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#1




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.) 
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#2




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




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 tailsup 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; 05232007 at 06:19 PM. Reason: spelling 
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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. 


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http://en.wikipedia.org/wiki/Law_of_large_numbers 
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Last edited by BrainGlutton; 05232007 at 06:27 PM. 
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#8




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.) 
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#11




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 allhead run per trial. No matter how large your allhead run needs to be, there is a much larger number of coins which will make your allhead 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.) 
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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




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 carbonbased 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 preselected configuration purely at random. But there was no preselection involved. The universe could have happened to produce some other other kind of selfaware 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 mindboggling oddsbeating 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. 
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Here's a quote from severus in the previous thread that may be enlightening:
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#22




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




Pochacco your posting severus' post has the problem of the law of large numbers again if you take the monkey as a random typer.

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Basically, what he's saying is: In an infinite sample space, an event can have probability zero and yet still occur. 
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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




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.

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However, he has failed to establish that either life or creation is impossible in an infinite number of trials. 


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#31




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 5050. On an infinite number of tries there will be an infinite number where the distribution of throws is 9010 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. 
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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. 
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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. 
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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. 
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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 dogeatdog 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. 
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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; 05232007 at 08:46 PM. 
#39




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; 05232007 at 08:49 PM. 


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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. 
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Last edited by Indistinguishable; 05232007 at 09:05 PM. 
#42




You can't flip a coin an infinite number of times.
Regards, Shodan 
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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? 
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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 smartass 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




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; 05232007 at 10:20 PM. 
#46




Oh, and, actually, smartass 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; 05232007 at 10:24 PM. 
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Thank you. 
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#49




Any surface has an infinite number of points.



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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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