Double Headed Coins and skepticism

[Piggy]

Indeed, and the question is just as pertinent there. Your position is analogous to asserting that such a perfectly calm state is physically impossible rather than simply very unlikely.

You have no basis for that claim, just as you have no basis for your claim about coin flips. We, on the other hand, have a very strong basis for ours (that it is possible).

You would, if you could demonstrate that your mathematical model was robust enough to ensure accuracy when applied to the real-world physical situation.

And yes, I also have no problem with the proposition that in the real world there is a limit to the potential extent of expanses of calm water in the oceans.

I know of no observational or experimental evidence which would lead us toward any other conclusion.

 

[Piggy]

I hope you realize that the above is utter nonsense. There is nothing about real sequences of coin flips that indicates anything other than that they behave as expected based on the standard laws of physics and probability. Long sequences of heads are rare, because that's what standard theory tells you. But it also tells you that they are not impossible, and it also explains why you've never seen one.

Not to put to fine a point on it, the position you are arguing for is unscientific, magical thinking based on your failure to comprehend basic statistics.

But I don't misunderstand the stats.

I simply disagree that we have reason to believe that they describe the physical set-up, which is extremely messy and exposed to more variables than we can possibly know.

 

[sol invictus]

I know of no observational or experimental evidence which would lead us toward any other conclusion.

Ever heard of hydrodynamics? Chemistry? Physics? We have massive experimental evidence that the laws of physics, chemistry, etc. correctly describe the world, and there is nothing - not a hint - of something in those laws that would support what you are saying.

But I don't misunderstand the stats.

I simply disagree that we have reason to believe that they describe the physical set-up, which is extremely messy and exposed to more variables than we can possibly know.

And yet, you're perfectly comfortable asserting that all that uncertainty adds up to one bizarre-and-entirely-unsupported-by-any-evidence-or-logic conclusion: that runs of 100 heads are impossible. Somehow you just magically know that's true.

 

[Simon Bridge]

There is nothing about real sequences of coin flips that indicates anything other than that they behave as expected based on the standard laws of physics and probability.

Have you seen this.

Humans tend to think long runs are unlikely due to psychological factors ... related to pattern recognition false-positives and our tendency to view our experiences in an unspoken context. As a result, it is quite easy to distinguish genuine coin-tosses from a made up sequence.

Aside: is Piggy trolling?

Long sequences of heads are rare, because that's what standard theory tells you.

I'd have thought that the theory is a standard theory because it is a strong model for the behavior of reality? In practical physics, the sort of effect Piggy is talking about just looks like a bit of a skew in the uncertainty distribution ... if it shows up at all. At best he is saying that modelling a physical coin toss as a perfect binary number generator fails to account for all the physics that may apply to the situation. Which is true - but so what? To be taken seriously he needs to show how he would change the math in the article (remember the thread topic?) to better account for the physical situation. I would assert that it has no significant effect on the statistics over the time periods considered in the model.

This is an advantage of using a quantitative model...

 

[marplots]

By the way, can you prove that a string of fair flips of fair coins is adequately described by your math, that it is in fact a system which will eventually result in all possible full-length combinations?

Note that "fair flips", "fair coins", and strings are all concepts from math. You haven't escaped a mathematical description, but adopted it. Monkeys are better. As far as turbulence goes, this is demanded by probability theory, or the result wouldn't be considered random for the coin flipping.

I think you are mistaking turbulence (at least for the coins) of a single toss as being equivalent to turbulence for the whole string of tosses. But turbulence balances out. Random in one direction is met and canceled by random in the other direction. This is why we expect to get about an equal number of heads and tails.

However, if you've already gotten 9 heads in a row, the same turbulence means it is equally likely to get another head as it is to get a tail. And this is so for every step on the chain.

I agree it is surprising. At one time, I played a lot of Yahtzee. I recall, seeing a "natural" Yahtzee more than once (5 6-sided dice all matching). I was quite surprised when I calculated the odds to be 100,000 to 1 against that happening. Yet I have no reason to think there was any cheating, overt, covert or psychic. **** really does happen.

Do not circumvent the autocensor.

Replying to this modbox in thread will be off topic  Posted By: kmortis

 

[Simon Bridge]

But I don't misunderstand the stats.

I simply disagree that we have reason to believe that they describe the physical set-up, which is extremely messy and exposed to more variables than we can possibly know.

Um - the statistical models are not supposed to be complete descriptions of physics. We know for a fact that they do not describe, exactly, the actual physical setup. I believe I have even said so explicitly in my article (remember the topic of the thread?)

However, we do know that we can use statistical models to very closely predict the outcomes of experiments and real-world events, as well as predict the limits in which the models will remain effective.

On topic: it is not the purpose of the model proposed in my article to model the physics of flipping a coin. I wanted to model the short-term statistics of a boolean event. It does not have to be a coin... I explicitly want to be able to generalize the model and admit, in the body of the article, that the model is not exhaustive.

How would the issues you have raised affect the way that article needs to be written?

 

[Piggy]

Ever heard of hydrodynamics? Chemistry? Physics? We have massive experimental evidence that the laws of physics, chemistry, etc. correctly describe the world, and there is nothing - not a hint - of something in those laws that would support what you are saying.

Yeah, heard of them.

Now, can you demonstrate how these disciplines show that your idealized statistical model is sufficiently robust to describe the actual physical system, and that we can safely conclude that it is the kind of system which will exhibit all mathematically possible combinations at very large scales?

I would appreciate it.

And yet, you're perfectly comfortable asserting that all that uncertainty adds up to one bizarre-and-entirely-unsupported-by-any-evidence-or-logic conclusion: that runs of 100 heads are impossible. Somehow you just magically know that's true.

It's hardly bizarre to note that long strings of coin flips generate a rugged result space, and that smooth result spaces at great extension are typical of rigged setups.

So that leaves the question of how much smoothness is actually achievable in the real universe we live in, when we're talking about extensive sequences of fair tosses.

I expect the roughness to be a feature at all large ranges, because I have no reason to believe anything will change as the system expands to include longer and longer series.

Can't prove it, I openly admit that.

Clear evidence to the contrary will, of course, change my mind.

 

[Piggy]

Aside: is Piggy trolling?

Absolutely not. I don't have the time or inclination for that.

 

[Piggy]

To be taken seriously he needs to show how he would change the math in the article (remember the thread topic?) to better account for the physical situation.

A man's gotta know his limitations.

I don't know of anyone who has the math to describe real-world physical systems in detail.

If you can demonstrate that extremely long expanses of smoothness in result-space are possible in real-world systems like this, well, I'm a convert.

But to ask for math that no one has? That's a bit too much.

So if this is the question before us:

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?

Reasonably, I'd put 100 tosses well within that frame.

And a counter-argument from an idealized statitics which we cannot prove is sufficiently robust to describe the system -- specifically, it cannot be proven that the system in question (fair tosses of fair coins) is the kind of system that will eventually result in all mathematically possible combinations at the full scale -- really doesn't hold water.

It is important to note that nobody has yet demonstrated that the stats match the physical system, and that we don't have a monkeys-and-keyboards effect going on here to some extent... in other words, that the actual results will not be plagued with repetition of small-scale cycles that prevent the complete range of mathematically possible combinations from actually playing out.

 

[Piggy]

However, we do know that we can use statistical models to very closely predict the outcomes of experiments and real-world events, as well as predict the limits in which the models will remain effective.

"Very closely" is the key.

When there is not a perfect match, we are wise to be suspicious of the extremes, and stop short of declaring that they're accurate.

For instance, if I have a highly accurate map, I trust it in most cases, but if I see two features very finely overlap, I would not be wise to conclude that those two features do in fact overlap in the landscape. It could be an artifact of the model.

In other words, I must become suspicious that I've reached the limits of the model's effectiveness.

Do you have any evidence that the statistical model proposed in this case is effective all the way out to the edges, where the long stretches of heads/tails live? (ETA: I don't mean your stats in the article, here, but the one proposed in this thread to demonstrate that very long stretches of heads are in fact possible on a fair coin.)

If you do, then, like I say, I'm a convert.

On topic: it is not the purpose of the model proposed in my article to model the physics of flipping a coin. I wanted to model the short-term statistics of a boolean event. It does not have to be a coin... I explicitly want to be able to generalize the model and admit, in the body of the article, that the model is not exhaustive.

How would the issues you have raised affect the way that article needs to be written?

I don't see that my objection on the issues of coin-tossing raised in this thread really has an impact on the article, given the conclusions you're drawing. (Sorry if it's somwhat tangential, but I still think it fits within the OP -- if not, it can certainly be moved.)

I will say two things about the article, tho. (Aside from the fact that I generally like it.)

First, you need an apostrophe in the phrase "atheists statement" in the 2nd paragraph of "The God Question". Minor, but worth noting.

Second, if "the idea is to help people less used to scientific thinking understand some of the claims of sceptics" (should be "skeptics", btw) then I think the article generally fails because it is clearly not aimed at or accessible to folks who are not accustomed to scientific thinking.

If that really is your purpose, I'd rewrite from the ground up. If you decide to keep the article as-is (which I see no reason not to) then I'd ditch that goal.

 

[Simon Bridge]

At one time, I played a lot of Yahtzee. I recall, seeing a "natural" Yahtzee more than once (5 6-sided dice all matching). I was quite surprised when I calculated the odds to be 100,000 to 1 against that happening. Yet I have no reason to think there was any cheating, overt, covert or psychic. **** really does happen.

We may have to sit this lot down and play a lot of dice-intensive games together to make the point stick.

But: um, there are 6 ways to get all the same number on 5 dice, and 7776 (6x6x6x6x6) ways to get any combination, right, so the probability of one natural yahtzee would be: 6/7776 ? Nothing like the 1/100000 quoted.

There is not enough information to work out the probability of more than one natural yahtzee in a game, because I don't know how many opportunities there were to roll this. If you rolled 5 dice 1000 times in your life - you'd expect to see 6/7.776 or roughly one natural yahtzee sometime ... how many times have you rolled those dice?

However - if the roller had said "I will get yahtzee on this roll" and did - that would be remarkable... if they kept repeating it you'd end up very suspicious very fast.

My point is, the circumstances determine the prior:

In the normal game where you know all the players they all have good reputations, the dice have always been well behaved before, and so on, you have a good reason to estimate a very high prior ... so it would be rational to want to see a lot of evidence before concluding that something other than normal chance was happening. You could work out how rational by running the bayesian analysis from the article (but for non-0.5 probabilities)

However, when someone comes in with the intention to roll 5 of a kind from the outset you'd be more suspicious and estimate a lower prior, resulting an in earlier rejection of chance... but also an earlier conclusion of cheating. You may not know how, that's a subject for further testing (outside the scope of this thread). The more opportunity they have to cheat, the lower the prior.

 

[Simon Bridge]

"Very closely" is the key.

When there is not a perfect match, we are wise to be suspicious of the extremes, and stop short of declaring that they're accurate.

The trick is to quantify the point where this has occurred. The point of the article was, in part, to illustrate this.

Do you have any evidence that the statistical model proposed in this case is effective all the way out to the edges, where the long stretches of heads/tails live?

I don't need any - it is not intended to. It is intended to work better than forward probabilities for short runs.

But, as it happens, there is a lot of evidence that bayesian analysis works well for long odds against real-world models. There is a large body of literature on the subject and why it produces good models... not just in mathematics, but in physics too. I first ran into the method, used formally, when studying inverse problems in practical physics. It is used, for example, in situations where the researcher has to alter the physical model, in real time, as the data is collected. Its very good at that.

If you do, then, like I say, I'm a convert.

Likewise - if you propose a better model, and justify it - look up "scientific method". Bayes theorum is usually applied recursively, its just in this example it makes no difference to the math. It is very good at situations where the process of coming to a conclusion is important and very often allows the researcher to make quite silly initial assumptions and still reach a sound conclusion (with uncertainties).

I don't see that my objection on the issues of coin-tossing raised in this thread really has an impact on the article, given the conclusions you're drawing. (Sorry if it's somwhat tangential, but I still think it fits within the OP -- if not, it can certainly be moved.)

Yep - you are being waaay too abstract and that part of your discussion should be moved. Please. Thank you.

I will say two things about the article, tho. (Aside from the fact that I generally like it.)

First, you need an apostrophe in the phrase "atheists statement" in the 2nd paragraph of "The God Question". Minor, but worth noting.

Well, since this is about God, can I have 12 apostrophes?

Second, if "the idea is to help people less used to scientific thinking understand some of the claims of sceptics" (should be "skeptics", btw) then I think the article generally fails because it is clearly not aimed at or accessible to folks who are not accustomed to scientific thinking.

Nah - its more about giving people used to scientific thinking another tool to understand the processes. It would be part of explaining scientific thinking to others rather than a complete description. Note: I also do not derive bayes theorum or explain the concept of probability. It is difficult to know where to pitch these things.

If that really is your purpose, I'd rewrite from the ground up. If you decide to keep the article as-is (which I see no reason not to) then I'd ditch that goal.

Good comment - I have a tendency for my goals to change mid-article without me noticing. That's one of the things I was hoping to have pointed out. Thanks.

 

[Piggy]

Yep - you are being waaay too abstract and that part of your discussion should be moved. Please. Thank you.

That's fine. I won't continue to discuss that topic here.

It's not the kind of thing I'd start a thread on, myself, so I'll just drop it unless someone else wants to begin a new thread.

Btw, are you familiar w/ biological research on the subject? Our brains appear to go through a process very similar to what you are describing. Our "gut feelings" as well as decisions such as what we want to order off a menu are based on non-conscious processes that give us a sense of how certain we are (or aren't) that a particular choice is the right one.

 

[bokonon]

... in other words, that the actual results will not be plagued with repetition of small-scale cycles that prevent the complete range of mathematically possible combinations from actually playing out.

I think it's the repetition of small-scale cycles that guarantees the complete range in this case.

The fact that two heads in a row is a possibility is what makes 4 heads in a row a possibility. How often will two 4-in-a-row sequences occur back-to-back? The statistics gives us the probability.

It's a kind of magical thinking and argument from incredulity that you're invoking here -- that at some point, the statistics will break down, and back-to-back sequences of just these small-scale cycles will become impossible. But I suppose we all have our blind spots.

 

[Alan]

I have just flipped an Australian 50-cent piece 100 times and recorded the results.

THHHTHTHTTTHHTTTTTHTTTTTHHHTHTHHHTHTTHTTTTHTHTHHHHHHTTHHTTHTHHHHHTHHTTHTTTTTTHHTTHHHTHTHHTTHTHHTHTHH

I highlighted the bit I'm talking about here: If just one of those 50/50 flips went the other way, I would have had 11 in a row within the first two minutes. There are other similar runs in there. Now, did those ones that broke a chain not have approximately a 50/50 chance of being heads/tails?

Edit: also, I counted that I got exactly 50 heads and 50 tails. whee!

 

[sol invictus]

Yeah, heard of them.

Now, can you demonstrate how these disciplines show that your idealized statistical model is sufficiently robust to describe the actual physical system, and that we can safely conclude that it is the kind of system which will exhibit all mathematically possible combinations at very large scales?

Of course not. But the burden of proof isn't on me. You're the one making the extraordinary claim - you're asserting that the known laws of physics cannot be used to describe coin flips and/or ocean surfaces, and moreover, that the correct description ensures that 100 head sequences/very flat surfaces are completely impossible. The burden of proof is squarely on your shoulders - and so far, you've provided absolutely none (which is why people are asking if you're trolling).

It's hardly bizarre to note that long strings of coin flips generate a rugged result space, and that smooth result spaces at great extension are typical of rigged setups.

You've just conflated the two questions I tried to get you separate. One question is whether 100 head sequences are possible. A different question is whether 100 head sequences are more likely to be generated by chance from fair coin flips or due to some sort of rigging.

 

[sol invictus]

Do you have any evidence that the statistical model proposed in this case is effective all the way out to the edges, where the long stretches of heads/tails live?

One of the many misunderstandings you evidently suffer from is this idea that long sequences of heads are an "edge" to the space of sequences, or are in any physically relevant way distinguished from any other sequence. If the coin is fair, physics couldn't care less which side humans consider heads and which we consider tails (it's just a label, an arbitrary convention).

Look - suppose we play the following game. I make a table with 100 entries. Each entry is either F or NF. Let's say there are 50 Fs and 50 NFs in some more or less random order. Now, here's the game - I flip a coin 100 times. Before each flip I consult the corresponding entry in my table. If it says F, I flip the coin over after catching it, before uncovering it and reading it. If it says NF, I don't flip it before uncovering it.

Do you stil believe that a sequence of 100 heads is impossible, given that setup? Note that if 100 heads is impossible, then whatever sequence you'd get by starting with 100 heads and turning all the Fs into tails is also impossible, since that's what I would have gotten had I not flipped the Fs. But since my table of Fs and NFs was arbitrary, that means all sequences are impossible, which is obvious nonsense.

Therefore, 100 head sequences are possible.

 

[Alan]

I am confident that Piggy is not trolling, and did not think it was the case for a second.

Could we perhaps agree that it is just very, very, very unlikely to happen any particular time instead of literally, truly, 100% impossible?

Additionally, if I was to predict any 100-flip sequence, the chance of me getting it right would be comparable to getting 100 heads?

 

[Alan]

Or let's say I toss a coin until I am on a streak of heads and 'feel' like the streak has to break soon.

Then, I go up to somebody and bet with them that it's going to come up with tails. Would the tosses beforehand affect the tosses in this bet and give me an unfair advantage?

 

[marplots]

We may have to sit this lot down and play a lot of dice-intensive games together to make the point stick.

But: um, there are 6 ways to get all the same number on 5 dice, and 7776 (6x6x6x6x6) ways to get any combination, right, so the probability of one natural yahtzee would be: 6/7776 ? Nothing like the 1/100000 quoted.

Yes, thank you. I see I misused the exponent key on my calculator -- how embarrassing. So it's actually about one out of a thousand. Not nearly as surprising as I thought.

However - if the roller had said "I will get yahtzee on this roll" and did - that would be remarkable... if they kept repeating it you'd end up very suspicious very fast.

The problem with the first part (not the repeating) is that someone can make the claim a thousand times -- how is the one correct claim then remarkable?

My point is, the circumstances determine the prior:

In the normal game where you know all the players they all have good reputations, the dice have always been well behaved before, and so on, you have a good reason to estimate a very high prior ... so it would be rational to want to see a lot of evidence before concluding that something other than normal chance was happening. You could work out how rational by running the bayesian analysis from the article (but for non-0.5 probabilities)

However, when someone comes in with the intention to roll 5 of a kind from the outset you'd be more suspicious and estimate a lower prior, resulting an in earlier rejection of chance... but also an earlier conclusion of cheating. You may not know how, that's a subject for further testing (outside the scope of this thread). The more opportunity they have to cheat, the lower the prior.

Completely agree. There is a balance we are trying to evaluate -- h1 there is cheating, or h2 there is no cheating. In no scheme are we ever completely sure, but we can, with varying confidence, weigh each hypothesis against the other. I'd also like to add that in the real world, there may be an h3,h4 and so on which will confound the matter.

So, for instance, in neo-cheating, the idea is to use a mixed strategy that falls more nearly toward expectations but still allows the cheater to come out ahead. In other words, a "partial cheat" who disappears into the natural variation of the "normal" game.