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Old 10th February 2011, 08:38 PM   #1
Simon Bridge
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Double Headed Coins and skepticism

A False Coin - hopefully the link works - I wrote considering the oft-repeated argument (by theists) that the atheist cannot know there is no God because the atheist has not been everywhere. This questions how one can be certain of a negative result only through a finite amount of negative data. eg. How hard do I look for something, and fail, before I can confidently conclude the thing is not there to find?

The usual replies involve teapots orbiting Jupiter. But I figured it should be possible to quantify the effect.

It's been a while though - so if someone would check the math and maybe suggest how to tidy up the rhetoric?

The idea is that I am tossing a coin and it keeps coming up heads - how many tosses before you conclude, reasonably, I'm cheating? How do your initial prejudices affect the answer to this question?

I used Baye's theorum to work out how suspicious you should be given how trustworthy you think I am at the outset and how many heads I've tossed in a row.

(I could use advice on a different word than "toss" for generating a boolean random event using a coin, "throw"? ... English people seem insistent that I am a "tosser" and, now I've finished, I have "tossed off". <sigh>)
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Old 10th February 2011, 08:45 PM   #2
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Originally Posted by Simon Bridge View Post
I wrote considering the oft-repeated argument (by theists) that the atheist cannot know there is no God because the atheist has not been everywhere.
There's a much easier debunking of that argument.

We do not need to travel everywhere to know there is no God, because God is not that kind of thing. God is not, say, a rock on a distant planet.

To say we need to travel everywhere in order to say there is no God is like saying we need to look behind the fridge to be sure our house isn't infested with galaxy clusters.
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Old 11th February 2011, 05:28 AM   #3
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Originally Posted by Piggy View Post
There's a much easier debunking of that argument.

We do not need to travel everywhere to know there is no God, because God is not that kind of thing. God is not, say, a rock on a distant planet.

To say we need to travel everywhere in order to say there is no God is like saying we need to look behind the fridge to be sure our house isn't infested with galaxy clusters.
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Old 11th February 2011, 10:11 AM   #4
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Originally Posted by Piggy View Post
There's a much easier debunking of that argument.

We do not need to travel everywhere to know there is no God, because God is not that kind of thing. God is not, say, a rock on a distant planet.

To say we need to travel everywhere in order to say there is no God is like saying we need to look behind the fridge to be sure our house isn't infested with galaxy clusters.
I look under the couch to see if jesus is hiding there amongst the dust bunnies. If I ever found the bugger, I could tell folks I'd found jesus, and they'd leave me alone. I might even let the bugger crash in my house for a while, as long as he didn't mooch too much from the fridge. Does that count?
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Old 11th February 2011, 10:25 AM   #5
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Originally Posted by Simon Bridge View Post
The idea is that I am tossing a coin and it keeps coming up heads - how many tosses before you conclude, reasonably, I'm cheating? How do your initial prejudices affect the answer to this question?
This is basic statistics. It involves a null hypothesis and a confidence interval.
We can set the confidence interval at a rather high level (say, p=.0001). Then we determine what range of responses are possible from a fair coin, where the identified result falls within this range of responses.
Even as few as 15 heads in a row would be outside the p=.0001 confidence interval for a fair coin.
What's less clear to me is how this response relates to proof or disproof of God.
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Old 11th February 2011, 10:39 AM   #6
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You know, i wouldn't say it is so much in watching your results , as in actually looking into how you could be cheating.

As a D&D player ( usually the dm.) i have seen some downright miraculous statistical anomalies. And have not questioned any of them, simply due to the fact that i could eliminate all methods of cheating. ( or at the very least, make them noticeable to me.).

Personally , assuming you were tossing the coin in a way i could observe, and there was no method of cheating possible. Then i would not question any number of " heads" ( i mean hell i have seen a 100 sided dice roll 100 10 times in a row. ). Now the second you have the ability to cheat though, the ol red flag goes up pretty quick.

To pertain to religion, well it should be obvious, they have all kinds of ways to cheat. Simply the definition of god allows them to cheat every time evidence is shown contrary to its existence. Really, religion in general, and christianity in specific is pretty much made out of ways to cheat logic. ( You can't see god, god won't ever do anything like parting a sea again, god is loving but allowed to do things that would make Hitler cringe because they are 'his way', as just a few examples). Which is what puts up the red flag as to b.s. to me. It is not so much the claim to always have heads ( or to use a d and d analogy, always roll a 20) , but the fact that their entire structure is set up to make cheating ( in regards to following logic.) very easy to do.
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Old 11th February 2011, 02:12 PM   #7
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Originally Posted by Simon Bridge View Post
A False Coin - hopefully the link works - I wrote considering the oft-repeated argument (by theists) that the atheist cannot know there is no God because the atheist has not been everywhere.
That argument is really more about burden of proof, and it is one that skeptics confront pretty frequently. How can skeptics say there is no Bigfoot or Loch Ness Monster or UFOs?

I prefer to avoid taking that bait and just point out that they're making the positive claim so it's up to them to make a convincing case. Nonexistence is the default against which claims of existence are compared.




Originally Posted by Simon Bridge View Post
This questions how one can be certain of a negative result only through a finite amount of negative data. eg. How hard do I look for something, and fail, before I can confidently conclude the thing is not there to find?
There is no 'rule' about this.
There are only cultural standards for confidence intervals.

And this is a practical problem for testing paranormal claims. Even skeptics will disagree about the threshold confidence interval for accepting a paranormal claim. In particular, since it is not an established mature field and there is no existing standard yet.
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Old 12th February 2011, 01:06 AM   #8
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Originally Posted by Simon Bridge View Post
<snip>
The idea is that I am tossing a coin and it keeps coming up heads - how many tosses before you conclude, reasonably, I'm cheating? How do your initial prejudices affect the answer to this question?<snip>
That is discussed (and is a plot point) during the first 10 minutes of Rosencrantz & Guildenstern Are Dead. (A spin-off of Shakespeare's Hamlet, and a very good movie in it's own right....... if you like Hamlet, of course!)


Originally Posted by sadhatter View Post
<snip>i have seen a 100 sided dice roll 100 10 times in a row. <snip>
That's...... unbelievable!
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Old 13th February 2011, 09:05 PM   #9
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I knew I'd regret bringing up the god angle ... I know there are lots of arguments and approaches, thank you. Has anyone got a comment to make on the paper?


Only thing I got here is that I need to explain in some detail why normal statistical hypothesis-testing is not good enough for the sorts of questions you get looking into paranormal claims.

It also doesn't work well to my purpose ... the idea is to lay groundwork for understanding more complex problems rather than produce an optimal solution to a specific problem. I put all this in the paper so I wouldn't have to write so much here.

There was a thing in skeptical inquirer about this I think:
http://www.abelard.org/briefings/bay...are_conditions

Oh yeah ... and:
http://www.script-o-rama.com/movie_s...ad-script.html

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Old 13th February 2011, 09:57 PM   #10
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Originally Posted by AvalonXQ View Post
This is basic statistics. It involves a null hypothesis and a confidence interval.
We can set the confidence interval at a rather high level (say, p=.0001). Then we determine what range of responses are possible from a fair coin, where the identified result falls within this range of responses.
Even as few as 15 heads in a row would be outside the p=.0001 confidence interval for a fair coin.
What's less clear to me is how this response relates to proof or disproof of God.
This is a bit like saying that a screwdriver is just a fancy hammer, because you have personally found that if you hit screws hard enough with a hammer they go in.

There's a major difference between "The odds of you doing that with a fair coin are X%" and "the odds of you cheating are Y%".
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Old 13th February 2011, 10:12 PM   #11
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Originally Posted by sadhatter View Post
You know, i wouldn't say it is so much in watching your results , as in actually looking into how you could be cheating.

As a D&D player ( usually the dm.) i have seen some downright miraculous statistical anomalies. And have not questioned any of them, simply due to the fact that i could eliminate all methods of cheating. ( or at the very least, make them noticeable to me.).

Personally , assuming you were tossing the coin in a way i could observe, and there was no method of cheating possible. Then i would not question any number of " heads" ( i mean hell i have seen a 100 sided dice roll 100 10 times in a row. ). Now the second you have the ability to cheat though, the ol red flag goes up pretty quick.

To pertain to religion, well it should be obvious, they have all kinds of ways to cheat. Simply the definition of god allows them to cheat every time evidence is shown contrary to its existence. Really, religion in general, and christianity in specific is pretty much made out of ways to cheat logic. ( You can't see god, god won't ever do anything like parting a sea again, god is loving but allowed to do things that would make Hitler cringe because they are 'his way', as just a few examples). Which is what puts up the red flag as to b.s. to me. It is not so much the claim to always have heads ( or to use a d and d analogy, always roll a 20) , but the fact that their entire structure is set up to make cheating ( in regards to following logic.) very easy to do.
No you haven't.
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Old 13th February 2011, 10:30 PM   #12
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Actually, my above post is a good example of Simon's problem: at what point do we conclude that an internet poster is exaggerating (or misremembering) an incredibly improbable event?

If I claimed to have seen a fair coin land heads 10 times in a row, no one would be that impressed (although some would probably wonder how I knew the coin and tosses were fair). If I claimed to have seen it land heads 20 times, a few (or many of you) would outright not believe it (or be convinced it wasn't a fair coin/toss). Only the biggest sap to ever post on the forums would believe it was a fair coin and toss if I claimed to have seen it land heads 50 times in a row. And that last result is much more likely than getting a 100 on a D100 ten times in a row. Hence, the skepticism of SadHatter's claim.

Of course, given a fair coin and toss
HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH
is just as likely as
THHHHTHHTHHHTTHHTTTTTTHHHHHHTHHHHHT
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Old 13th February 2011, 10:34 PM   #13
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Originally Posted by sadhatter View Post
( i mean hell i have seen a 100 sided dice roll 100 10 times in a row. ).
The chances of that happening are 1/10010.

That's 1 in 100,000,000,000,000,000,000.

You could roll your D100 dice once a second for a million years and you would still be extremely unlikely to have a single sequence of 10 consecutive 100s.

So I'm quite confident that either you are exaggerating, misremembering, or there was some sort of cheating involved.
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Old 14th February 2011, 09:29 AM   #14
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Originally Posted by Roboramma View Post
The chances of that happening are 1/10010.

That's 1 in 100,000,000,000,000,000,000.

You could roll your D100 dice once a second for a million years and you would still be extremely unlikely to have a single sequence of 10 consecutive 100s.

So I'm quite confident that either you are exaggerating, misremembering, or there was some sort of cheating involved.
And there is the rub. I wouldn't expect one to believe because there would be quite a chance of cheating. Specifically on my end, i could easily be exaggerating, or have been cheated myself.

It is not the result, but the situation that determines how valid something is.

As an aside, to rebut claims of cheating, simply for the lulz, and that i have some spare time. It is statistics, if it can happen it is bound to happen to someone. During my almost decade of playing the game i have seen some rather wacky rolls. From the top of my head, some of the better ones have been:

A d4 landing on a point ( this was actually done over the course of a year, in an attempt to show it could happen.)

A d20 from the time of its use that rolled nothing higher than a 3 for over 50 rolls ( was then microwaved and threw out a window.)

two six sided dice landing on top of each other.

You roll enough dice and you see some wacky crap. I am sure other pen and paper rp'ers have seen just as wacky events.
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Old 14th February 2011, 10:00 AM   #15
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One has to be very careful with logic that says "I don't believe you saw such and such because the odds are 1/N". You cannot compute odds like that after the fact - it's a misuse of probability.

Suppose sadhatter had seen a die roll 1 ten times in a row rather than 100. S/he'd be telling us about that instead... which means the relevant odds are not those of that specific event, but the cumulative odds of any event of sufficient interest to sadhatter that s/he would tell us about it.

That said, I don't believe it either.
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Old 14th February 2011, 10:01 AM   #16
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Originally Posted by Simon Bridge View Post
I knew I'd regret bringing up the god angle ... I know there are lots of arguments and approaches, thank you. Has anyone got a comment to make on the paper?

Only thing I got here is that I need to explain in some detail why normal statistical hypothesis-testing is not good enough for the sorts of questions you get looking into paranormal claims.

I probably wrote my response in a rambling way, sorry. I was trying to address the paper.

The main point I was trying to make is that prior probabilities aside, we will still quibble about the confidence interval thresholds. There is no cultural standard that builds common ground between advocates and skeptics.

The second problem that I didn't mention is that there is often a disagreement about what constitutes the null hypothesis. In your paper it looks like 'phenomenon is naturally occurring'? The reason this is important to paranormal investigation is that when the skeptics succesfully expose a cheat, it's very consistently been about good controls rather than good stats.
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Old 14th February 2011, 10:08 AM   #17
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Originally Posted by Simon Bridge View Post
I wrote considering the oft-repeated argument (by theists) that the atheist cannot know there is no God because the atheist has not been everywhere.
I think "been everywhere" is the wrong argument, because for most people you're likely to encounter, the God they're arguing for is presumed to be ubiquitous. The claim is more likely to be "the atheist does not know everything," which is perhaps more fair. Why would you be more willing to accept that there are billions of neutrinos streaming through you every second (you just can't detect them) than that there is a divine spirit maintaining the distance between electrons and protons (you just can't detect it)?

Certain other values of "God" are more easily dismissed: A god which will override the laws of physics in your favor if you pray, for instance, can be falsified by showing that no such override occurs.

As far as the coin is concerned, the odds of a string of consecutive heads is halved with each fair throw of a fair coin. The odds of throwing 6 heads in a row is one in 64. If we began a string of throws with 6 heads in a row, that would pass my own personal threshold for being confident that either the coin or the process was weighted, though of course I wouldn't be certain. If that was followed with six more heads in a row, I'd claim certainty.
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Old 14th February 2011, 09:18 PM   #18
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Originally Posted by sol invictus View Post
Suppose sadhatter had seen a die roll 1 ten times in a row rather than 100. S/he'd be telling us about that instead... which means the relevant odds are not those of that specific event, but the cumulative odds of any event of sufficient interest to sadhatter that s/he would tell us about it.

I'm fairly certain that the total number of significant interest events would still be a drop in a bucket. At least in this scenario.

Like in your example, changing the number that appears 10 times in a row. That would be any of the 100 numbers.

100 in 100,000,000,000,000,000,000 is, relatively speaking, nearly the same as 1 in 100,000,000,000,000,000,000.

Add in 900 more "significant events" (which you would think would cover them all); like consecutives (rolling a 1 then a 2 then 3.... up to 10), and it would still be 1000 in 100,000,000,000,000,000,000.

What's an extra zero or two amongst friends?

Originally Posted by sol invictus View Post
That said, I don't believe it either.
I notice that in sadhatters newest post s/he doesn't deny that this 10, 100s in a row on a 100 sided die happened.
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Old 14th February 2011, 09:36 PM   #19
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At a certain point, we need to do what our brains are evolved to do, which is to be suspicious of severe statistical anomalies.

If you're asking for a philosophical or mathematical bright line, you can take your ball and go home.

But if a coin lands on heads even 100 times in a row, we all know it's not a fair coin. The real world we live in simply does not work that way.
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Old 14th February 2011, 09:40 PM   #20
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Originally Posted by Piggy View Post
At a certain point, we need to do what our brains are evolved to do, which is to be suspicious of severe statistical anomalies.

If you're asking for a philosophical or mathematical bright line, you can take your ball and go home.

But if a coin lands on heads even 100 times in a row, we all know it's not a fair coin. The real world we live in simply does not work that way.
It can happen. Our brains are not good at calculating statistics.
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Old 14th February 2011, 09:49 PM   #21
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Originally Posted by Alan View Post
It can happen. Our brains are not good at calculating statistics.
What do you mean by "can"?
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Old 14th February 2011, 09:50 PM   #22
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Let's say it turns up THHTHTTHHHTTHTTTTHTHTHHTHTTHHHTTHTTTTHTHHTHHTHTTHH HTTHTTHTTHHHTHHTHTTHHHTTHTTTTHHHTHHTHHTTHHHTTHTTHT HH.

My understanding is it could have just as easily been all heads as it could have been in that order. But if it turned up in that order, it wouldn't look suspicious. Even though the chance of it happening is the same.

It can turn up in that above combination, just like any of them (including all one or the other, or perfectly alternating, or anything).

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Old 14th February 2011, 09:57 PM   #23
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Originally Posted by Alan View Post
Let's say it turns up THHTHTTHHHTTHTTTTHTHTHHTHTTHHHTTHTTTTHTHHTHHTHTTHH HTTHTTHTTHHHTHHTHTTHHHTTHTTTTHHHTHHTHHTTHHHTTHTTHT HH.

My understanding is it could have just as easily been all heads as it could have been in that order. But if it turned up in that order, it wouldn't be suspicious.

It can turn up in that above combination, just like any of them (including all one or the other, or perfectly alternating, or anything).
Pretty sad definition of "can", in a world where we observe that fair coins don't actually have runs of anywhere near 100, nor do we expect them to.

What you're doing is abandoning your actual experience of the world in favor of abstractions and idealizations that don't appear to describe the universe we really live in.

Bad strategy, that.

And btw, in real-world terms, our brains turn out to be not bad statisticians. Even infants somehow understand that something's wrong when events occur that are way outside expected probabilities.
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Old 14th February 2011, 10:02 PM   #24
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Very unlikely things happen all the time. Things equally as unlikely as a run of 100 heads. Like any one other combination after 100 coin tosses.


We are terrible statisticians. For example, are you aware of the Monty Hall problem? Or the question of how many people need to be in a room for there to be a greater than 50% chance of two of them sharing the same birthday (not including the year)?

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Old 14th February 2011, 10:16 PM   #25
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Originally Posted by Alan View Post
Very unlikely things happen all the time. Things equally as unlikely as a run of 100 heads. Like any one other combination after 100 coin tosses.


We are terrible statisticians. For example, are you aware of the Monty Hall problem? Or the question of how many people need to be in a room for there to be a greater than 50% chance of two of them sharing the same birthday (not including the year)?
We're not very good statisticians when it comes to artificial situations or unimportant business such as the number of folks in a room sharing a birthday. But for everyday work, our brains are Johnny on the spot.

I've seen some wild coincidences. But nothing approaching 100 consecutive heads on a fair coin.

And there's no doubt in my mind that you would back off a bet if you saw a coin exhibit such behavior. I sure would. And if you wouldn't, you're more gullible than I'd care to admit.

That's because we know how coins really behave. We know they're well enough balanced that they're not actually going to land on one side or another 100 times in a row if there's no manipulation going on.

Drawing Ts and Hs on paper doesn't change that, of course, because we don't live in a world of Ts and Hs and white plains and mathematical abstractions.

We live in a world of metal and air, and experience that lets us know when things are out of whack.

Our instincts do not let us down in this case.

We may not understand precisely why, but we're not wrong when we conclude that a coin flipped to 100 consecutive heads is fixed. And no number of Ts and Hs on a sheet of paper changes that. Evolution serves us better.
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Old 14th February 2011, 10:54 PM   #26
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Originally Posted by Alan View Post
Very unlikely things happen all the time. Things equally as unlikely as a run of 100 heads. Like any one other combination after 100 coin tosses.
As Sol pointed out we are talking about significant events, not "any combination".

Odds of 100 heads is (somewhere around) 1 in 1,270,000,000,000,000,000,000,000,000,000. Even if all 7,000,000,000 people on the planet flipped all day for years it still probably wouldn't happen.

ETA: ...and before the thread goes on the same derail that a similar thread did months ago: Yes, the odds get better if you start counting again after every miss (tails) instead of starting the count again after every 100 flips. It's still very unlikely though.
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Old 14th February 2011, 10:57 PM   #27
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I'm not saying that it is impossible that a person can cheat. I am saying that it is possible that the person is not cheating.

I, for one, was talking about any combinations.

What's the chance of any other combination after 100 coin tosses? It's low, therefore that combination probably wouldn't happen either.

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Old 14th February 2011, 11:00 PM   #28
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Well if you change "possible" to "very, very, very small possibility" then, yes, I agree.

Obviously the smaller the possibility becomes, the closer it gets to impossible.

This is a very similar discussion to the .999... repeating = 1 thread.
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Old 14th February 2011, 11:43 PM   #29
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Originally Posted by OnlyTellsTruths View Post
I'm fairly certain that the total number of significant interest events would still be a drop in a bucket. At least in this scenario.

Like in your example, changing the number that appears 10 times in a row. That would be any of the 100 numbers.

100 in 100,000,000,000,000,000,000 is, relatively speaking, nearly the same as 1 in 100,000,000,000,000,000,000.

Add in 900 more "significant events" (which you would think would cover them all); like consecutives (rolling a 1 then a 2 then 3.... up to 10), and it would still be 1000 in 100,000,000,000,000,000,000.
Yes, so probably any of those anomalies would be very unlikely to happen to any given individual. But if none had happened to sadhatter, s/he wouldn't be posting about them - but someone else might. And if none of them had happened to anyone that could post here, some other - equally unlikely, but different - anomaly might have happened to someone. Etc.

Ex post facto statistics is a very, very tricky business.
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Old 14th February 2011, 11:51 PM   #30
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It's a good thing sadhatter kept his story believable. If he had said he rolled 39 17 55 80 48 51 96 23 52 88 31, I would have called him out on it, because that's even less likely.
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Old 15th February 2011, 12:00 AM   #31
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The word "possible" covers things that are very unlikely to happen so I'm perfectly satisfied with the word. I don't think somebody could read what I wrote and think that I was saying that it wasn't very unlikely. Just that other possible strings of the same length are equally unlikely to occur.
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Old 15th February 2011, 12:05 AM   #32
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Originally Posted by Yashichi View Post
It's a good thing sadhatter kept his story believable. If he had said he rolled 39 17 55 80 48 51 96 23 52 88 31, I would have called him out on it, because that's even less likely.
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Old 15th February 2011, 12:16 AM   #33
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Originally Posted by Yashichi View Post
It's a good thing sadhatter kept his story believable. If he had said he rolled 39 17 55 80 48 51 96 23 52 88 31, I would have called him out on it, because that's even less likely.

Well, he'd either have to be writing them down or have a very good memory!

Again, we all understand that any and all sets have the same odds of occurring. We are talking about significant sets. For instance if he had predicted those 11 (from Yashichi's post) beforehand it would then become significant.

10 numbers in a row has to happen (unless the die breaks!). Any specific 10 in a row is very unlikely.
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Old 15th February 2011, 12:26 AM   #34
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As far as varying levels of "possible" go, I'm going to steal Modified's great example from the
Impossible or just unlikely
thread:


Someone wins the big weekly one in a billion lotto.

Now compare that to the same person winning it every single week of their life.

The first is nearly impossible. The second is so close to impossible that it basically equals impossible.

Again, this brings to mind the .999... repeating = 1 thread.
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Old 15th February 2011, 12:53 AM   #35
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Originally Posted by OnlyTellsTruths View Post
As far as varying levels of "possible" go, I'm going to steal Modified's great example from the
Impossible or just unlikely
thread:


Someone wins the big weekly one in a billion lotto.

Now compare that to the same person winning it every single week of their life.

The first is nearly impossible. The second is so close to impossible that it basically equals impossible.

Again, this brings to mind the .999... repeating = 1 thread.
Considering that things just as unlikely happen regularly, I wouldn't feel comfortable putting it in the 'nearly impossible' category.
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Old 15th February 2011, 01:00 AM   #36
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Originally Posted by sol invictus View Post
Yes, so probably any of those anomalies would be very unlikely to happen to any given individual. But if none had happened to sadhatter, s/he wouldn't be posting about them - but someone else might. And if none of them had happened to anyone that could post here, some other - equally unlikely, but different - anomaly might have happened to someone. Etc.

Ex post facto statistics is a very, very tricky business.
I accept that, clearly if there are six billion people on this planet, and some fraction of them are rolling dice regularly, some of those people will see some pretty strange rolls come up, and those are the ones that we'd expect to post on forums. But I still don't believe it, and the reason for that, and I think yours too, has to do with the crazy probabilities involved.

I did fail to include the possibility that it just happened to be an unfair die, though. It's possible for that to be the case and for the players not to know about it.
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Old 15th February 2011, 02:11 AM   #37
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Originally Posted by Alan View Post
Considering that things just as unlikely happen regularly, I wouldn't feel comfortable putting it in the 'nearly impossible' category.
You think that things just as unlikely as the same person hitting the one in a billion lottery every single week of their entire life happen "regularly"?

We're talking about one single person hitting one in a billion, hundreds of times in a row.

ETA: I see you are talking about the first part, apologies. Still, I would personally call one in a billion 'nearly impossible'.
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Old 15th February 2011, 05:38 AM   #38
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There is another thread at the moment on the topic of whether something is impossible or just unlikely.
http://www.internationalskeptics.com...d.php?t=200580

I think that there are examples in that thread of things that I would call "almost impossible". One in a billion doesn't come close to that. But this is a subjective matter of what to call things and it isn't important.

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Old 15th February 2011, 06:03 AM   #39
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Post deleted -- I made the same error as OTT. Sorry.
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Old 15th February 2011, 06:12 AM   #40
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Btw, I don't think this is actually a problem of pure probabilities. It's a problem of probabilities plus real physics.
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