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

_{Salvador Cordova

June 23, 2013

Mathematics}

Share: Facebook; Twitter; LinkedIn; Flipboard; Print; Email

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

OK, well, that's reassuring, even if only a few people answered!Elizabeth B Liddle_{June 27, 2013
June
06
Jun
27
27
2013
12:55 AM
12
12
55
AM
PDT}

Sorry, I misread.DiEb_{June 26, 2013
June
06
Jun
26
26
2013
11:25 PM
11
11
25
PM
PDT}

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
June
06
Jun
26
26
2013
04:53 PM
4
04
53
PM
PDT}

That's a very nice point DiEb! I had envisaged only tests of 500 tosses.Elizabeth B Liddle_{June 26, 2013
June
06
Jun
26
26
2013
02:59 PM
2
02
59
PM
PDT}

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
June
06
Jun
26
26
2013
02:47 PM
2
02
47
PM
PDT}

Which do you vote for, Joe?Elizabeth B Liddle_{June 26, 2013
June
06
Jun
26
26
2013
02:31 PM
2
02
31
PM
PDT}

@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
June
06
Jun
26
26
2013
02:29 PM
2
02
29
PM
PDT}

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
June
06
Jun
26
26
2013
02:27 PM
2
02
27
PM
PDT}

#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
June
06
Jun
26
26
2013
02:10 PM
2
02
10
PM
PDT}

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
June
06
Jun
26
26
2013
01:20 PM
1
01
20
PM
PDT}

Anyone?Elizabeth B Liddle_{June 26, 2013
June
06
Jun
26
26
2013
12:14 PM
12
12
14
PM
PDT}

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
June
06
Jun
26
26
2013
05:02 AM
5
05
02
AM
PDT}

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
June
06
Jun
26
26
2013
12:58 AM
12
12
58
AM
PDT}

keiths @53: I think you have some good thoughts there. Perhaps we can pursue your example just one step further. It isn't just that it is my SSN. If my friend is over one evening and I ask him his SSN, and then I roll one-by-one each number in order to match his SSN, we know something is up. In other words, it doesn't have to be meaningful to me, it just has to have some independent meaning apart from the roll itself. Same thing would be the case if, instead of an SSN, I just asked my friend to call out 9 numbers in a row and then I roll them one by one. We'd be suspicious. Even though those number don't have any particular "meaning." The only "meaning" is that they were specified independent of the roll itself. Now if I had a whole room full of people, it might be unlikely to roll someone's SSN, but we would be slightly less surprised. If we had a stadium of people, we probably would think "that's interesting," but wouldn't be terribly surprised. Finally, if there is no specification at all beforehand -- I have the entire SSN database at my disposal -- then not only is it not surprising if I roll someone's number, I would expect to. In your example of Delbert, there was no number called beforehand, no individual named. I just roll and then look up whomever it happened to land on in the database. Not at all surprising that it landed on someone. Why? Because there is no specification; in other words, no independent meaning to the number. In the case of a coin toss, there are a small number of obvious specific cases (i.e., specifications) that we can easily have in mind: all heads, all tails, some kind of repeating pattern (e.g., HTHT all the way), 250 heads followed by 250 tails, etc. Probably no more than a handful of specific scenarios (or specifications) that we would all readily recognize. They are already recognizable to us as having some independent significance beforehand. Thus, when we see them, we are surprised. And we should be just as surprised at seeing 500 heads as if we had asked someone to write down a series of H's and T's up to 500 and then we roll, and amazingly, roll the exact list they wrote on their paper. So it all comes back to the specification. And that is the key to the coin toss example and the key to the design inference generally. ----- 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.Eric Anderson_{June 25, 2013
June
06
Jun
25
25
2013
10:50 PM
10
10
50
PM
PDT}

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
June
06
Jun
25
25
2013
10:46 PM
10
10
46
PM
PDT}

BTW, the odds of flipping a coin 500 times and getting some pattern is 1.Joe_{June 25, 2013
June
06
Jun
25
25
2013
07:58 PM
7
07
58
PM
PDT}

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
June
06
Jun
25
25
2013
07:42 PM
7
07
42
PM
PDT}

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
June
06
Jun
25
25
2013
07:21 PM
7
07
21
PM
PDT}

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
June
06
Jun
25
25
2013
07:14 PM
7
07
14
PM
PDT}

keiths- How are you defining "resolution"?Joe_{June 25, 2013
June
06
Jun
25
25
2013
07:02 PM
7
07
02
PM
PDT}

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.
A resolution of the 'all-heads paradox'keiths_{June 25, 2013
June
06
Jun
25
25
2013
06:40 PM
6
06
40
PM
PDT}

This coin toss stuff is a pretty good litmus test of rationality. Anyone who says 500 heads in a row is just as probable as any other sequence of 500 coin tosses is right. And they are completely missing the point. What is it about the 500 heads in a row that demands our attention, that demands an explanation? That is the key question here. 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." There is a huge difference between the one and the other. Everyone knows it. A small child knows it. For a speaker to use "every outcome is just as improbable" to try and refute design serves two purposes: (i) it gives the speaker a rhetorical hook to latch onto in the attempt to avoid admitting any possibility of design, and (ii) it allows everyone else to see that the speaker is a fool. Anyone who thinks that 500 heads is no different from any other sequence needs to take a break from posting, go on a couple of long walks, think about it carefully, and sincerely ask themselves the following question: "We know all the sequences are equally improbable. Yet we also know there is something unique or unusual or special about 500 heads (or the Taj Mahal compared to a pile of rubble, or whatever other reasonable example you want). Why is that? What is it that makes it unique? What aspect, or quality, or characteristic is at play?" Once you have answered that question, you will be able to understand the design inference.Eric Anderson_{June 25, 2013
June
06
Jun
25
25
2013
05:45 PM
5
05
45
PM
PDT}

See: The Law of Large Numbers vs. KeithS, Eigenstate and my TSZ Criticsscordova_{June 24, 2013
June
06
Jun
24
24
2013
07:29 AM
7
07
29
AM
PDT}

Jerad:
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
June
06
Jun
24
24
2013
06:26 AM
6
06
26
AM
PDT}

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
June
06
Jun
24
24
2013
06:25 AM
6
06
25
AM
PDT}

F/N: I think it useful to clip the following from BA's tread, as a point of reference: _______________________ [Clipping 48 in the DDS mendacity thread, for record:] >>It seems people have a major problem appreciating: (a) configuration spaces clustered into partitions of vastly unequal statistical weight,and (ii) BLIND sampling/searching of populations under these circumstances. It probably does not help, that old fashioned Fisherian Hyp testing has fallen out of academic fashion, never mind that its approach is sound on sampling theory. Yes it is not as cool as Bayesian statistics etc, but there is a reason why it works well in practice. It is all about needles and haystacks. Let’s start with a version of an example I have used previously, a large plot of a Gaussian distribution using a sheet of bristol board or the like, baked by a sheet of bagasse board or the like. Mark it into 1-SD wide stripes, say it is wide enough that we can get 5 SDs on either side. Lay it flat on the floor below a balcony, and drop small darts from a height that would make the darts scatter roughly evenly across the whole board. Any one point is indeed as unlikely as any other to be hit by a dart. BUT THAT DOES NOT EXTEND TO ANY REGION. As a result, as we build up the set of dart-drops, we will see a pattern, where the likelihood of getting hit is proportionate to area, as should be obvious. That immediately means that the bulk of the distribution, near the mean value peak, is far more likely to be hit than the far tails. For exactly the same reason why if one blindly reaches into a haystack and pulls a handful, one is going to have a hard time finding a needle in it. The likelihood of getting straw so far exceeds that of getting needle that searching for a needle in a haystack has become proverbial. In short, a small sample of a very large space that is blindly taken, will by overwhelming likelihood, reflect the bulk of the distribution, not relatively tiny special zones. (BTW, this is in fact a good slice of the statistical basis for the second law of thermodynamics.) The point of Fisherian testing is that skirts are special zones and take up a small part of the area of a distribution, so typical samples are rather unlikely to hit on them by chance. So much so that one can determine a degree of confidence of a suspicious sample not being by chance, based on its tendency to go for the far skirt. How does this tie into the design inference? By virtue of the analysis of config spaces — populations of possibilities for configurations — which can have W states and then we look at small, special, specific zones T in them. Those zones T are at the same time the sort of things that designers may want to target, clusters of configs that do interesting things, like spell out strings of at least 72 – 143 ASCII characters in contextually relevant, grammatically correct English, or object code for a program of similar complexity in bits [500 - 1,000] or the like. 500 bits takes up 2^500 possibilities, or 3.27*10^150. 1,000 bits takes up 2^1,000, or 1.07*10^301 possibilities. To give an idea of just how large these numbers are, I took up the former limit, and said now our solar system’s 10^57 atoms (by far and away mostly H and He in the sun but never mind) for its lifespan can go through a certain number of ionic chemical reaction time states taking 10^-14s. Where our solar system is our practical universe for atomic interactions, the next star over being 4.2 light years away . . . light takes 4.2 years to traverse the distance. (Now you know why warp drives or space folding etc is so prominent in Sci Fi literature.) Now, set these 10^57 atoms the task of observing possible states of the configs of 500 coins, at one observation per 10^-14 s. For a reasonable estimate of the solar system’s lifespan. Now, make that equivalent in scope to one straw. By comparison, the set of possibilities for 500 coins will take up a cubical haystack 1,000 LY on the side, about as thick as our galaxy. Now, superpose this haystack on our galactic neighbourhood, with several thousand stars in it etc. Notice, there is no particular shortage of special zones here, just that they are not going to be anywhere near the bulk, which for light years at a stretch will be nothing but straw. Now, your task, should you choose to accept it is to take a one-straw sized blind sample of the whole. Intuition, backed up by sampling theory — without need to worry over making debatable probability calculations — will tell us the result, straight off. By overwhelming likelihood, we would sample only straw. That is why the instinct that getting 500 H’s in a row or 500 T’s or alternating H’s and T’s or ASCII code for a 72 letter sequence in English, etc, is utterly unlikely to happen by blind chance but is a lot more likely to happen by intent, is sound. And this is a simple, toy example case of a design inference on FSCO/I as sign. A very reliable inference indeed, as is backed up by literally billions of cases in point. Now, onlookers, it is not that more or less the same has not been put forth before and pointed out to the usual circles of objectors. Over and over and over again in fact. And in fact, here is Wm A Dembski in NFL:
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
June
06
Jun
24
24
2013
01:49 AM
1
01
49
AM
PDT}

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
June
06
Jun
24
24
2013
01:04 AM
1
01
04
AM
PDT}

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
June
06
Jun
24
24
2013
12:18 AM
12
12
18
AM
PDT}

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
June
06
Jun
23
23
2013
11:45 PM
11
11
45
PM
PDT}

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
June
06
Jun
23
23
2013
11:20 PM
11
11
20
PM
PDT}

1 2 3 Next

You must be logged in to post a comment.

Leave a Reply