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#1
05-23-2007, 07:06 PM
 kanicbird Guest Join Date: May 1999 Posts: 18,435
How many times do you have to flip a coin to ensure a heads

Believe it or not this comes from all the talk about theistic and atheistic faith issues that have come up lately.

Here is the premises:

If you flip a infinite number of coins a infinite number of times you will get a infinite number of coins that will never come up heads.

-but-

If you flip any one coin a infinite number of times it will hit heads and tails equally.

These 2 can't coexist.

There is a saying that if you take a room full of monkeys and give them a typewriter and a infinite amount of time you will eventually get a copy of Hamlet. No we all know this could never happen either as monkeys will destroy the paper at a much higher probability then completing a single letter (barring evolution).

There seems like there is some limit on probability, when the chance of something happening is so so so small it is not just effectively zero it is zero and some things just can never happen*.

* Now this is where the concept of God comes in, things like creation and life may have zero chance of happening on there own without a 'God'.

(Not that I'm hanging my theists hat on this, as I don't think God designed the system for Him to be proven, but I would like to see the responses.)
#2
05-23-2007, 07:11 PM
 Mikemike2 Guest Join Date: Sep 2001 Location: Calgary Posts: 919
If statement A is correct then statement B is incorrect. The coin flipped in statement B could be one of the infinite amount of coins that will never come up heads. So there is no problem with both of these statements coexisitng. We just have a hard time wrapping our heads around how big a number infinite is
#3
05-23-2007, 07:16 PM
 begbert2 Guest Join Date: Jan 2003 Location: Idaho Posts: 9,035
I completely don't get your point.

A coin can be fair (that is, have an equal intrinsic probability of turning up heads or tails on each successive flip) and still land tails-up on each individual flip for any finite* number of coin flips. There's no contradiction here; the odds of it happening get infinitesimally small as the sequence of flips gets long, but the occurrence of a heads never becomes impossible (assuming a fair coin). It also never becomes certain. If it's a fair coin, it's even odds each flip, no matter what's happened before.

Now, as I flip a coin and each time it comes up tails, I will gradually begin to doubt that it's a fair coin, since it doesn't seem to be behaving the way I'd expect a fair coin to act. But no amount of flipping will *prove* that a heads is impossible, and no number of flips is enough to guarantee that a heads will show. That's not the nature of coin flips.

So out of curiousity, how does this relate to theology?

* It is of course impossible to to flip a coin an infinite number of times. Don't toy with him, Mikemike2.

Last edited by begbert2; 05-23-2007 at 07:19 PM. Reason: spelling
#4
05-23-2007, 07:17 PM
 Nancarrow Guest Join Date: Oct 2004 Posts: 1,697
Quote:
 Originally Posted by kanicbird Believe it or not this comes from all the talk about theistic and atheistic faith issues that have come up lately. Here is the premises: If you flip a infinite number of coins a infinite number of times you will get a infinite number of coins that will never come up heads.
I'm pretty sure that's incorrect. Because you're flipping them an infinite number of times, NONE of them would never come up heads.

If you just flipped them a finite number of times, no matter how many that is, then sure, an infinite number of them would never come up heads.
#5
05-23-2007, 07:19 PM
 kanicbird Guest Join Date: May 1999 Posts: 18,435
Quote:
 Originally Posted by Mikemike2 If statement A is correct then statement B is incorrect. The coin flipped in statement B could be one of the infinite amount of coins that will never come up heads. So there is no problem with both of these statements coexisitng. We just have a hard time wrapping our heads around how big a number infinite is
Statement B has been mathematically proven, called the law of large numbers.

http://en.wikipedia.org/wiki/Law_of_large_numbers
#6
05-23-2007, 07:26 PM
 BrainGlutton BANNED Join Date: Mar 2003 Location: Tampa, Florida Posts: 78,508
Quote:
 Originally Posted by kanicbird * Now this is where the concept of God comes in, things like creation and life may have zero chance of happening on there own without a 'God'.
What you're addressing here is the basic unanswered question of cosmology: "Why is there something rather than nothing?" But invoking God does not answer that question. If God exists, God is something. Why does God exist rather than not existing? (And please don't waste our time with the ontological argument! It's bullshit!)

Last edited by BrainGlutton; 05-23-2007 at 07:27 PM.
#7
05-23-2007, 07:26 PM
 kanicbird Guest Join Date: May 1999 Posts: 18,435
Quote:
 Originally Posted by begbert2 So out of curiousity, how does this relate to theology?
The contention that certain things like creation & life are too small probabilistically to happen on their own (with out a 'designer'), they fall below a quantized threshold that makes things with such a small chance of happening actually zero chance of happening.
#8
05-23-2007, 07:32 PM
 begbert2 Guest Join Date: Jan 2003 Location: Idaho Posts: 9,035
Oh, I get it. The answer is that "really improbable" does not mean "impossible", no matter how improbable it is.

Plus, well, really improbable things become more probable with the amount of tries you have at them (assuming all you're fighting against is probability). Given how many suns and maybe planets there are, and how long the universe has been around, hasn't there been enough chances even for some really improbable things to occur, even if only once or twice?

(Plus maybe we're not the only universe ever to happen. We dunno.)
#9
05-23-2007, 07:36 PM
 Der Trihs Guest Join Date: Aug 2005 Location: California Posts: 38,325
Quote:
 Originally Posted by kanicbird The contention that certain things like creation & life are too small probabilistically to happen on their own (with out a 'designer'), they fall below a quantized threshold that makes things with such a small chance of happening actually zero chance of happening.
Except there's no evidence of that. Scientists who try cooking up life's precursors in simulations of Earth's early environment find it quite easy. From what I've read such arguments usually try to talk about a large, complicated DNA molecule or even a cell coming together by pure chance; not some simple molecules becoming self replicating, and only after much evolution becoming life. Also, they often ignore evolution's abilty to weed out alternates, and ignore the natural tendancy of atoms and molecules to order themselves in various ways - something life exploits a LOT.
#10
05-23-2007, 07:37 PM
 Mikemike2 Guest Join Date: Sep 2001 Location: Calgary Posts: 919
Quote:
 Originally Posted by kanicbird Statement B has been mathematically proven, called the law of large numbers. http://en.wikipedia.org/wiki/Law_of_large_numbers
I did say "if". I believe in the law of large numbers, but as long as were are dealing with the term infinite, there can be no answers as there is no point at which to measure.
#11
05-23-2007, 07:39 PM
 Larry Borgia Charter Member Join Date: Mar 2000 Location: Washington DC Posts: 9,894
You'll never have an infinite number of coins, nor will you be able to flip a coin an infinite number of times, so these paradoxes are irrelevant for the real world.

To expect a run of n heads you'll need 2^n coins. Flipping 1024 coins ten times each will get you one run of heads. Or rather, if you have a large number of trials flipping 1024 coins ten times each you'll average one all-head run per trial. No matter how large your all-head run needs to be, there is a much larger number of coins which will make your all-head run probable.

In order to determine the likelihood of an improbable event we need a clear understanding of just how improbable it is. We don't know that with life, but given the age and size of the universe, I'd bet the probability of life is pretty high. (We do know it isn't zero, at least.)
#12
05-23-2007, 07:44 PM
 begbert2 Guest Join Date: Jan 2003 Location: Idaho Posts: 9,035
Quote:
 Originally Posted by Larry Borgia (We do know it isn't zero, at least.)
Actually the theistic angle on the OP does assume it's naturally impossible -thus making God (or something like it) necessary. If it's merely unlikely, God is unnecessary.
#13
05-23-2007, 07:49 PM
 The Hamster King Member Join Date: Jun 2000 Location: Los Angeles Posts: 10,138
Quote:
 Originally Posted by kanicbird If you flip a infinite number of coins a infinite number of times you will get a infinite number of coins that will never come up heads.
The phrase "never come up heads" is meaningless if you flip a coin an infinite number of times.
#14
05-23-2007, 07:55 PM
 kanicbird Guest Join Date: May 1999 Posts: 18,435
Quote:
 Originally Posted by Larry Borgia You'll never have an infinite number of coins, nor will you be able to flip a coin an infinite number of times, so these paradoxes are irrelevant for the real world.
Yes, and the universe is finite and had a limited amount of time available to develop life - so there is less chance of a complex thing happening if there is finite space and time then infinite space and time.
#15
05-23-2007, 07:58 PM
 kanicbird Guest Join Date: May 1999 Posts: 18,435
Quote:
 Originally Posted by Larry Borgia You'll never have an infinite number of coins, nor will you be able to flip a coin an infinite number of times, so these paradoxes are irrelevant for the real world.
Yes, and the universe is finite and had a limited amount of time available to develop life - so there is less chance of a complex thing happening if there is finite space and time then infinite space and time.

If it can be proven that life (not to mention 'creation' itself) is impossible given infinite time and space, I would contend that it is also impossible with finite time and space
#16
05-23-2007, 07:59 PM
 Kimstu Guest Join Date: Dec 1999 Posts: 19,661
Remember also that the physical universe as we understand it was never "trying" to produce any particular form of life, much less the one that happened to evolve.

It's like picking up a bridge hand of 13 cards dealt randomly from a pack of 52. For any particular hand, the odds of being dealt exactly that hand at random are about 1 in 400000000000000000000. (Yes, 20 zeroes. )

That's orders of magnitude bigger than the number of seconds in the lifetime of the universe. Which means that if you were dealing one bridge hand per second from the birth of the universe to the present moment, the odds would still be thousands to one against your ever getting that particular hand.

And yet, that's the hand that was dealt. Does this demonstrate statistically that there must be an Intelligent Designer of bridge hands who voluntarily stepped in and gave you that particular hand, rather than random chance?

By no means. The thing is that if you're dealing bridge hands at random, you're going to deal out some configuration of 13 cards each time. Each random configuration is just as unlikely as any of the others.

Similarly, the current precise configuration of the physical universe, including this particular set of carbon-based life forms existing on this particular planet orbiting this particular star, is indeed massively improbable. But then, so are all the other possible random configurations of the physical universe.

It would indeed be unrealistically unlikely that the universe could have evolved into any one pre-selected configuration purely at random. But there was no pre-selection involved. The universe could have happened to produce some other other kind of self-aware life form on a different planet near a different star; and if it had (or has), those other life forms would/will probably be just as awed as we are at the mind-boggling odds-beating improbability of their very existence. That doesn't mean that there is any statistical reason to think that anything but chance and physical laws were involved in the process.
#17
05-23-2007, 07:59 PM
 The Hamster King Member Join Date: Jun 2000 Location: Los Angeles Posts: 10,138
Quote:
 Originally Posted by kanicbird There seems like there is some limit on probability, when the chance of something happening is so so so small it is not just effectively zero it is zero and some things just can never happen*.
I raised this issue in February in this thread about improbable configurations of matter in an infinite universe.

Quote:
 * Now this is where the concept of God comes in, things like creation and life may have zero chance of happening on there own without a 'God'.
In order to make such an argument you need to determine the probability of either event occurring spontaineously. Just saying "well gosh it SEEMS really, really unlikely" doesn't cut it.
#18
05-23-2007, 08:02 PM
 begbert2 Guest Join Date: Jan 2003 Location: Idaho Posts: 9,035
Quote:
 Originally Posted by kanicbird If it can be proven that life (not to mention 'creation' itself) is impossible given infinite time and space, I would contend that it is also impossible with finite time and space
Sure. But it hasn't been so proven, now has it? Doesn't that make this something of a strawman?
#19
05-23-2007, 08:10 PM
 kanicbird Guest Join Date: May 1999 Posts: 18,435
Quote:
 Originally Posted by Pochacco The phrase "never come up heads" is meaningless if you flip a coin an infinite number of times.
This is correct, I just realized my initial premise A was in error, the number of coins left converges on zero much faster then the number of flips approaching infinity, so yes you won't have a single coin that fits that.
#20
05-23-2007, 08:14 PM
 The Hamster King Member Join Date: Jun 2000 Location: Los Angeles Posts: 10,138
Here's a quote from severus in the previous thread that may be enlightening:

Quote:
 Originally Posted by severus When we're dealing with an uncountable probability space, saying that an event happens with probability 1 (we also say it happens "almost surely") doesn't mean that it must happen. Similarly, saying that an event happens with probability 0 does not mean that it cannot happen. For example, pick a number at random from the set of reals between 0 and 1. What is the probability of you picking exactly 1/2? The probability is 0. But it doesn't mean that this cannot happen. Similarly, the probability of you picking something else than 1/2 is 1, but as we can see, this isn't what will necessarily happen. This is related to what we know in popular culture as the infinite monkey theorem. If you have a monkey typing at random on a typewriter, for an infinite amount of time, any given finite string of text will be found somewhere within the output of your typing monkey with probability 1. However, it doesn't mean that it must be found in the monkey's output. After all, the monkey could very well only be typing the letter 'a'. The probability of this happening is 0, of course, but it is a valid result. This is because the space of all infinite sequences of letters is uncountable.
#21
05-23-2007, 08:16 PM
 BrainGlutton BANNED Join Date: Mar 2003 Location: Tampa, Florida Posts: 78,508
Quote:
 Originally Posted by kanicbird Yes, and the universe is finite and had a limited amount of time available to develop life - so there is less chance of a complex thing happening if there is finite space and time then infinite space and time.
So what? See post #6.
#22
05-23-2007, 08:16 PM
 kanicbird Guest Join Date: May 1999 Posts: 18,435
Kimstu Yes I understand something will happen, but the law of big numbers will tend towards a certain outcome, so those cards will tend to approach a equal distribution as the number of trials increase towards infinity.
#23
05-23-2007, 08:23 PM
 kanicbird Guest Join Date: May 1999 Posts: 18,435
Pochacco your posting severus' post has the problem of the law of large numbers again if you take the monkey as a random typer.
#24
05-23-2007, 08:25 PM
 kanicbird Guest Join Date: May 1999 Posts: 18,435
Quote:
 Originally Posted by begbert2 Sure. But it hasn't been so proven, now has it? Doesn't that make this something of a strawman?
It's a thought experiment. As I pointed out above I don't expect it to prove the existance of God, as God will not allow that.
#25
05-23-2007, 08:27 PM
 The Hamster King Member Join Date: Jun 2000 Location: Los Angeles Posts: 10,138
Quote:
 Originally Posted by kanicbird Pochacco your posting severus' post has the problem of the law of large numbers again if you take the monkey as a random typer.
I'm not sure what you mean by this.

Basically, what he's saying is: In an infinite sample space, an event can have probability zero and yet still occur.
#26
05-23-2007, 08:28 PM
 The Hamster King Member Join Date: Jun 2000 Location: Los Angeles Posts: 10,138
Quote:
 Originally Posted by kanicbird As I pointed out above I don't expect it to prove the existance of God, as God will not allow that.
Why not?
#27
05-23-2007, 08:33 PM
 Kimstu Guest Join Date: Dec 1999 Posts: 19,661
Quote:
 Originally Posted by kanicbird Kimstu Yes I understand something will happen, but the law of big numbers will tend towards a certain outcome, so those cards will tend to approach a equal distribution as the number of trials increase towards infinity.
In other words, if you deal out bridge hands at random for an infinite number of times, all possible hands will appear equally often (i.e., infinitely many times). Right.

But what does that have to do with the question of the universe and the existence of life? The universe is not infinitely old (not according to all currently accepted cosmological theories, anyway), so no analogy involving an infinite number of trials is applicable to events in the universe.
#28
05-23-2007, 08:35 PM
 Sapo Guest Join Date: Dec 2006 Location: Puerto Rico Posts: 6,786
I think Kimstu has it just right. We have only one universe where things happened the way we are seeing them happen. The probability of this particular universe happening is zero (well, not really but close enough), yet it happened. Had any other universe come to happen, we would be here sitting on our purple mushrooms poking at clay telepathomatics with our noses debating why we don't sit on chairs typing on computers.
#29
05-23-2007, 08:40 PM
 The Hamster King Member Join Date: Jun 2000 Location: Los Angeles Posts: 10,138
Quote:
 Originally Posted by Kimstu ... so no analogy involving an infinite number of trials is applicable to events in the universe.
I think the point that kanicbird was trying to make is that if something is impossible in an infinite number of trials, then it must be impossible in a finite number of trials as well.

However, he has failed to establish that either life or creation is impossible in an infinite number of trials.
#30
05-23-2007, 08:49 PM
 Kimstu Guest Join Date: Dec 1999 Posts: 19,661
Quote:
 Originally Posted by Pocaccho I think the point that kanicbird was trying to make is that if something is impossible in an infinite number of trials, then it must be impossible in a finite number of trials as well.
But isn't that wrong? For example, if we flip an infinite number of fair coins, it's impossible that they all come up heads. Yet if we flip any finite number of fair coins, there is a calculable finite non-zero probability that they all come up heads. Ergo, impossible in the infinite case, possible in the finite case (though still very improbable if the finite number is large).
#31
05-23-2007, 08:49 PM
Both the premises in the OP are true (or, at least, the probability that they are not true is infinitesimal), and they are not contradictory. Yes, in an infinite number of tries there will be an infinite number of heads. On an infinite number of tries there will also be an infinite number of tails. On an infinite number of tries there will be an infinite number where the distribution is 50-50. On an infinite number of tries there will be an infinite number where the distribution of throws is 90-10 in favour of heads. It all depends on which infinite subset of that infinite set of throws you look at.

Note that none of the above says that the proportion of heads (assuming a fair coin) will approach 50% closer and closer as the number of throws gets larger and larger (i.e. as the number of throws tends to infinity).

There's lots of interesting paradoxes when you start trying to play with infinities themselves rather than finite quantities that grow without bound, e.g. there are just as many integers as there are integers that are divisible by 2, just as many integers as rational numbers (x/y where x and y are integers), etc. etc.
#32
05-23-2007, 09:05 PM
 treis Guest Join Date: Jul 2001 Posts: 9,188
Quote:
 Originally Posted by kanicbird Believe it or not this comes from all the talk about theistic and atheistic faith issues that have come up lately. Here is the premises: If you flip a infinite number of coins a infinite number of times you will get a infinite number of coins that will never come up heads. -but- If you flip any one coin a infinite number of times it will hit heads and tails equally. These 2 can't coexist. There is a saying that if you take a room full of monkeys and give them a typewriter and a infinite amount of time you will eventually get a copy of Hamlet. No we all know this could never happen either as monkeys will destroy the paper at a much higher probability then completing a single letter (barring evolution). There seems like there is some limit on probability, when the chance of something happening is so so so small it is not just effectively zero it is zero and some things just can never happen*. * Now this is where the concept of God comes in, things like creation and life may have zero chance of happening on there own without a 'God'. (Not that I'm hanging my theists hat on this, as I don't think God designed the system for Him to be proven, but I would like to see the responses.)
Infinity is a funny thing. For example, did you know that there are as many even whole numbers as there are whole numbers? Or that there are fewer whole numbers than numbers between 0 and .000000000000000000001?
#33
05-23-2007, 09:07 PM
 The Hamster King Member Join Date: Jun 2000 Location: Los Angeles Posts: 10,138
Quote:
 Originally Posted by Kimstu But isn't that wrong? For example, if we flip an infinite number of fair coins, it's impossible that they all come up heads. Yet if we flip any finite number of fair coins, there is a calculable finite non-zero probability that they all come up heads. Ergo, impossible in the infinite case, possible in the finite case (though still very improbable if the finite number is large).
Ah, but I think you're comparing apples and oranges. There are two different events here -- in the first case you're looking for an INFINITE number of coins to turn up heads, and in the second you're looking for a FINITE number of coins to turn up heads. Not the same event at all. The phrasing "all the coins" hides the fact that we're talking about two different sets.

You'll note that in his original formalation of the problem kanicbird was careful to compare one coin flipped an infinite number of times with an infinite number of coins flipped an infinite number of times to avoid this problem.
#34
05-23-2007, 09:14 PM
 DrCube Guest Join Date: Oct 2005 Location: Caseyville, IL Posts: 6,705
Quote:
 Originally Posted by kanicbird If it can be proven that life (not to mention 'creation' itself) is impossible given infinite time and space, I would contend that it is also impossible with finite time and space
Umm, I'm not that well versed in probability, but I'm pretty sure there is no way you can prove that. The universe, life, "creation" -- all obviously did happen. I'm not sure if that makes them dead certainties, but I'm quite positive that they cannot be impossible. Else how could we be here talking about it?
#35
05-23-2007, 09:16 PM
 Mangetout Charter Member Join Date: May 2001 Location: England Posts: 56,984
Quote:
 Originally Posted by kanicbird The contention that certain things like creation & life are too small probabilistically to happen on their own (with out a 'designer'), they fall below a quantized threshold that makes things with such a small chance of happening actually zero chance of happening.
Kind of a hijack and possibly doesn't need saying anyway, but I've seen these arguments many times and in my experience, they always make ridiculous assumptions, such as:
-Trying to calculate the probability of an entire cell forming spontaneously and entirely out of random chemicals (which of course bears absolutely no resemblance to anything any proper scientist might say about the possible origins of life0
-Ignoring chemical affinities - i.e. assigning every atom in the thought experiment an equal probability of bonding with every other atom, regardless of the element
-Pretending the universe would only have the opportinuty to take one shot at a time at creating life in one specific location (i.e. ignoring scaling)

And that's in addition to any simple errors of mathematics, assumed values of variables, etc.
#36
05-23-2007, 09:26 PM
 David Simmons Charter Member Join Date: Nov 2001 Posts: 12,684
Quote:
 Originally Posted by kanicbird Believe it or not this comes from all the talk about theistic and atheistic faith issues that have come up lately. Here is the premises: If you flip a infinite number of coins a infinite number of times you will get a infinite number of coins that will never come up heads. -but- If you flip any one coin a infinite number of times it will hit heads and tails equally. These 2 can't coexist.
Unfortunately claiming that the two can't coexist is a result of using the logic that's true for numbers on infinity which is not a number. You simply can't do mathematical operations with infinity and the logic of mathematics doesn't apply to infinity.

a + n cannot equal a.

∞ + n equals ∞

The second statement isn't a mathematical statement even though it is in mathematical form. All is says is that the result from adding any finite number to an endless quantity is still endless.
#37
05-23-2007, 09:35 PM
 Chief Pedant Guest Join Date: Feb 2007 Posts: 6,961
Quote:
 Originally Posted by kanicbird Now this is where the concept of God comes in, things like creation and life may have zero chance of happening on there own without a 'God'. (Not that I'm hanging my theists hat on this, as I don't think God designed the system for Him to be proven, but I would like to see the responses.)
You are essentially saying that what we cannot currently (or perhaps ever) explain leaves only God(s) as a possible explanation. I'm pretty sure that's how the whole concept of religion began in the first place. We are well on our way to explaining 99% of what was a mystery 25,000 years ago. I wouldn't hang my hat on the last fractional percent of what is unexplained.

In any case, I do agree that God did not "design the system for Him to be proven" unless he wants it proved that there is an uninvolved Malicious Designer for what is obviously a broken and troubled dog-eat-dog creation full of random and hurtful capricious events.

Back to watching chimps slaughter one another and baboons eat flamingos alive on the Nature channel...the news from Iraq and Darfur is just too depressing.
#38
05-23-2007, 09:44 PM
 Indistinguishable Guest Join Date: Apr 2007 Posts: 10,520
Quote:
 Originally Posted by Kimstu But isn't that wrong? For example, if we flip an infinite number of fair coins, it's impossible that they all come up heads. Yet if we flip any finite number of fair coins, there is a calculable finite non-zero probability that they all come up heads. Ergo, impossible in the infinite case, possible in the finite case (though still very improbable if the finite number is large).
It's perfectly well possible to flip an infinite number of fair coins and have them all come up heads. The probability of this is 0, but a probability of 0 is not the same thing as being impossible (as remarked upon above by Pochacco and his quoting of severus).

In fact, no matter what particular exact outcome you get after you flip an infinite number of fair coins (e.g., first coin tails, next three heads, next two tails, ...), its a priori probability was 0. Which should really put the nail into the coffin of the idea that a probability of 0 is the same thing as impossibility.

Last edited by Indistinguishable; 05-23-2007 at 09:46 PM.
#39
05-23-2007, 09:49 PM
 Indistinguishable Guest Join Date: Apr 2007 Posts: 10,520
As to the Law of Large Numbers which keeps getting invoked, all its various formulations are actually only of the form "... happens with probability 1", rather than "... happens for sure". Just as a probability of 0 isn't the same thing as impossibility, a probability of 1 isn't the same thing as completely guaranteed.

Last edited by Indistinguishable; 05-23-2007 at 09:49 PM.
#40
05-23-2007, 09:53 PM
 Kimstu Guest Join Date: Dec 1999 Posts: 19,661
Quote:
 Originally Posted by Indistinguishable In fact, no matter what particular exact outcome you get after you flip an infinite number of fair coins (e.g., first coin tails, next three heads, next two tails, ...), its a priori probability was 0. Which should really put the nail into the coffin of the idea that a probability of 0 is the same thing as impossibility.
Okay, I see, thanks.

And the probability of getting a particular configuration in any finite number of tries is not zero, so it's even more obviously not impossible.
#41
05-23-2007, 10:01 PM
 Indistinguishable Guest Join Date: Apr 2007 Posts: 10,520
Quote:
 Originally Posted by kanicbird Here is the premises: If you flip a infinite number of coins a infinite number of times you will get a infinite number of coins that will never come up heads. -but- If you flip any one coin a infinite number of times it will hit heads and tails equally. These 2 can't coexist.
You're right that those two premises can't coexist, but the laws of probability don't guarantee either one of them. The laws of probability do tell us that the second premise holds for any particular fair coin with probability 1; however, the first premise doesn't even have that support. (With countably many coins, the laws of probability, as conventionally formulated, actually give the first premise a probability 0 of holding [to those who care about this sort of thing, I can rant all day about the arbitrariness of the countable additivity condition in Kolmogorov's axioms]; with an uncountable infinity of coins, the situation is different, but depends strongly upon the particular probability distribution you assume, there not being a particular distinguished one to pick in this case).

Last edited by Indistinguishable; 05-23-2007 at 10:05 PM.
#42
05-23-2007, 10:25 PM
 Shodan Charter Member Join Date: Jul 2000 Location: Milky Way Galaxy Posts: 35,275
You can't flip a coin an infinite number of times.

Regards,
Shodan
#43
05-23-2007, 10:36 PM
 Shamozzle Guest Join Date: Apr 2007 Posts: 2,263
Quote:
 Originally Posted by Indistinguishable ... Which should really put the nail into the coffin of the idea that a probability of 0 is the same thing as impossibility.
Can you explain to me how a probability of 0 is not the same as an impossibility?

If I have a die with no "1" on it, the probability that I roll a "1" is zero. It is possible to role a "1". What am I missing?
#44
05-23-2007, 10:54 PM
 Kimstu Guest Join Date: Dec 1999 Posts: 19,661
Quote:
 Originally Posted by Shamozzle Can you explain to me how a probability of 0 is not the same as an impossibility? If I have a die with no "1" on it, the probability that I roll a "1" is zero. It is [impossible] to [roll] a "1". What am I missing?
Nothing, in the case of a die with a finite number of sides. Rolling a "1" with a die having no "1" on it is not a possible outcome, and thus has zero probability.

But what I think Indistinguishable is getting at is that the situation gets more complicated in the case of infinite possible outcomes.

If you have a fair die with 6 numbered sides, the probability of rolling any of those 6 numbers on one roll is 1/6. If there are 12 sides, the probability of rolling any of them is 1/12. And so on.

Thus if you have a fair die with an infinite number of sides, the probability of rolling any one of them must be 1/(infinity), which is zero. Yet, if you do roll that die (leaving aside the physical question of how you possibly could roll a die with an infinite number of sides, and no smart-ass remarks about spherical dice, please), you will in theory get one of those sides as the outcome.

So this is a situation where an outcome may have probability zero and yet be possible anyway.
#45
05-23-2007, 11:19 PM
 Indistinguishable Guest Join Date: Apr 2007 Posts: 10,520
Yeah, what Kimstu said. To be explicit, all impossible events have probability zero, but not all events of probability zero are impossible. One illustrative example would be a dartboard. You toss your dart and you hit the board at, say, x = 0.06892062362..., y = 0.908246236222... What was the probability of hitting that exact point? Well, zero, under the conventional model (infinitely many points on the dartboard, each with equal probability of getting hit; that probability is supposed to be a real number, and the probabilities can't add up to more than 1; therefore, each point has probability 0 of getting hit). All the same, you did actually hit that exact point. So, since you pulled off an event of probability 0, clearly, events of probability 0 are not impossible.

I would imagine the Internet would be filled with pages illustrating the distinction between "probability 0" and "impossible" (and between "probability 1" and "guaranteed"), since it's an all too common mistake to conflate them, but am sorely disappointed to find that Google fails me in this regard. At any rate, Wikipedia has a bit on the distinction.

Last edited by Indistinguishable; 05-23-2007 at 11:20 PM.
#46
05-23-2007, 11:22 PM
 Indistinguishable Guest Join Date: Apr 2007 Posts: 10,520
Oh, and, actually, smart-ass remarks about spherical dice are perfect. Take a perfect sphere, toss it up in the air, have it come to the ground, and note which exact point it lands on. Every particular exact point on the sphere has a probability of 0 of being the one it lands on (by the same reasoning as in the above dartboard example), yet, nonetheless, one of them does end up as such. Another example where an event of probability 0 comes to pass.

Last edited by Indistinguishable; 05-23-2007 at 11:24 PM.
#47
05-23-2007, 11:33 PM
 Czarcasm Charter Member Charter Member Join Date: Apr 1999 Location: Beervania Posts: 53,764
Quote:
 Originally Posted by Indistinguishable Oh, and, actually, smart-ass remarks about spherical dice are perfect. Take a perfect sphere, toss it up in the air, have it come to the ground, and note which exact point it lands on. Every particular exact point on the sphere has a probability of 0 of being the one it lands on (by the same reasoning as in the above dartboard example), yet, nonetheless, one of them does end up as such. Another example where an event of probability 0 comes to pass.
Sonofabitch-this actually makes sense to me.
Thank you.
#48
05-24-2007, 12:09 AM
 Larry Borgia Charter Member Join Date: Mar 2000 Location: Washington DC Posts: 9,894
Quote:
 Originally Posted by Indistinguishable Oh, and, actually, smart-ass remarks about spherical dice are perfect. Take a perfect sphere, toss it up in the air, have it come to the ground, and note which exact point it lands on. Every particular exact point on the sphere has a probability of 0 of being the one it lands on (by the same reasoning as in the above dartboard example), yet, nonetheless, one of them does end up as such. Another example where an event of probability 0 comes to pass.
Dude, you're freaking me out. The only thing I can think of is that real spheres don't have an infinite number of points to land on.
#49
05-24-2007, 12:22 AM
 Shamozzle Guest Join Date: Apr 2007 Posts: 2,263
Any surface has an infinite number of points.
#50
05-24-2007, 12:29 AM
 Shamozzle Guest Join Date: Apr 2007 Posts: 2,263
Quote:
 Originally Posted by Kimstu Nothing, in the case of a die with a finite number of sides. Rolling a "1" with a die having no "1" on it is not a possible outcome, and thus has zero probability. But what I think Indistinguishable is getting at is that the situation gets more complicated in the case of infinite possible outcomes. If you have a fair die with 6 numbered sides, the probability of rolling any of those 6 numbers on one roll is 1/6. If there are 12 sides, the probability of rolling any of them is 1/12. And so on. Thus if you have a fair die with an infinite number of sides, the probability of rolling any one of them must be 1/(infinity), which is zero. Yet, if you do roll that die (leaving aside the physical question of how you possibly could roll a die with an infinite number of sides, and no smart-ass remarks about spherical dice, please), you will in theory get one of those sides as the outcome. So this is a situation where an outcome may have probability zero and yet be possible anyway.
Ok, I get that. The terminology is a bit disturbing though. Semantics, I guess.

So is this true, then:

With an infinite number of monkeys hammering on an infinite number of typewriters, not only is "Hamlet" being hammered out by one of the monkeys, but, in fact, an infinite number of "Hamlets" are being initiated at any given moment in time, and further, that every and any publication of past, present, or future is being initiated an infinite number of times per given moment in time?

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