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

JDH,
If you consider 1/1.17*10^149 “highly unlikely” and not “improbable” you don’t seem to really know a thing about numbers and what an exponential means. For instance – even in 1 billion trials something that had a chance of occurring of 1/1.17*10^149 of happening it would still be improbable.
The probability of getting exactly 250 heads is the number of ways that can happen divided by the number of possible outcomes which is 2^500. So (1.17 x 10^149) / (2^500) is the correct probability. I figured it's about 0.035. About 3.5 times out of 100.
Jerad – And I perfectly understand the idea that every specifically ordered sequence is unique and has equal chance of occurring. So don’t pretend that I don’t. But we can never claim that anything with a 1/1.17 * 10^149 chance of ever occurring. It is not reasonable to do so.
I don't even know who you are. I was just responding to Sal's post. I wasn't pretending anything. I"m sorry if my use of actual quote marks instead of the blockquote html tag confused the issue. Perhaps you were responding to things other people said? Maybe try reading the thread?Jerad_{June 23, 2013
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Jerad - And I perfectly understand the idea that every specifically ordered sequence is unique and has equal chance of occurring. So don't pretend that I don't. But we can never claim that anything with a 1/1.17 * 10^149 chance of ever occurring. It is not reasonable to do so.JDH_{June 23, 2013
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Jerad, First of all, I did not read the original post, but I don't need to in order to critique the utter foolishness of your argument. If you consider 1/1.17*10^149 "highly unlikely" and not "improbable" you don't seem to really know a thing about numbers and what an exponential means. For instance - even in 1 billion trials something that had a chance of occurring of 1/1.17*10^149 of happening it would still be improbable. Please do not write dribble. Something that has a probability of happening of 1/1.17*10^149 is impossible. If it happened, then you do not have a fair coin. There is no other reasonable opinion. The fact that you don't understand this makes me wonder how you reason.JDH_{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's called "matching a prespecification". And it is pretty fundamental to design detection.Joe_{June 23, 2013
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Jerad, Specify a pattern and then flip the coin to try to get a match.Joe_{June 23, 2013
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Sal If it helps I would agree that if I tossed a coin 500 times and it came down heads every time then there something unfair about the tossing mechanism. I don't think that was commentator's point, the comma indicates the quote continued, but there is such a mass of comments there I cannot be bothered to sort out what everyone was saying. 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.Mark Frank_{June 23, 2013
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"Thank you gentleman for you comments, but you could for once agree I was right and I am in good agreement with operational practice that we would say 500 coins heads is inconsistent with fair coins such that you would reject the chance hypothesis if you saw it." I would NOT say it was 'inconsistent with fair coins'. I would agree it was highly unlikely. As far as rejecting the chance hypothesis . . . what is your hypothesis exactly? Do you mean 'fair' hypothesis? And how are you going to test your alternate hypothesis. It matters. If each coin flip is a data point then 500 such data points is pretty good. If the whole sequence of 500 flips is a 'data point' then I'd say one point is not enough to reject. You must be specific in your statements. You kind of imply that it's the whole sequence you're interested in. "But that is an aside. I must admit, I’m rather stunned at the reluctance to reject the chance hypothesis for 500 coins heads. Even though chance is formally possible as an explanation, operationally speaking, would any of you personally accept it? Just curious…" As I said, if I got 500 heads in a row I'd probably check to see if the coin was really fair. I'd also check the flipping technique. All part of making sure the procedure really was 'fair'. AND, if my data points were the individual flips then it would depend on the p-value I picked, etc. But checking the probabilites of each flip is different from checking the probability of a sequence of flips. "So why won’t you agree with me? I provided a similar analysis above with the combinations of 500 coins that are 50% heads ignoring order and showed that they are 1.17 x 10^149 times more likely than all coins heads. You only provided the case for a 2-coins set and I provided one for a 500-coin set, which in principle could be extended to an N-coin set. You could have pointed out you’re just agreeing with what I laid out with C(n,r) " We all agree that it's incredibly more likely that you'll get 250 heads in 500 throws when you don't specify the order. All we are saying is that any particular sequence is equally unlikely and that 500 heads is just one of those particular sequences. The probability of getting EXACTLY 250 heads and 250 tails in any order is fairly small. Calculate it out.Jerad_{June 23, 2013
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I’ll add my agreement with Mark. One has to distinguish between a sequence and a combination. For two coins, HH HT TH TT are equally probable sequences. But if one only looks at the combination, ignoring the order, then mixed (HT or TH) is twice as likely as all heads (HH), because it counts two distinct sequences.
So why won't you agree with me? I provided a similar analysis above with the combinations of 500 coins that are 50% heads and showed that they are 1.17 x 10^149 times more likely than all coins heads. You only provided the case for a 2-coins set and I provided one for a 500-coin set, which in principle could be extended to an N-coin set. You could have pointed out you're just agreeing with what I laid out with C(n,r) :roll: I'm practically quoting standard practice in statistics and discrete math you guys can't be forthcoming and say, "I agree with Sal". Nothing I said in the OP is outside of reasonable practice. Anyway, thank you for commenting, but it seems even the appearance of agreement with a creationist on any non-trivial topic (even textbook math) is avoided by Darwinists....scordova_{June 23, 2013
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Mark Frank:
I am not quite sure what the comment was getting at but remember that every specific sequence of coins has exactly the same probability of arising given a fair coin. Why is a sequence of 500 heads more inconsistent with a fair coin than a sequence that is a jumble of heads and tails?
LoL! Mark if you call the sequence ahead of time and then toss the coin and get a match, then that would be highly improbale given a chance only scenario. Unbelieveable how freaking dull these anti-design people. are.Joe_{June 23, 2013
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Thank you gentleman for you comments, but you could for once agree I was right and I am in good agreement with operational practice that we would say 500 coins heads is inconsistent with fair coins such that you would reject the chance hypothesis if you saw it. The way I used the math is not at all idiosyncratic, and in fact is based on standard practice. For various reasons in operational practice we would group together strings that are consistent with 50% heads. A similar issue arose when people were able to beat dice games in the 1970s by intelligently designing their throwing technique. This resulted in the passing of a law in Nevada to prevent such non-random throws. See: Couple Accused of Dice Sliding in Wynn Las Vegas
By why does the fact that a sequence belongs to the larger class of sequences with roughly 50% heads make it less surprising than a sequence that is all heads?
It is unsurprising unless the sequence had statistics in inconsistent with the chance hypothesis or was specified by a recognizable pattern. Examples: 1. H T H T H T...... 2. Champernowne sequence 3. identical to another set of coins or record of coins that we are aware of (very similar issue to the FBI case I mentioned here: Coordinated Complexity, the key to refuting postdiction and single target objections). But that is an aside. I must admit, I'm rather stunned at the reluctance to reject the chance hypothesis for 500 coins heads. Even though chance is formally possible as an explanation, operationally speaking, would any of you personally accept it? Just curious... Thanks again for coming across the ailse from skeptical zone. I know we all have intense disagreements, and thank you for keeping the discussion civil...scordova_{June 23, 2013
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I'll add my agreement with Mark. One has to distinguish between a sequence and a combination. For two coins, HH HT TH TT are equally probable sequences. But if one only looks at the combination, ignoring the order, then mixed (HT or TH) is twice as likely as all heads (HH), because it counts two distinct sequences. The issue then is whether eigenstate (the TSZ responder) was correct in taking your post to be about a sequence of events, where the order (sequencing) does matter.Neil Rickert_{June 23, 2013
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Mark is correct. Any particular sequence of heads and tails has the same 2^-500 probability of occurring. The distribution of the outcomes addresses the number of kinds of outcomes not the probability of any given one. If I got 500 heads in a row I'd be very surprised and suspicious. I might even get the coin checked. But it could happen. You might not ever see that particular outcome even if you flipped coins your whole life. But you might not ever get HTHTHTHTHT . . . . either. Or THTHTHTHTH . . . . Or HHTTHHTTHHTT . . . . (and those three sequences are at 0 sigma). Or any other particular sequence. 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.Jerad_{June 23, 2013
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Sal I am not quite sure what the comment was getting at but remember that every specific sequence of coins has exactly the same probability of arising given a fair coin. Why is a sequence of 500 heads more inconsistent with a fair coin than a sequence that is a jumble of heads and tails? There are of course many more sequences that have roughly the same number of heads and tails than there are sequences that are all heads. By why does the fact that a sequence belongs to the larger class of sequences with roughly 50% heads make it less surprising than a sequence that is all heads? There is an answer to this question but it involves Bayesian thinking which is anathema to the ID community.Mark Frank_{June 22, 2013
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