss example, an irrational belief would be represented by a prior of 1 or 0. I probably need a more detailed section on investigation... I was kinda hoping that the discussion here could guide me as to what I needed to talk about ... like I've probably been too pedantic in some bits and too vague in others.

While we quibble over confidence thresholds, the effect of the prior makes the exact threshold unimportant - technically the usual fight is over what constitutes evidence. The MDC experience shows us that it is very difficult to set up an unambiguous trial.

The second problem that I didn't mention is that there is often a disagreement about what constitutes the null hypothesis. In your paper it looks like 'phenomenon is naturally occurring'?

Yeah - I sat down and worked out the usual "high-school" hypothesis test for the same problem and realised that I had not properly specified the problem. The actual hypothesis test is too implicit and the paper wanders between "cheating" and "double-headed coin".

The actual analysis sets up a false dichotomy by only taking into account the possibility of cheating by double-headed-coin... though what I really ant to do is eliminate fair chance as a cause for the observed results.

The reason this is important to paranormal investigation is that when the skeptics succesfully expose a cheat, it's very consistently been about good controls rather than good stats.

Though that's not how they argue. Good point though.

What I am bringing up in the article is probably a small point for skeptics, but one that is often misunderstood by, um, most people. Since a great deal is popularly written on the points you have brought up, I thought I'd concentrate on this one. So - good putting the paper in context of the broader field, but I'm not writing a book here (later?)

I think the math works better if the model is of many coins in a bag, a certain proportion of which are double-headed. I draw one coin from the bag and start tossing. It also makes the language a bit more neutral.

 

[Simon Bridge]

re: run of 10x00 result on d100:

No you haven't. If you have, it's prima facie evidence the die is badly weighted.

That's a 1 in 100,000,000,000,000,000,000 event, or one in one hundred quintillion.

However, the same calculation holds for any particular sequence of numbers rolled. Why is this particular one so special?

Of course, in D&D, the 00 roll usually indicates a failure (you have to roll d100 less than[/] the number to succeed or it's roll generating percentages does not work) ... whatever, this number is special if anything else has another meaning ... 1% chance of failure and 99% otherwise. So a long run on failures would be a remarkable event.

Note: in D&D irregularities in the dice are part of the game ... players agree to use a particular set for the game, and this is fair because all players have access to the same dice - and so the same bias.

To answer the OP, after 5 in a row, I'd say, "Hold on. I want you to do 10 test flips, please." If your opponent seems a touch unsavory, demand it before you start. Of course, a decent prestidigitator could easily swap the fraud one in and out.

So if you saw 5 heads in a row, you'd reserve final judgement until you'd seen a further 10 "test flips" ... which suggests that you'd feel iffy after 5 flips and certain after 15. However, I suspect you are reading stuff into the example that I have not stated ...

Can you quantify your confidence though? If you saw 4 heads in a row would your reaction be different? How much by? You are guessing right? Why should anyone believe your guess as opposed to someone who wants to be suspicious after two flips, or twenty?

Your opinion, though, agrees quite well with the calculations if you think the chance a cheater is involved is about 1% ... but if you want to reject it outright at the 5th toss, then that would be rational if you give equal weight to the possibility of cheating.

Have you read the article?

Quite frankly, I don't even know if it's possible to weight a coin so it lands on one side more than the other more than a fractional increase. Even the best would be pushing it to do 10 in a row, I'm guesstimating.

I'll also add that it will probably bounce in ways that look funny.

Fortunately I did not intend you to consider a weighted coin, only the possibility that it is double-headed. Since the example is illustrative, though, we can extrapolate the math to the possibility that the coin has more then two sides - math is like that - or, to put it in more common language, "to situations which involve other kinds of random-number generators"

 

[Simon Bridge]

I was hoping someone would check the math ... do I take it that everyone who has read the paper agrees with the math?

 

[Simon Bridge]

Could you clarify this a bit, as far as I know, and based on my experience, unless we are trained, our brains are awful statisticians.

If you want me to cite evidence: Las Vegas
If that's too artificial for you: Prayer, Homeopathy, Lucky Charms (not the cereal).

It's a little glib isn't it? He's talking about the kinds of stuff we are evolved to deal with - like spotting faces against a random background: we are good at it - a bit too good - but if you consider how often we detect a pattern that really is there (reading this post, identifying a photograph for eg) against the false positives (spotting BVM/elvis/spacecraft in a cloud), the odds are pretty good. That's not exactly doing statistics (what are the odds a particular influx of light on the retina represents something I need to pay attention to?) any more than catching a ball is exactly doing differential equations.

 

[Malerin]

re: run of 10x00 result on d100:

However, the same calculation holds for any particular sequence of numbers rolled. Why is this particular one so special?

Because it disconfirms, to an extremely high degree, the null hypothesis of random chance. To put it another way, there are many, many combinations that are consistent with random chance. 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.

 

[Piggy]

Could you clarify this a bit, as far as I know, and based on my experience, unless we are trained, our brains are awful statisticians.

If you want me to cite evidence: Las Vegas
If that's too artificial for you: Prayer, Homeopathy, Lucky Charms (not the cereal).

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.

 

[Alan]

So it's impossible?

 

[Piggy]

So it's impossible?

What, 100 consecutive heads or tails on a fair coin from a fair toss here on earth? Yeah, that's not possible. The world we really live in doesn't work that way.

 

[sol invictus]

What, 100 consecutive heads or tails on a fair coin from a fair toss here on earth? Yeah, that's not possible. The world we really live in doesn't work that way.

2^100 is about 10^30, which is about how many viruses there are on earth at any given moment. So if there's some improbable event that applies to viruses and has probability equal to flipping tails 100 times on a fair coin, it happens to one every day (or however long viruses live).

Therefore it's not correct to say such events are impossible. If you want a cutoff on probabilities below which you can say the event never happens, it needs to be much smaller than that.

And anyway, as you know all sequences of flips are equally improbable. Since coins have been flipped far more than 100 times in the history of the world, the sequence of all coin flips that so far have taken place has a probability that's 2^{-N}<<2^{-100}, where N is the number of flips.... and yet that event (flipping that sequence) happened.

 

[Alan]

I'm trying to see why you think that, Piggy. I think it's just a loose definition of "impossible", but I'll ask this just in case:

If you toss a coin and in comes up heads, is it more likely to come up tails the next time? And the time after that?

 

[Piggy]

2^100 is about 10^30, which is about how many viruses there are on earth at any given moment. So if there's some improbable event that applies to viruses and has probability equal to flipping tails 100 times on a fair coin, it happens to one every day (or however long viruses live).

Therefore it's not correct to say such events are impossible.

I don't live on paper.

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.

 

[sol invictus]

If you toss a coin and in comes up heads, is it more likely to come up tails the next time? And the time after that?

In theory, no.

In practice, coins are more likely to land facing the same way up as they were before they were flipped, so... probably yes, depending on the technique of the coin flipper.

 

[Piggy]

I'm trying to see why you think that, Piggy. I think it's just a loose definition of "impossible", but I'll ask this just in case:

If you toss a coin and in comes up heads, is it more likely to come up tails the next time? And the time after that?

That's the wrong question to ask. It's looking at the landscape through a soda straw.

Yes, the coin is stateless. It doesn't know what happened on the previous flip.

The pertinent question is: What will happen if a human being flips a fair coin 100 times here on the surface of the earth?

What we observe is that runs of 100 do not occur. In fact, nothing even close occurs.

And that's because the physics of the coin and the atmosphere and gravity and human muscles ensures a certain level of random variation.

As I said, we may not yet be able to explain why this is so, but there's no doubt that it is so.

And referring to a fictional, abstract, pen-and-paper world of pure statistics doesn't change that fact, because we do not live in that world.

 

[sol invictus]

I don't live on paper.

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.

You've added "fair flips" - without that caveat, you're probably wrong. It's possible to learn to control coin flips to at least some degree. With that caveat, you're not living in the real world you claimed to live in - because no coin flip is fair.

 

[Piggy]

You've added "fair flips" - without that caveat, you're probably wrong. It's possible to learn to control coin flips to at least some degree. With that caveat, you're not living in the real world you claimed to live in - because no coin flip is fair.

That's incorrect. Of course coin flips are "fair" in the common sense of the word -- that is, the flipper is not exerting any control which could reasonably pre-determine the outcome, and over enough flips the heads and tails even out.

 

[sol invictus]

That's the wrong question to ask. It's looking at the landscape through a soda straw.

Yes, the coin is stateless. It doesn't know what happened on the previous flip.

The pertinent question is: What will happen if a human being flips a fair coin 100 times here on the surface of the earth?

What we observe is that runs of 100 do not occur. In fact, nothing even close occurs.

And that's because the physics of the coin and the atmosphere and gravity and human muscles ensures a certain level of random variation.

As I said, we may not yet be able to explain why this is so, but there's no doubt that it is so.

And referring to a fictional, abstract, pen-and-paper world of pure statistics doesn't change that fact, because we do not live in that world.

Do you recognize the irony in your position? You claim to know that sequences of 100 coin flips never come up all tails in the "real world", and that we're talking about some kind of detached-from-reality theory world.

But how many times have you flipped a coin 100 times in a row? Obviously not enough times to conclude any such thing - and therefore, you're the one using theory. And using it incorrectly, I might add.