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

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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,

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:

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

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:

$\large \large C(n,r) = \frac{n!}{r!(n-r)!} = \frac{500!}{250!(500-250)!} \approx 1.17 \times 10^{149}$, 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.

OK, well, that's reassuring, even if only a few people answered!Elizabeth B Liddle
June 27, 2013
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June 26, 2013
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If you look into a sequence of coin tosses, the odds are 2:1 that your pattern of 236 Heads and 264 Tails will appear before a pattern of 500 Heads (See e.g. Li, S., 1980: A martingale approach to the study of occurrence of sequence patterns in repeated experiments. The Annals of Probability 8, 1171–1176), thus the answer would be B.
Answer B is: Less than the probability that someone will throw an all-heads sequence? If odds are 2 to 1 that Liz's sequence will appear before all coins heads, then that means it has a higher probability than all coins heads, not lower. Thus for the Answer is A not B for non-tests of 500, Answer C for tests of 500 tosses.scordova
June 26, 2013
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That's a very nice point DiEb! I had envisaged only tests of 500 tosses.Elizabeth B Liddle
June 26, 2013
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Is the probability that this sequence will ever be thrown again by anyone in the history of the universe: A:Greater than the probability that someone will throw an all-heads sequence? B:Less than the probability that someone will throw an all-heads sequence? C:The same as the probability that someone will throw an all-heads sequence?
DiEB just gave away the store. It most likely will never happen but in the magical world of infinities your pattern is more likely to appear than 500 heads.Joe
June 26, 2013
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Which do you vote for, Joe?Elizabeth B Liddle
June 26, 2013
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@Elizabeth: How is the experiment performed? If you look into a sequence of coin tosses, the odds are 2:1 that your pattern of 236 Heads and 264 Tails will appear before a pattern of 500 Heads (See e.g. Li, S., 1980: A martingale approach to the study of occurrence of sequence patterns in repeated experiments. The Annals of Probability 8, 1171–1176), thus the answer would be B. If you perform only tests of 500 tosses, than the answer is C. Isn't the theory of probability marvelous?DiEb
June 26, 2013
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Is the probability that this sequence will ever be thrown again by anyone in the history of the universe:
You didn't throw it, so the next time it is thrown will be the first. Over on TSZ they are discussing the physics of it and you took that out. And then you act as if that's OK. But anyway, the odds of 500 heads in a row is 1 in 2^500. The odds of getting a pattern is 1. The safe answer is the odds of hitting your pattern is less than or equal to 2^500.Joe
June 26, 2013
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#62 Lizzie - actually I think it is B. I imagine there are people out there working on ways to cleverly produce a sequence of 500 heads e.g. by special means of tossing the coin - no one is likely to be working on ways to produce the sequence you tossed.Mark Frank
June 26, 2013
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Is the probability that this sequence will ever be thrown again by anyone in the history of the universe: A:Greater than the probability that someone will throw an all-heads sequence? B:Less than the probability that someone will throw an all-heads sequence? C:The same as the probability that someone will throw an all-heads sequence?
I vote for C. But, sadly, the attention seems to have moved on. gpuccio might respond but I suspect Sal and Barry have moved to greener pastures.Jerad
June 26, 2013
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Anyone?Elizabeth B Liddle
June 26, 2013
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Incidentally, I should add that I’m not sure all ID folks who use coin toss examples are quite as careful with their descriptions or setup of the scenario as they should be. That may be part of the disconnect we sometimes see. But I think an objective person can easily see that there is something “special” about particular outcomes and should spend some time thinking about the why.
I agree with both these points. I'd also like to be clear that people don't think that there is some arcane physical law that means that the more heads you've tossed the more likely it is that the next one will be tails. It is not the "Laws of Physics" that make more equal ratios more probable than less equal ratios. I think that is a real danger inherent in using the word "probability" too loosely. As I constantly ask my students: "probability of what?" In the context of Fisherian inference and the binomial theorem, sequences with fairly equal ratios are more "probable" is simply that they are more of them! Same with sequences that are "special" - there are far more non-special sequences than special sequences, however we describe "special" - whether it's your phone number, or stripes, or prime numbers, or even the sequence you last threw. I'd like, as a test question, to ask people this question: I just threw (well, "threw" - I used Excel) this 500 toss sequence: T H H T H T H T H T T T H H H T T T H T H H T H T T H H T H T T T T H H T T H H H H T T H H H T H T T H T H T T H T H H T T T T T T T H T T H T H T T T H T H H T H T H H H H H T H H T T H H H T T T T T T T H T T H T T T H T H H H H H T H T T T H T T T H H H T T T T T T T H T T H H H T H T H H T H H H H T H H T H T T T H H H T T T T T H T H T T T T T H T T T H H H T T H H T T H H H H H T T H H H T T H T T H T T T H H H H H T H H T T H H T H T H T H T T T H T T H T T T T T T T H H H T T T T T T T H T H T T H H H T T T H H T T H H T T H T T T H H H T T H T H T T H T T T H T H T H H H T H H T T T T H T H T H T H T T T T T T T H H H H T T T T T H H H T H H T T T T H H H T T H T T H T H T T H H T H H H T H T T T T H H H H T T T H H H H H T H H H T T T T T H H H T H T T T H H T T H T H H T H H H T H T H H T T T T T T H T T H H H H T H H H H H T H T H H T H T H T H T H T H T H H T H H T H H T T H T T T T H H T T H H T T H T T H H T H T H H H T H H T T T H T H T H T T T H H H H T H T H T H T H There are 236 Heads and 264 Tails. Is the probability that this sequence will ever be thrown again by anyone in the history of the universe: A:Greater than the probability that someone will throw an all-heads sequence? B:Less than the probability that someone will throw an all-heads sequence? C:The same as the probability that someone will throw an all-heads sequence? Answers to this question might at least reconnect some of the apparent disconnects! Or possibly reveal more fundamental differences than I think exist)Elizabeth B Liddle
June 26, 2013
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Eric,
I think you have some good thoughts there.
Thanks. Yes, the numbers don't matter; their meaning doesn't matter; and even the fact that they belong to my set of "numbers that are meaningful to me for any reason or just because I say so" doesn't really matter. What matters is that they form a small set, so that getting one of those numbers is improbable under the assumption of fairness. In other words, a surprise. But here the paradox rears its head again. As I put it in the OP at TSZ:
In fact, there are millions of different ways to carve up the 9-digit numbers into two sets, one huge and one tiny. Should we always be surprised when we get a number that belongs to a tiny set? No, because every number belongs to some tiny set, properly defined. So when I’m surprised to get my own SSN, it can’t be merely because my SSN belongs to a tiny set. Every number does. The answer, I think, is this: when we roll Delbert’s SSN, we don’t actually conclude that the die was fair and that the rolls were random. For all we know, we could roll the die again and get Delbert’s number a second time. The outcome might be rigged to always give Delbert’s number. What we really conclude when we roll Delbert’s number is that we have no way of determining whether the outcome was rigged. In a one-off experiment, there is no way for us to tell the difference between getting Delbert’s (or any other random person’s) SSN by chance versus by design or some other causal mechanism. On the other hand, rolling our own SSN does give us a reason to be suspicious, precisely because our SSN already belongs to the tiny set of “numbers that are meaningful to me”.
That last paragraph could use some improvement. I'm working on a better description of my resolution of the paradox, but it will have to wait until tomorrow.keiths
June 26, 2013
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June 25, 2013
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To say that “500 heads is just as probable as any other sequence” and think that this is somehow a refutation of the design inference is just as illogical as saying “every arrangement of stones is just as improbable as the next; therefore the Taj Mahal was not designed.”
I would not think that is a refutation of the design inference except in the case where it is used as a justification of the design inference. Nor would I equate 500 heads in a row with something that was clearly NOT created via a series of random events like the Taj Mahal. IF a coin toss is fair, i.e. truly random, then each event is independent of the ones that came before and therefore there can be no conscious pattern or structure being created. Something like the Taj Mahal or the Pyramids or Stonehenge (all of which are much more complicated than a sequence of coin tosses) were clearly created in a sequence where every new stone was placed in a non-random process, in a particular relationship with all that had gone before. The raw materials have been displaced from their natural sources and have been worked in ways consistent with the working techniques known to be used by the intelligent agents known to be around at the time. In all those cases we have a good idea of how and who made them. And pretty good ideas of why as well.Jerad
June 25, 2013
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BTW, the odds of flipping a coin 500 times and getting some pattern is 1.Joe
June 25, 2013
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For example:
My point is that for anyone to roll their SSN (or another personally significant 9-digit number) is highly unlikely, because the number of personally significant 9-digit numbers is low for each of us while the space of possible 9-digit numbers is large. If someone sits down and rolls their SSN, then either: 1. The die was fair, the rolls were random, and they just got lucky. It’s pure coincidence. 2. Something else is going on.
If I rolled my SSN I would be totally amazed. How many other people's SSN have numbers greater than 6? All heads is a pre-specification, keiths. The probability of getting 1 head is 1/2. 2 heads in a row is 1/4. 3 heads in a row is 1/8. 10 heads is 1/1024. The odds of 500 heads in a row is 1/2^500. I don't know if that is a paradox, but it is what it is.Joe
June 25, 2013
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I know how it is normally defined. I was asking how you are defining it. The point being keiths couldn't resolve a paradox if his life depended on it. And he cannot resolve that which he clearly doesn't understand.Joe
June 25, 2013
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Joe,
How are you defining “resolution”?
To resolve a paradox is to explain it in a way that makes complete sense, but that Joe couldn't follow even if his life depended on it.keiths
June 25, 2013
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keiths- How are you defining "resolution"?Joe
June 25, 2013
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Eric,
What is it about the 500 heads in a row that demands our attention, that demands an explanation? That is the key question here.
June 25, 2013
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June 25, 2013
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See: The Law of Large Numbers vs. KeithS, Eigenstate and my TSZ Criticsscordova
June 24, 2013
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But there is no mathematical problem with getting 500 heads in a row even though it is exceedingly unlikely.
If you really believe that then start flipping a coin and let us know when you have flipped 500 heads in a row.Joe
June 24, 2013
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keiths:
All sequences are equally probable, as even you now admit.
If you really believe that then start flipping a coin and let us know when you have flipped 500 heads in a row.Joe
June 24, 2013
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p. 148: “The great myth of contemporary evolutionary biology is that the information needed to explain complex biological structures can be purchased without intelligence. My aim throughout this book is to dispel that myth . . . . Eigen and his colleagues must have something else in mind besides information simpliciter when they describe the origin of information as the central problem of biology. I submit that what they have in mind is specified complexity, or what equivalently we have been calling in this Chapter Complex Specified information or CSI . . . . Biological specification always refers to function . . . In virtue of their function [[a living organism's subsystems] embody patterns that are objectively given and can be identified independently of the systems that embody them. Hence these systems are specified in the sense required by the complexity-specificity criterion . . . the specification can be cashed out in any number of ways [[through observing the requisites of functional organisation within the cell, or in organs and tissues or at the level of the organism as a whole] . . .” p. 144: [[Specified complexity can be defined:] “. . . since a universal probability bound of 1 [[chance] in 10^150 corresponds to a universal complexity bound of 500 bits of information, [[the cluster] (T, E) constitutes CSI because T [[ effectively the target hot zone in the field of possibilities] subsumes E [[ effectively the observed event from that field], T is detachable from E, and and T measures at least 500 bits of information . . . ”
(And, Stephen Meyer presents much the same point in his Signature in the Cell, 2009, not exactly an unknown book.) Why then do so many statistically or mathematically trained objectors to design theory so often present the strawman argument that appears so many times yet again in this thread? First, it cannot be because of lack of capacity to access and understand the actual argument, we are dealing with those with training in relevant disciplines. Nor is it that the actual argument is hard to access, especially for those who have hung around at UD for years. Nor is such a consistent error explicable by blind chance, chance would make them get it right some of the time, by any reasonable finding, given their background. So, we are left with ideological blindness, multiplied by willful neglect of duties of care to do due diligence to get facts straight before making adverse comment, and possibly willful knowing distortion out of the notion that debates are a game in which all is fair if you can get away with it. Given that there has been corrective information presented over and over and over again, including by at least one Mathematics professor who appears above, the collective pattern is, sadly, plainly: seeking rhetorical advantage by willful distortion. Mendacity in one word. If we were dealing with seriousness about the facts, someone would have got it right and there would be at least a debate that nope, we are making a BIG mistake. The alignment is too perfect. Yes, at the lower end, those looking for leadership and blindly following are jut that, but at the top level there is a lot more responsibility than that. Sad, but not surprising. This fits a far wider, deeply disturbing pattern that involves outright slander and hateful, unjustified stereotyping and scapegoating. Where, enough is enough.>> ______________ I trust that we can now proceed with a more reasonable approach. G'day KFkairosfocus
June 24, 2013
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Sal, Here are the facts: 1. You quotemined eigenstate in your OP. 2. You claimed that eigenstate's statement was wrong, even including an emoticon to express your incredulity at his supposed error. 3. Here is eigenstate's full statement -- the one you chose to quote in the OP -- with the parts you cut out 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 was correct. All sequences are equally probable, as even you now admit. You were wrong to dispute his statement. The error is entirely yours, and the responsibility for retracting your claim thus rests entirely with you.keiths
June 24, 2013
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Eigenstate is correct. Every possible sequence is equally probable
I never said otherwise, but I did say:
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,
So are you saying KeithS that all coins 500 fair coins heads is consistent with the binomial distribution? Yes or no? Are you saying patterns that are 10 sigma or more from expectation are consistent with theory?
Every possible sequence is equally probable
True, but that's not the point I was making nor contesting, and it is rebutting an argument I didn't make, I said nothing of the sequences being or not being equiprobable (a false insinuation by you and eigenstate), I said:
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,
If all sequences are equiprobable, is there any sequence, based on the sequence alone which you can reject the chance hypothesis? Yes, and it has nothing to do with it being equiprobable, but with its inconsistency against expectation of fair coins (in your words, advance knowledge). All sequences are equiprobable, but not all sequences are consistent expectation plus or minus a few sigma. You and eigenstate are misreading, misconstruing, misattributing arguments to me which I didn't make. I never said or implied that all heads is more probable than any other specific sequence. That is you and eigenstate's false insinuation... Eigenstate has to make a retraction:
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,
NO it is not. Equiprobability is not the criteria for rejection or acceptance of the chance hypothesis, it is the deviation from expectation. If you had a theory that predicted an expectation value and then you ran an experiment that yielded results 10 sigma from expectation, are you saying you wouldn't find that disconcerting? That this wouldn't raise alarm. This is exactly what I was getting at when I said:
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,
And now that I went into even more detail with the binomial distribution in the OP, you have no excuse to keep repeating your misreading and misrepresentations of what I said or implied.
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,
Apparently eigenstate will have to live with that declaration from now on. :-)scordova
June 24, 2013
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was to imply that a priori, we know that the coin was fair and that it had been normally tossed.
I implied we know a priori the coin is fair, I did not say the tossing was normal or even that tossing was used to make the pattern, all I said was the coins were discovered in the all heads state and that the pattern is INCONSISTENT with a process of random tossing (the implication being that a process other than random tossing is responsible for the pattern). I said in https://uncommondescent.com/intelligent-design/siding-with-mathgrrl-on-a-point-and-offering-an-alternative-to-csi-v2-0/
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?
Did I say 1 coin tossed 500 times? No. Did I mention that we saw a tossing process? No. I stated "if we saw 500 fair coins" -- that means 500 coins that we observe in an observable state, not 1 coin flipped 500 times. So next time I'll just say we open a box and find fair 500 coins in the all heads state just to emphasize we didn't see a tossing process. Thank you any way for your criticism, and future iterations of this description will hopefully preclude the mis-interpretations floating around.scordova
June 23, 2013
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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?
I understand necessary and sufficient conditions. And 'if and only if' statements. You are allowed to find a divinity where you will of course. But there is no mathematical problem with getting 500 heads in a row even though it is exceedingly unlikely.Jerad
June 23, 2013
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