SSDD: a 22 sigma event is consistent with the physics of fair coins?

_{Salvador Cordova

June 23, 2013

Mathematics}

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SSDD – Same Stuff, Different Darwinist. This time someone said at skeptical zone:

if you have 500 flips of a fair coin that all come up heads, given your qualification (“fair coin”), that is outcome is perfectly consistent with fair
coins,

Comment in The Skeptical Zone

So if someone has 500 fair coins, and he finds them all heads, that is consistent with expected physical outcomes of random flips? 😯 I don’t think so!

Correct me if I’m wrong but if you have 500 fair coins, the expectation is 250 coins will be heads, not 500. Now if you have 261 of the 500 coins heads, that is still within a standard deviation of expectation, and thus would still be a reasonable outcome of a random process. But 500 coins heads out of 500 fair coins? No way!

Given:

p = probability of heads: 0.5
n = number of coins: 500

Then the standard deviation for binomial distributions yields:

$\sigma = \sqrt{np(1-p)} = \sqrt{ 500 (0.5) (1-0.5)} \approx 11$

So 261 coins heads is (261 -250)/11 = 1 standard deviations (1 sigma) from expectation from a purely random process of coin flips.

So 272 coins heads is (272 -250)/11 = 2 standard deviations (2 sigma) from expectation from a purely random process of coin flips.

….

So 500 coins heads is (500-250)/11 = 22 standard deviations (22 sigma) from expectation! These numbers are so extreme, it’s probably inappropriate to even use the normal distribution’s approximation of the binomial distribution, and hence “22 sigma” just becomes a figure of speech in this extreme case…

There are many configurations that are 250 coins heads. The number is:

$C(n,r) = \frac{n!}{r!(n-r)!} = \frac{500!}{250!(500-250)!} \approx 1.17 \times 10^{149}$ , thus there are many coin configurations most consistent with the expectation of 250 coins heads (50%), whereas only 1 configuration of all heads for the least consistent behavior for a fair coin.

Bottom line, the critic at skeptical zone is incorrect. His statement symbolizes the determination to disagree with my reasonable claim that 500 fair coins heads is inconsistent with a random physical outcome.

SSDD.

Comments

What a very peculiar conversation! Mung:
No, according to Elizabeth, you have to know the distribution before you can assign a probability. Yes, she really said that.
Would you like to link to just where I said that, Mung? What I have said, and will say again, is that unless you know the probability distribution of your data, you can't assign a probability to any one observation. Do you disagree? I assume not! Sal:
What KeithS, no refutation of the calculations I provided above. You think a 22-sigma deviation from the mean is “consistent” with expected behavior?
Yes, of course it is. Every single sequence you toss has a 1/2^500 probability of being tossed. You know this, and so does Barry (he sais so in the OP). We all know this. This argument is about zilch! None of us even disagrees (except possibly Granville). All toss sequences are equally consistent with the laws of physics, and all are equally improbable. However, some classes of sequence are less probable than others, so if you get a member of that class you may be rather surprised, and, rightly, suspect skulduggery. But you should be neither more nor less suspicious if you get all heads, than if you get alternating heads or tails, or a sequence of prime numbers in binary, or the sequence you wrote down beforehand (including the one I just posted). The point is that what is extremely unlikely is a member of the tiny set of sequences that you consider "special" - you are much more likely to get a member of the much larger class of sequences that are not "special". So while every single sequence has a 1/2^500 probability of being tossed, the probability of seeing one of the class of "special" numbers is (size of set of special numbers)/2^500 probability, while the probability of seeing some other number is (500-size of set of special numbers)/2^500 JDH:
What I would like each one of you to consider is how do you find evidence for the presence of a superior intellect. Can you do a “controlled” experiment? No way. You can’t do a controlled experiment to prove the existence of a superior intelligence. This should be obvious, because you don’t control the superior intelligence. You can’t order Him to do something, you can only observe what He supposedly does.
I absolutely agree. Sal
No dice, KeithS, you’ll have to assail the calculation deviation from expectation I provided, and you know you aren’t going to do that since those are textbook derivations. You and Eigenstate are the ones promoting an idiosyncratic interpretation of statistics. You’re the guys who’ll have to admit error and stop trying to save face.
Your only error, Sal, and I think was just mistyping, was to imply that a priori, we know that the coin was fair and that it had been normally tossed. In other words, you inadvertently eliminated design a priori, leaving fluke. However, that probably wasn't what you meant. Either you meant that the coin was fair, but the tossing was dodgy, or the tossing was fair but the coin was dodgy. In which case, getting 500 heads would certainly lead you to conclude "dodgy", with a confidence of 500 sigma. But not, obviously, if you knew for a fact that it was all kosher. The whole point of null hypothesis testing is not that it tells you your observation is "impossible" under your null, but that it Extremely Unlikely. Setting your rejection region at 22 sigma doesn't suddenly turn "Unlikely" into "Impossible", simply because 2^500 is deemed to be the maximimum number of events in the universe. Or do people think that it does? If it did, no sequence would ever be tossed, because every single sequence, as Barry says, has a probability of 1/2^500! As I said, we all pretty much agree on this.Elizabeth B Liddle_{June 23, 2013
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Keiths, In another thread, keiths said or quoted,
Such a leap would be justified only if he already knew that homochirality couldn’t be explained by any non-chance, non-design mechanism (such as Darwinian evolution).
Since when is Darwinian evolution a non-chance mechanism. This is what I believe is the most foolish thing about Darwinism. It really believes that you can get purpose out of non-purpose, directed out of unguided. Natural selection is not a magic potion that builds design out of non-design. I firmly believe it is impossible to generate purpose and information from random steps. The best you can do with random steps is a fluctuation from the mean. You can't have direction come about from unguided random steps. It is mathematically impossible. If you think it is possible, please prove it to me with clear mathematics. Not with hand-waving and assertion. Oh and don't point to self organization. Self-organization is an information destroying process, not an information increasing process.JDH_{June 23, 2013
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Jerad, I hope I don't sound condescending in my reply. I do have respect for all of my opponents intellects, I just think they believe things that are foolish. I hope this belief of mine does not make my writing offensive. Anyway - here is where understanding the terms necessary and sufficient is really useful. 500 heads in a row of a proven fair coin is certainly a sufficient proof of divine intervention. ( i.e. P => Q ) It is not a necessary condition. ( i.e. !P does not => !Q ) In other words I would choose to believe divine intervention has taken place if 500 heads of a fair coin were flipped in a row. I would not choose to believe in divine intervention if and only if 500 heads of a fair coin were flipped in a row. Is that clear?JDH_{June 23, 2013
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Hi JDH,
If 500 heads in a row of a fair coin would not convince you that “design” is taking place, then the debate is over. It’s not because I cannot come up with evidence, it’s that you will not believe any evidence that I bring forth.
See my comment here.keiths_{June 23, 2013
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And one thing else I want people to recall. Many of the laws of physics - at least the ones we can observe at the macroscopic level are probabilistic in nature. Second law of Thermodynamics, Diffusion, Resistance, Radioactivity etc. these are all well established laws which work because the probability of observing an exception is essentially nil. In my opinion your insistence that there is a small but finite probability that 500 heads in a row could happen is not clarification of a mathematical curiosity, it is a misunderstanding of physics and how it works.JDH_{June 23, 2013
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If 500 heads in a row of a fair coin would not convince you that “design” is taking place, then the debate is over. It’s not because I cannot come up with evidence, it’s that you will not believe any evidence that I bring forth.
If 500 heads in a row convinces you of some kind of divinity does that mean if you never see it you won't believe?Jerad_{June 23, 2013
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With eigenstate’s approach, the chance hypothesis is never rejectable in principle! And that is completely in opposition to operational practices where deviations from expectation count for something.
That is untrue. Set up a proper hypothesis testing trial and then you WILL get a rejection of a null hypothesis with a defined confidence interval.
What KeithS, no refutation of the calculations I provided above. You think a 22-sigma deviation from the mean is “consistent” with expected behavior?
Sal, you've already shot your argument in the foot. The only way for all possible 500 sequences of Hs and Ts to be equally likely is for them all to be consistent with a fair coin being flipped.Jerad_{June 23, 2013
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Lizzie, KeithS, eignestate, Jared, and anyone else "listening." I will tell you what so bothers me about this whole thread. I believe firmly that God has made it obvious that He designed the world. I think there is abundant evidence for this everywhere. What I would like each one of you to consider is how do you find evidence for the presence of a superior intellect. Can you do a "controlled" experiment? No way. You can't do a controlled experiment to prove the existence of a superior intelligence. This should be obvious, because you don't control the superior intelligence. You can't order Him to do something, you can only observe what He supposedly does. This is the fallacy of trying to design a controlled experiment about prayer. If God exists he can choose to have the answer to the experiment be anything He wants. But why should He subject himself to the experimenters. He is God. He intervenes when He wants to, in a way that is consistent with His agenda, not ours. The only way to scientifically determine if there is a superior intelligence is to investigate what you suppose He has done, ( like create the world ) and see if it shows design. While I see design all over the place ( and according to his own quotes Mr. Dawkins at least see the appearance of it ) the real stickler is how can I get others to admit that the acknowledged appearance of design is because it was designed. I can only show that the chances of a natural event being the cause of the apparent design is improbable. And here is the real problem. I don't think any of the evidences I could present ( e.g. fine-tuning, consciousness, free-will ) are as understandable as a simple binary event done "n" times. If 500 heads in a row of a fair coin would not convince you that "design" is taking place, then the debate is over. It's not because I cannot come up with evidence, it's that you will not believe any evidence that I bring forth.JDH_{June 23, 2013
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Sal, Of course it is more likely for a 500-flip sequence to contain some combination of 250 heads versus all 500 heads. Both Eigenstate and I have said exactly that: link, link But absolutely nothing about that contradicts what eigenstate said in the quote you are disputing. Here's what we get if we undo your quotemine. The portions you cut are highlighted in bold):
Maybe that’s just sloppily written, but if you have 500 flips of a fair coin that all come up heads, given your qualification (“fair coin”), that is outcome is perfectly consistent with fair coins, and as an instance of the ensemble of outcomes that make up any statistical distribution you want to review. That is, physics is just as plausibly the driver for “all heads” as ANY OTHER SPECIFIC OUTCOME.
Eigenstate is correct. Every possible sequence is equally probable and therefore equally consistent with the physics of fair coins. You were wrong to challenge him (and also wrong to quotemine him) and it is only appropriate that you acknowledge this in an appendix to your OP.keiths_{June 23, 2013
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What KeithS, no refutation of the calculations I provided above. You think a 22-sigma deviation from the mean is "consistent" with expected behavior? :roll:scordova_{June 23, 2013
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Sal, If you flip a fair coin 500 times, there are 2^500 possible sequences. Each specific sequence is possible, and each specific sequence is equally (im)probable. They are all equally consistent with the physics of fair coins. If any of them were inconsistent with the physics of fair coins, it would mean that they could not happen. But that's ridiculous. Sure, a sequence of all heads is improbable, as is any other specific sequence. But impossible? No way. The probability of getting all heads is small but nonzero -- just like every other sequence. Eigenstate was correct:
if you have 500 flips of a fair coin that all come up heads, given your qualification (“fair coin”), that is outcome is perfectly consistent with fair coins,
He was correct because the all-heads sequence, like every other specific sequence, is equally consistent with the physics of fair coins. After all, that's what it means for the coins to be fair.keiths_{June 23, 2013
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Since eigenstate was correct all along, I think you should append a notice to your OP stating that he was correct and that your challenge was mistaken.
No dice, KeithS, you'll have to assail the calculation deviation from expectation I provided, and you know you aren't going to do that since those are textbook derivations. You and Eigenstate are the ones promoting an idiosyncratic interpretation of statistics. You're the guys who'll have to admit error and stop trying to save face.scordova_{June 23, 2013
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keiths:
All heads is perfectly consistent with the physics of fair coins, as is any other specific sequence.
A "fair" coin with a "heads" on each side is perfectly consistent with "the physics of fair coins." A "fair" coin with a "tails" on each side is perfectly consistent with "the physics of fair coins." A "fair" coin with three sides with "blech" on each side is perfectly consistent with "the physics of fair coins." thanks keithsMung_{June 23, 2013
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Sal, Follow the link in my last comment. I've explained it on the other thread.keiths_{June 23, 2013
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Even ignoring the fact that you quotemined him, his statement is correct as it stands. All heads is perfectly consistent with the physics of fair coins, as is any other specific sequence.
No it is not, unless of course you figure a 22-sigma deviation from expectation is "perfectly consistent" with expectation of fair coins. You're assessing probabilities with respect to other sequences, but in practice, one also assesses probabilities with respect to expectation. With eigenstate's approach, the chance hypothesis is never rejectable in principle! And that is completely in opposition to operational practices where deviations from expectation count for something. If you stand by eigenstate's comments, then on what grounds will you ever reject the chance hypothesis short of you seeing someone rigging an apparatus, etc.? Answer: NEVER, because in eigenstate's world, what matters to him is every sequence is just as probable as the next, whereas in operational practice, deviations from expectation value count for something.scordova_{June 23, 2013
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Sal, The point of your OP was to disagree with this statement by eigenstate:
if you have 500 flips of a fair coin that all come up heads, given your qualification (“fair coin”), that is outcome is perfectly consistent with fair coins,
Even ignoring the fact that you quotemined him, his statement is correct as it stands. All heads is perfectly consistent with the physics of fair coins, as is any other specific sequence. The reason you would find an all heads outcome to be suspicious is not because it is inherently improbable, but because it is both signficant and improbable. See this comment on the other thread for details. Since eigenstate was correct all along, I think you should append a notice to your OP stating that he was correct and that your challenge was mistaken.keiths_{June 23, 2013
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Neil Rickert:
One has to distinguish between a sequence and a combination. For two coins, ... are equally probable sequences.
No, according to Elizabeth, you have to know the distribution before you can assign a probability. Yes, she really said that.Mung_{June 23, 2013
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Unfortunately Sal worded it confusingly. He told us that the coin WAS fair, and seemed to imply that they WERE flipped by fair means.
I did not say that, I said if 500 fair coins were found to be all heads. We would reject that a random coin tosses was the means that created it. But the next iteration, I'll try to make it even more clear. Suppose you opened a box containing the coins, and all of the 500 of them were in the heads state... Thank you for the criticism, and I'll change my wording in response to you criticism. Thanks to all for reading and commenting on my thread.scordova_{June 23, 2013
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Jerad agrees of course.Jerad_{June 23, 2013
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But Elizabeth you are wrong. Let’s break down each statement. 1. Coins are fair. ( This is consistent with the “physics of fair coins”. ) 2. Flipping a fair coin creates some sequence of heads and tails ( This is consistent with the “physics of fair coins”. ) 3. A human being can read that sequence and classify the number of heads and the number of tails, and write it down in a different representation ( “HTHH..”) ( This is consistent with the “physics of fair coins”. ) 4. The set of flips matches exactly any pre determined sequence. ( This is NOTconsistent with the “physics of fair coins”). I can only conclude that the reason you don’t see the difference between statements 1-3 and statement 4 is because you are willingly closing your eyes and ears to reasonable, logical argument.
No, the conclusion you should draw is that the point is so trivial, people are missing it! Clearly the chances of getting a predetermined (written down in advance, for instance) sequence from a series of coinflips is so infinitessimal, whatever the sequence (and, as Barry rightly says, no one sequence is any more probable than any other), that were we to see it done, we'd rightly assume some kind of skulduggery. Sal said, or meant, something slightly more interesting, which is that even if we hadn't predetermined the sequences, if we were to observe one of a tiny class of sequences that is easily described (all heads, all tails, alternating heads and tails, runs of prime numbers of heads, whatever), we'd suspect skullduggery. You agree, I agree, Sal agrees, Barry agrees, Jerad agrees (I think), eigenstate agrees, we all agree. All heads, all tails, regular stripes, whatever, would make us go count the spoons. There's something weird with coin, or there's something weird with the toss. Unfortunately Sal worded it confusingly. He told us that the coin WAS fair, and seemed to imply that they WERE flipped by fair means. And that the result was all heads. Clearly that can't mean skullduggery, because Sal's already told us there wasn't any. And as all-heads is perfectly consistent with fair coins and fair tossing, there's no reason to invoke anything design. On the other hand if one of the coins turned into a flamingo and walked away, well, then, we'd have to seriously consider that the law of physics had been violated! Honestly, that's all the issue was. :) Schnapps? or, wait... Is it possible that some people really think that it is physically impossible ("against the laws of physics") for a series of 500 tosses to come down 500 heads? That would be interesting.Elizabeth B Liddle_{June 23, 2013
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Elizabeth @17 said:
All he’s saying is that given the coins are fair and that they were flipped by the and that they were flipped by the normal “physics of fair coins”, then the outcome is perfectly consistent with both fairness and physics.
But Elizabeth you are wrong. Let's break down each statement. 1. Coins are fair. ( This is consistent with the "physics of fair coins". ) 2. Flipping a fair coin creates some sequence of heads and tails ( This is consistent with the "physics of fair coins". ) 3. A human being can read that sequence and classify the number of heads and the number of tails, and write it down in a different representation ( "HTHH..") ( This is consistent with the "physics of fair coins". ) 4. The set of flips matches exactly any pre determined sequence. ( This is NOTconsistent with the "physics of fair coins"). I can only conclude that the reason you don't see the difference between statements 1-3 and statement 4 is because you are willingly closing your eyes and ears to reasonable, logical argument.JDH_{June 23, 2013
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All heads is a prespecification.
Agreed. However, my point is we can in special cases bypass the discussion of specification in rejecting the chance hypothesis. I did not have to invoke CSI to make a case for design in certain statistical situations...scordova_{June 23, 2013
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However, if the fairness of the coins is not a given, or the physics of the flipping is not a given, then you would have reason to suspect unfair coins, or unfair tossing, respectively.
OR that a human didn't flip the coins in the 500 coin set randomly, but rather arranged them intelligently. That inference is reasonable (not absolute), especially given we know the capabilities of human intelligent designers. I respect many will not be willing to make such extrapolations to the designs of life, but I'm astonished that eigenstate would object to my claim of "all coins heads" as inconsistent with random processes on fair coins. I've shown in this essay, I was correct in saying "all coins heads" is inconsistent with the physics of fair coins from the standpoint of probability and statistics as used in operational practice, especially the notion of expectation values and deviations from expectation. I appreciate the TSZ crowd joining in the conversation, but I have to point out, it seems evolutionists really don't want to even give the appearance of agreeing a creationist even on uncontroversial matters. I could have said, "finding 500 coins heads, violates expectation of fair coins, therefore finding 500 fair coins all heads is not reasonably the result of chance." And that should have been the end of it, but because I'm a creationist, the point has to be belabored.... That's ok, it give us a chance to talk about math in detail, and that is a good thing, in and of itself.scordova_{June 23, 2013
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"Pure chance is sufficient to explain human beings, the probability of the exact arrangement of atoms in a human is no more improbable than the exact arrangement of atoms in a given pile of rubble." Is that Granville Sewell's definition of 'probabilism'?Gregory_{June 23, 2013
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What is far more interesting and relevant is why do we reject the fair coin hypothesis in this case or indeed a large range of other interesting strings – some with 50% heads? It clearly is not because this particular string is more improbable than other strings – they are all equally improbable. Nor can it be that other strings belong to larger classes of strings – the all heads string belongs to the very large class of strings with more than 260 heads.
One may attribute it to the convenience descriptions that we humans use to do statistics and science. One may even argue it is purely subjective that we concoct notions of expectation values in order to make inferences about the universe tractable to our brains. To paraphrase Laplace on Probabilities
Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective situation of the beings who compose it – an intelligence sufficiently vast to submit these data to analysis – it would embrace in the same formula the movements of the greatest bodies of the universe and those of the lightest atom; for it, nothing would be uncertain and the future, as the past, would be present to its eyes .... but since we’re not God, we need probability.
We have the sort of math we do because of our human uncertainty of outcomes. Hence we think of the world in terms of expectation values since we have uncertainty... So let us even suppose for the sake of argument, operational practice in statistics and physics is a matter of intellectual convenience in a world of uncertainties. This does not negate the fact when intelligent agents act in a way that violate our convenience descriptions of what we deem chance processes, we will reasonably (not absolutely) infer a human-like intelligence is in operation, or at least something that looks like a human-like intelligence. And if I may make a nuanced description. I was not talking about 500 sequential coin flips of one coin, I was talking about a set of 500 fair coins. If we found 500 fair coins all heads we would reasonably (not absolutely) 1. reject the chance hypothesis 2. accept some mechanism was responsible for the 22-sigma deviation from expectation value Because we know human designers are capable of directly or indirectly making all the coins heads, we could reasonably infer a human-like intelligence made the coins heads since the such intelligent designers (humans) have sufficient capability to do so. I respect that some will not be willing to extrapolate such reasoning to designs in life since the Intelligent Designer of life is not seen in operation today. That's a respectable position, but one I don't share personally. I have gone through a lot of trouble to suggest, on scientific grounds alone, one such Intelligent Designer can be postulated: Quantum Enigma of Consciousness and the Identity of the Designer. On scientific grounds this would be viewed as speculative. Some would say it's not even science. I respect that. But life to me doesn't seem to accord with a chance hypothesis, it violates mathematical expectation from what we know of chemistry and physics. In principle, our understanding of physics and chemistry could change, but I personally don't think that will happen. Thank you for sharing your thoughts, and it is nice to see you here at UD.scordova_{June 23, 2013
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All heads is a prespecification.Joe_{June 23, 2013
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Sal, you simply missed eigenstate's point, which was fairly trivial. I think you just mistyped. You wrote:
For example, consider if we saw 500 fair coins all heads, do we actually have to consider human subjectivity when looking at the pattern and concluding it is designed? No. Why? We can make an alternative mathematical argument that says if coins are all heads they are sufficiently inconsistent with the Binomial Distribution for randomly tossed coins, hence we can reject the chance hypothesis. Since the physics of fair coins rules out physics as being the cause of the configuration, we can then infer design.
He wrote:
Maybe that’s just sloppily written, but if you have 500 flips of a fair coin that all come up heads, given your qualification (“fair coin”), that is outcome is perfectly consistent with fair coins, and as an instance of the ensemble of outcomes that make up any statistical distribution you want to review.
All he's saying is that given the coins are fair and that they were flipped by the and that they were flipped by the normal "physics of fair coins", then the outcome is perfectly consistent with both fairness and physics. However, if the fairness of the coins is not a given, or the physics of the flipping is not a given, then you would have reason to suspect unfair coins, or unfair tossing, respectively. It was just the way you phrased it. I presume you meant to say something like:
For example, consider if we saw 500 ~~fair~~ normally tossed coins all heads, do we actually have to consider human subjectivity when looking at the pattern and concluding ~~it is designed~~ that the coins were not fair? No. Why? We can make an alternative mathematical argument that says if coins are all heads they are sufficiently inconsistent with the Binomial Distribution for randomly tossed coins, hence we can reject the chance hypothesis. Since the physics of fair coins rules out physics as being the cause of the configuration, we can then infer design.
Elizabeth B Liddle_{June 23, 2013
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The reason why 500 straight coins would raise eyebrows, and most other results, while equally improbable, would not, is easy: because "all heads" is simply describable, and most others are not (many would be describable only in 500 bits, by actually listing the result). If we flip n fair coins, the probability that the result can be described in m bits, since there are at most 2^m such results, is less than 2^m/2^n. So if you flip a billion coins, and get "all heads" or "(only) all prime numbered coins are heads" you would rightly be surprised and suspect something other than chance. If the TSZ writer is correct, why do Darwinists need Darwinism? Pure chance is sufficient to explain human beings, the probability of the exact arrangement of atoms in a human is no more improbable than the exact arrangement of atoms in a given pile of rubble.Granville Sewell_{June 23, 2013
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scordova:
So why won’t you agree with me?
The disagreement is not about the analysis of combinations. It is about whether the issue was combinations or ordered sequences.Neil Rickert_{June 23, 2013
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The probability of getting EXACTLY 250 heads and 250 tails in any order is fairly small. Calculate it out.
About 3.57% according to the binomial distribution: See: http://stattrek.com/online-calculator/binomial.aspx enter probability: 0.5 number of trials : 500 number of successes : 250 67% of population is within one sigma from the mean, 95% within 2 sigma, and 99 44/100% within 3 sigma, etc. 22 sigma is such an extreme case, it is only a figure of speech since the binomial distribution is not well approximated by the normal distribution in such an extreme case.
If you write down a sequence of 500 Hs and Ts and start flipping coins you will probably not get that particular sequence in your life time.
nor the lifetime of the universe.scordova_{June 23, 2013
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