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Old 17th February 2011, 06:51 PM   #161
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Originally Posted by sol invictus View Post
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.
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Old 17th February 2011, 06:53 PM   #162
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Originally Posted by sol invictus View Post
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.
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Old 17th February 2011, 07:07 PM   #163
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Originally Posted by Piggy View Post
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.

Originally Posted by Piggy View Post
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.
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Old 17th February 2011, 07:08 PM   #164
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Originally Posted by sol invictus View Post
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?

Quote:
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...
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Old 17th February 2011, 07:12 PM   #165
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Originally Posted by Piggy View Post
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.

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Old 17th February 2011, 07:21 PM   #166
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Originally Posted by Piggy View Post
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?
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Old 17th February 2011, 07:24 PM   #167
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Originally Posted by sol invictus View Post
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.

Originally Posted by sol invictus View Post
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.
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Old 17th February 2011, 07:25 PM   #168
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Originally Posted by Simon Bridge View Post
Aside: is Piggy trolling?
Absolutely not. I don't have the time or inclination for that.
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Old 17th February 2011, 07:36 PM   #169
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Originally Posted by Simon Bridge View Post
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:

Quote:
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.
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Old 17th February 2011, 07:59 PM   #170
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Originally Posted by Simon Bridge View Post
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.

Originally Posted by Simon Bridge View Post
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.
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Old 17th February 2011, 08:00 PM   #171
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Originally Posted by marplots View Post
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.

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Old 17th February 2011, 08:17 PM   #172
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Originally Posted by Piggy View Post
"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.

Quote:
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.

Quote:
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).

Quote:
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.

Quote:
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?

Quote:
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.

Quote:
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.
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Old 17th February 2011, 08:27 PM   #173
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Originally Posted by Simon Bridge View Post
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.
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Old 17th February 2011, 08:53 PM   #174
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Originally Posted by Piggy View Post
... 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.
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Old 17th February 2011, 09:30 PM   #175
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I have just flipped an Australian 50-cent piece 100 times and recorded the results.

THHHTHTHTTTHHTTTTTHTTTTTHHHTHTHHHTHTTHTTTTHTHTHHHHHHTTHHTTHTHHHHHTHHTTHTTT TTTHHTTHHHTHTHHTTHTHHTHTHH

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!

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Old 17th February 2011, 09:45 PM   #176
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Originally Posted by Piggy View Post
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).

Quote:
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.
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Old 17th February 2011, 09:52 PM   #177
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Originally Posted by Piggy View Post
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.

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Old 17th February 2011, 10:18 PM   #178
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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?

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Old 17th February 2011, 10:42 PM   #179
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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?
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Old 18th February 2011, 06:30 AM   #180
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Originally Posted by Simon Bridge View Post
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.

Quote:
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?


Quote:
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.
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Old 18th February 2011, 06:43 AM   #181
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Simon's post has me wondering about another question. Suppose you meet someone who tells you they will roll, with a fair die, 5 of the same number in a row. They do so. Now, they remark that this is an amazing thing, but to really convince you of the special nature of this event, they will now roll another two times and get the same number as before each time.

Knowing the probability of this is about 1:46,000 (36 * 1,296); the die is fair, and they made the claim before they rolled -- you are very surprised and wondering about telekinetic abilities, or some unknown cheating method.

However, when you subsequently learn that this person has been doing this same thing an average of 10 times a day for the last 10 years (about 36,000 trials) you realize that at some point, someone would have the experience you had.

My question is whether having the additional knowledge has affected the actual math involved in determining cheating? How can history reach into the present and change the meaning of an event in this way?
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Old 18th February 2011, 07:31 AM   #182
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Originally Posted by Piggy View Post
Yet despite that fact, observation of actual monkey-and-keyboard setups demonstrates that they will never produce the string "The quick brown fox jumped over the lazy dog." no matter how long we give them to do it.
If that is true, it's necessarily the case the likelihood of pressing the "x" key after the "o" key (or at least some key after another) is less than it is after a different key.

In other words, if what you are arguing for with regards to coin tossing is true, it's necessarily the case that the likelihood of the coin turning up heads decreases as the number of consecutive heads increases.

Or, to put it more simply, you are arguing that the gambler's fallacy is not a fallacy.
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Old 18th February 2011, 08:10 AM   #183
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Originally Posted by Piggy View Post
Each pair is possible, and yet the extended string will never occur, because in reality they have a much narrower range of output than that at that length, with a lot of repetition of identical or similar patterns.
Except that this claim is just as stupid as your claim for tossing coins. You're just assuming some magical change in the laws of physics that suddenly jumps out and grabs the monkey's hand to stop it typing. It really is incredibly simple, so much so that I have to agree it looks like you're just trolling - if something is possible, then it's possible. Yes, it's a tautology. But that's exactly what you're denying. You're claiming that something that is absolutely physically impossible suddenly becomes impossible without any change in the physical setup or the laws of physics. Basically, you believe in God and miracles, you're just not calling it that.

In reality, it's not just possible for the monkeys to type any arbitrary phrase, it's guaranteed to happen, given enough time and enough monkeys. All the observations show is that each key does not have an equal probability of being pressed, therefore it will need more time and monkeys than if you assume equal probability. That's possibly the really sad part - you're spouting these observations of monkeys as if they support your point, but they're absolutely irrelevant to your claims. The monkey observations merely change the probability of key presses and therefore calculations dependent on them. The only difference is a quantitative one, not a qualitative one - it changes from one very small probability to a different small probability. If your god is going to step in and magically make those probabilities 0, it doesn't matter which one you actually start with. You're not just arguing from ignorance of reality, you don't even appear to understand your own argument.
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Old 18th February 2011, 08:20 AM   #184
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Originally Posted by Piggy View Post
It's a fascinating question, and in a way it's one I don't have an answer for, but in another way I do have an answer.

Let's go back to the monkey example for a moment.

There is nothing that prevents a monkey from pressing the "u" key after he presses the "q" key.

And there's nothing that prevents a monkey from pressing the space bar after pressing the "n" key. Or from pressing the dot key after pressing the "g" key.

Yet despite that fact, observation of actual monkey-and-keyboard setups demonstrates that they will never produce the string "The quick brown fox jumped over the lazy dog." no matter how long we give them to do it.

Each pair is possible, and yet the extended string will never occur, because in reality they have a much narrower range of output than that at that length, with a lot of repetition of identical or similar patterns.

Similarly, in the case of the metal cup dropped down the long flight of stairs, it may hit on the same spot on two consecutive steps, but if it's a random drop (and not a carefully orchestrated effort) it will not make identical strikes all the way down, no matter how many times you drop it, because our rough and bumpy world doesn't work like that.

Now, precisely why is that the case? That, I couldn't tell you.

But a truly fair toss of a fair coin, in the real world, when repeated many times in sequence, is exactly the kind of thing that we should expect to reflect the messiness of physical reality. If we find that this messiness is absent -- which would certainly be the case if we got 100 heads or tails in a row -- we would be wise to conclude that the exercise had somehow been shielded from it, or in other words, that it was rigged.
In this context, "almost surely" is a mathematical term with a precise meaning, and the "monkey" is not an actual monkey, but a metaphor for an abstract device that produces a random sequence of letters ad infinitum

How do you actually observe a metaphor?
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Old 18th February 2011, 08:23 AM   #185
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Originally Posted by Cuddles View Post
In reality, it's not just possible for the monkeys to type any arbitrary phrase, it's guaranteed to happen, given enough time and enough monkeys. All the observations show is that each key does not have an equal probability of being pressed, therefore it will need more time and monkeys than if you assume equal probability.
I don't believe this is true. Monkeys are far more likely to bang the same key over and over again than to press random keys sequentially, and they'd have to be pressing a variety of keys with frequencies approximating some target language to manage a phrase of even three words.

The situation with monkeys and typewriters is not really comparable to the situation with a fair coin fairly tossed. The constraints are different, and the range of possibilities is vastly different. The results of one have no bearing on the results of the other.
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Old 18th February 2011, 08:26 AM   #186
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Originally Posted by bokonon View Post
I don't believe this is true. Monkeys are far more likely to bang the same key over and over again than to press random keys sequentially, and they'd have to be pressing a variety of keys with frequencies approximating some target language to manage a phrase of even three words.
Nonsense. As long as each key has a non-zero probability to be pressed*, any arbitrary sequence will eventually be pressed given enough time. It will certainly take longer if certain keys or patterns are more common, but that was exactly my point - it will only take longer, there is no qualitative change.

* Piggy has explicitly stated this to be the case so I'm assuming it is true for the example.

Quote:
The situation with monkeys and typewriters is not really comparable to the situation with a fair coin fairly tossed. The constraints are different, and the range of possibilities is vastly different. The results of one have no bearing on the results of the other.
No, the two examples are identical. One deals with a case where two possible outcomes have the same probability, the other deals with a case where multiple outcomes all have different probabilities. That changes the exact numbers in the calculations, but the final conclusion is identical - any arbitrary sequence must eventually occur given enough trials.

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Old 18th February 2011, 08:32 AM   #187
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Originally Posted by Cuddles View Post
Nonsense. As long as each key has a non-zero probability to be pressed*, any arbitrary sequence will eventually be pressed given enough time. It will certainly take longer if certain keys or patterns are more common, but that was exactly my point - it will only take longer, there is no qualitative change.
If, for the sake of argument, we stipulate that a monkey will press every key it touches three times in a row, then the fact that each key has a non-zero probability to be pressed is irrelevant: the word "nonsense" can never appear in the output; the closest approximation would be nnnooonnnssseeennnssseee.

It's the nature of a fair coin fairly tossed that each independent event has a 50% probability. It's the nature of monkey brains and keyboards that repetition will be more common than variation, and adding more time doesn't guarantee "any arbitrary sequence" under those conditions.
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Old 18th February 2011, 08:49 AM   #188
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Originally Posted by bokonon View Post
If, for the sake of argument, we stipulate that a monkey will press every key it touches three times in a row, then the fact that each key has a non-zero probability to be pressed is irrelevant: the word "nonsense" can never appear in the output; the closest approximation would be nnnooonnnssseeennnssseee.

It's the nature of a fair coin fairly tossed that each independent event has a 50% probability. It's the nature of monkey brains and keyboards that repetition will be more common than variation, and adding more time doesn't guarantee "any arbitrary sequence" under those conditions.
Fair point, I was being too general there. My apologies. However, the point stands in the specific example since, as noted, Piggy has explicitly admitted that the sequences in question are possible. In a case like his, where the arbitrary sequence is not explicitly disallowed by the conditions, it must come up eventually.

Edit: To clarify a bit further - I am working with the assumption that after a key press, there is a finite, but different, probability for every other key to be pressed. Piggy's statement that the sequences in question are possible supports that assumption. It's possible to set up a situation where that is not the case, such as keys always being pressed three times, but that's not the situation we're talking about.

A related point worth noting is that Piggy claims his argument stems from preferring real world observations over maths. However, the observations he cites could not possibly prove that monkeys never type a certain sequence, all they can show is that they don't tend to do it in within a limited observation time. So he's actually engaging in exactly the same behaviour that he's complaining about in the first place, by taking a limited set of observations and extrapolating to a general case. Even if his extrapolation were correct (which it clearly isn't since it involves magic), it could not prove his ultimate point since in being correct it would prove that such extrapolation can be valid and therefore he's wrong to complain about it.

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Old 18th February 2011, 10:50 AM   #189
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Originally Posted by bokonon View Post
If, for the sake of argument, we stipulate that a monkey will press every key it touches three times in a row, then the fact that each key has a non-zero probability to be pressed is irrelevant: the word "nonsense" can never appear in the output; the closest approximation would be nnnooonnnssseeennnssseee.

It's the nature of a fair coin fairly tossed that each independent event has a 50% probability. It's the nature of monkey brains and keyboards that repetition will be more common than variation, and adding more time doesn't guarantee "any arbitrary sequence" under those conditions.
You can clearly set up a sets of rules that - if followed absolutely - will make certain sequences impossible. But what's absurd about Piggie's position is that it requires that there be certain absolute rules that must always be followed (99 heads must always be followed by a tails). That goes sharply against everything we know about science and reality.

In fact, it contradicts Aristotelean logic, as I proved by reductio ad absurdum in post #177.
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Old 18th February 2011, 10:58 AM   #190
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Originally Posted by Piggy View Post
No, that's not it. It's a long string of physical events occurring in a turbulent environment which reach the same end state. This puts it in a separate category from the types of strings of events -- of many various configurations -- which we would expect, for good reason, to result from actions within such an environment.
Yeah, I'm still seeing the same misunderstanding. Let me rephrase the scenario.

I describe in advance a sequence of 100 flips that are a mix of heads and tails.

The test flips are conducted, and they match my prediction.

Is the predicted sequence more or less likely than the test flips coming up 100 heads.

(my thought: it's the same probability)

The reason we'd be 'impressed' by a psi calling a mixed string in advance is the same reason we're pre-built to be impressed by a string of 100 heads: we've culturally pre-called this one as 'unnatural'.


Something else worth noting: people are really terrible at determining whether a string is random or composed because we think runs are a sign of intelligent input, but mixes are 'natural'. This is why perfect permutations are incredibly rare in long truly random strings, but runs are absolutely the norm.

This is how we caught some psi cheaters - their results were 'too mixed' - they lacked the telltale runs that are typical true randomness. They were obviously cleaned up by a human who thought runs would be suspicious; whereas, they're not really.




Just a comment about the monkeys - the analogy was originally really a joke and not intended to be a serious claim, so it's reasonable to reject it on that grounds.

But it should also be rejected for the following very good reason: monkeys are intelligent creatures with brains that do things with purpose. They are not going to create truly random results in the way a fair coin creates truly random results due to mechanics. And nobody really expects them to.



Magicians take advantage of this. An example is to ask a volunteer to provide a 'random' number between 1 and 10. Most of the time, they will choose 7, sometimes 3, rarely other numbers. Ask for a 'random' vegetable, and people choose carrot almost all the time. A few choose broccoli. If you buy P&T's "How To Play With Your Food" they provide a transparency of St. Carl the Carrot to do a camera trick, because it's that consistent.

The point is: we're really bad at recognizing random vs created. Again, piggy's model is exactly the same as [William Paley's Watchmaker Analogy].
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Old 18th February 2011, 12:04 PM   #191
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Originally Posted by marplots View Post
My question is whether having the additional knowledge has affected the actual math involved in determining cheating? How can history reach into the present and change the meaning of an event in this way?
It's not that history is reaching into the present to change the reality of what happened, but that you are able to better evaluate something with more complete information.

It is completely related to the opening post, because this is the impact of prior probability.

It expands to other skeptical topics. For example, healthfraud claims. It's valuable to look at a body of literature rather than just the one most recent study. How does the current study fit into the body of literature?

In your example, it's: How does the current toss fit into the overall toss history for this individual?

A more concrete example is my colleague at work who tells me he knows he's going to win the lottery every damn time he buys a ticket. One day he may eventually win, but he's one of millions who make exactly the same claim and lose week after week. When one of them eventually wins, it's not very impressive because the overwhelming majority of the predictions have been negative. ie: a broken clock is right twice a day, but this doesn't support the claim that it 'sometimes works.'
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Old 18th February 2011, 12:39 PM   #192
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Originally Posted by Piggy View Post
A large number of highly variable patterns are consistent with our observation of how fair tosses behave at a large scale.
Well, meaning they're the majority of outcomes and therefore more common. That's not saying anything special.

Essentially: you are arbitrarily lumping them together as a 'type' vs another 'type' which is straight runs.

That's a cultural preference - it does not represent a natural phenomenon.

Another poster divided the outcomes into 'looks like Pi' vs 'does not look like Pi' / somebody who reads morse code could interpret Heads as dashes and Tails as dots and have a category 'makes English words' vs 'does not make English words' &c.

But there's no underlying natural phenomenon. It's just playing Bible Code with coin tosses, and that includes straight runs as one outcome that is decreed 'special' because of cultural human meaning.
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Old 18th February 2011, 03:48 PM   #193
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Every thread on the probablity involved with coin tossing needs at least one reference to this guy:


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Old 18th February 2011, 06:05 PM   #194
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Originally Posted by dudalb View Post
Every thread on the probablity involved with coin tossing needs at least one reference to this guy:
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Old 18th February 2011, 06:18 PM   #195
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Originally Posted by blutoski View Post
Gargh. This sounds wrong. I'd like to explain why I think this reasoning is faulty.

Basically, it's inferring cause from results, rather than testing a pre-formed cause hypothesis. Bem just got pwnd for this. (See: [Why Psychologists Must Change the Way They Analyze Their Data: The Case of Psi])
Often times, in statistics, you get a fluky result and have to come up with a cause, ex post facto. I might not suspect a certain dealer is cheating (or have the thought even formed in my mind). However, after the dealer has dealt himself five royal flushes in a row, I would be convinced he's cheating.

Quote:
This is related to [pareidolia] in that we have a tendency to anthropomorphise 'recognizeable' natural events backwards from the event. ie: we see random numbers, and look for "a pattern" - we're good at finding patterns with completely open criteria.
It's always something you have to watch out for. On the flip side however (no pun intended), that doesn't mean we can never apply a hypothesis to an extremely unlikely event. We do it all the time.

Quote:
In the world of hypothesis testing, we need to stick to a pre-designated binary criteria, or we've stopped doing science and started doing metaphysics.

When a lottery number sequence comes up a winner, to me it's random. To the person who chose it, it's probably a very special set of numbers. Birthdates or something.
Ahh, now what if the same person wins the same lottery again? We might say it's a fluke. Given how many people play the lottery, there might be some repeat winners. But if they win again? And again? At some point, you become extremely skeptical of chance, and much more open to some rigging of the lottery.


Quote:
31415926535897932384 is 'obviously cheating' to somebody who recognizes pi, but not to other people. I'm sure mathematicians have a personal inventory of irrational numbers they would recognize but the rest of us wouldn't.
The inability to recogize the significance of a result doesn't mean the result can't disconfirm the null hypothesis of random chance. Let's go back to poker. Suppose my friend and I are watching a poker game. I know how to play, and my friend doesnt't. The dealer deals himself a royal flush. Then another. And another. My friend then asks why everyone has left the game. My friend's inability to recognize the results as meaningful doesn't mean they aren't meaningful. he just couldn't spot it.

Similarly, a Pi result from a bunch of ten sided dice is statistically significant, even though some people won't be able to recognize the significance. Imagine SETI receiving two signals from two systems with possibly habitable planets: one is a series of bursts of static with milli-second pauses between bursts (17489126). Very intriguing. The other is also a series of bursts of static with milli-second pauses: (314159265). Which signal would generate the most excitement?


Quote:
I think that's what piggy's trying to describe: perhaps a string of identicals is attracting his attention more than other strings because there aren't a lot of special combinations out there he would recognize easily on account of being, well, normal, and not having a math degree or something.

And it's not even restricted to math: the problem generalizes to other disciplines like biology. Behe makes a good living reverse-engineering natural sequences in DNA or shapes of molecules for evidence of intelligent cause.
I think Piggy's point is that while it's theoretically possible for a person to randomnly toss a fair coin heads a billion times in a row, he can confidently say that such a result will never happen. For all intents and purposes, it's impossible.
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Old 18th February 2011, 06:23 PM   #196
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Originally Posted by Malerin View Post
Ahh, now what if the same person wins the same lottery again? We might say it's a fluke. Given how many people play the lottery, there might be some repeat winners. But if they win again? And again? At some point, you become extremely skeptical of chance, and much more open to some rigging of the lottery.
Or put another way....

When I lived in Florida, there was this guy from Ohio who won the Lotto up there and moved south, but for some reason kept playing. He won a second jackpot in Florida.

Astronomical odds, right?

Well, yeah, but considering how many people play the lottery, it's actually not unusual.

However, if every draw every week for a whole year were to go to a previous lottery jackpot winner, there would be no doubt in anyone's mind that something was seriously wrong.
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Old 18th February 2011, 06:30 PM   #197
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Originally Posted by Alan View Post
Could we perhaps agree that it is just very, very, very unlikely to happen any particular time instead of literally, truly, 100% impossible?
I think, with respect to the wishes of the OP, if anyone wants to pursue this topic any further, a new thread should be started.

In fact, this issue has cropped up in several different forms over the years here on JREF.

Can we say that leprechauns do not exist, or must we be content to simply call them "extremely unlikely"? Is "strong atheism" unjustifiable? Is it really true that it's possible that a statue might wave its hand, or that a mixture of gasses might coincidentally segregate? Are we really obliged to concede that Sagan's dragon is merely "unproven" and not "false"?

My stance has consistently been that it is incorrect to hedge on these issues. And I have as yet seen no provable arguments to the contrary.

But this thread is not the place to hash that out.

If anyone's interested, please, let's move it outside.
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Old 18th February 2011, 06:31 PM   #198
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Originally Posted by Malerin View Post
The inability to recogize the significance of a result doesn't mean the result can't disconfirm the null hypothesis of random chance. Let's go back to poker. Suppose my friend and I are watching a poker game. I know how to play, and my friend doesnt't. The dealer deals himself a royal flush. Then another. And another. My friend then asks why everyone has left the game. My friend's inability to recognize the results as meaningful doesn't mean they aren't meaningful. he just couldn't spot it.
Again, there is a difference between the assertion "sequence A is more likely to be produced by cheating than it is by chance" and "producing sequence A by chance is impossible".

Quote:
I think Piggy's point is that while it's theoretically possible for a person to randomnly toss a fair coin heads a billion times in a row, he can confidently say that such a result will never happen. For all intents and purposes, it's impossible.
But (as we keep pointing out) that's quite obviously a false statement. If that particular sequence is impossible because it is very unlikely, then all sequences of the same length are impossible, because they are all equally unlikely - a patently absurd conclusion that can be trivially falsified.

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Old 18th February 2011, 06:34 PM   #199
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Originally Posted by sol invictus View Post
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).
Quite correct, physics doesn't know or care which side is heads and which side is tails. But that's entirely irrelevant to my argument.

Physics also does not care which side of an airplane we consider "up" or "down" but this has no relevance to the question of the dynamics of an airplane attempting to navigate a hurricane.
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Old 18th February 2011, 06:35 PM   #200
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Originally Posted by tsig View Post
In this context, "almost surely" is a mathematical term with a precise meaning, and the "monkey" is not an actual monkey, but a metaphor for an abstract device that produces a random sequence of letters ad infinitum

How do you actually observe a metaphor?
No, the monkey is a real flesh-and-blood monkey.
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