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Akita T
Caldari Navy Volunteer Task Force
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Posted - 2011.04.16 21:59:00 -
[91]
Edited by: Akita T on 16/04/2011 22:32:44
Edit : simplified to just the relevant things.
Quote: The same for 3 dimensions.
That's just the thing : NO, it is NOT the same for three dimensions. Your gut tells you that it is, but IT IS NOT.
Quote: Polya's proof does not limit you in your movements. It is the increasing amount of directions you can choose with each new dimension, which makes it less probable to return. It does not make it in anyway impossible for a walk to return.
As the number of steps in the 3D random walk approaches infinity, the number of walks that HAVE NOT returned to the origin actually approaches a number of roughly 0.6594626704 * (6^steps), which is infinite. The total number of possible walks is 6^steps, and as number of steps approaches infinity, that too approaches infinity.
Last time I checked, lim (constant * variable / variable) when variable goes to infinity is the constant, so the PERCENTAGE of walks that never return to the origin approaches ~0.66 !!!
Therefore, the probability that you DO return to the origin at least once is indeed only P = 0.3405373296..., NOT P = 1. Q.E.D.
Yes, it is counterintuitive in the fact that the number of paths for the random walk that DO get you back to the origin indeed also goes towards infinity, but the rate at which paths which do that increases as the numbers of steps increases does not go up fast enough when compared to the total number of possible paths to EVER get to P=1. _
CCP LEADERSHIP MENTALITY NEEDS TO CHANGE FAST ! "New junky features sell, old polished content doesn't" ? KILL IT WITH FIRE. |
Whitehound
The Whitehound Corporation Frontline Assembly Point
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Posted - 2011.04.16 22:32:00 -
[92]
Edited by: Whitehound on 16/04/2011 22:35:08
Originally by: Akita T That's just the thing : NO, it is NOT the same for three dimensions. Your gut tells you that it is, but IT IS NOT.
What I mean by it being the same is that it is wrong... It is the same wrong interpretation of being somehow limited in your movement.
You then switch in your interpretation from infinite movements to infinite walks of infinite movements and back, but it is not all just the same. In your question do you also not ask for the probability of infinite walks, but you are asking for the probability of a single walk:
Quote: "GIVEN AN INFINITE AMOUNT OF TIME, what is the probability you will EVENTUALLY land on the origin point at least once again for each of the three cases ?"
Where do you see yourself mentioning an infinite number of walks of infinite movements? All I see in your question is you asking for the probability over infinite time...
What you think you are talking about - an infinite set of walks of infinite movements - has got nothing to do with your question. --
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Akita T
Caldari Navy Volunteer Task Force
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Posted - 2011.04.16 22:40:00 -
[93]
Edited by: Akita T on 16/04/2011 22:42:53
Originally by: Whitehound
Quote: "GIVEN AN INFINITE AMOUNT OF TIME, what is the probability you will EVENTUALLY land on the origin point at least once again for each of the three cases ?"
Where do you see yourself mentioning an infinite number of walks of infinite movements? All I see in your question is for the probability over infinite time... What you think you are talking about - an infinite set of walks of infinite movements - has got nothing to do with your question.
The two statements should be equivalent. If roughly 66% of all possible infinite length 3D walks *never* go back to the origin, that should be the same as saying that you have a 66% probability of never making it back to the origin given a random selection of just one of the infinite numbers of possible random walks.
If I say "this operation carries a 66% risk of death", and you DO get the operation, it is OBVIOUS that you will EITHER live OR die. AFTER the fact, you end up with either P=0 or P=1 for "alive". But still, the probability of death for the operation is P=0.66, not 1, nor 0.
_
CCP LEADERSHIP MENTALITY NEEDS TO CHANGE FAST ! "New junky features sell, old polished content doesn't" ? KILL IT WITH FIRE. |
Whitehound
The Whitehound Corporation Frontline Assembly Point
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Posted - 2011.04.16 23:22:00 -
[94]
Originally by: Akita T Isn't that pretty much the same thing anyway ?
No... The probability for an infinite number of movements (to reach a given point) is not the same thing as the probability for an infinite number of walks of infinite movements (to reach a given point). The later implies a set of walks, where each walk differs from another. In the special case of looking at an infinite number of the same walk will it be the same thing, too. --
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Akita T
Caldari Navy Volunteer Task Force
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Posted - 2011.04.16 23:28:00 -
[95]
Well then, I concede that there is a nonzero probability you are right and I am wrong given that specific formulation of the question.
_
CCP LEADERSHIP MENTALITY NEEDS TO CHANGE FAST ! "New junky features sell, old polished content doesn't" ? KILL IT WITH FIRE. |
Katie Tanaka
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Posted - 2011.04.17 03:38:00 -
[96]
Originally by: Whitehound No... The probability for an infinite number of movements (to reach a given point) is not the same thing as the probability for an infinite number of walks of infinite movements (to reach a given point). The later implies a set of walks, where each walk differs from another.
The nonsense continues; escalating now to a level that seeks to challenge the definition of "probability" itself.
I guess we need to abandon the Central Limit Theorem and go "back to basics".
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Akita T
Caldari Navy Volunteer Task Force
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Posted - 2011.04.17 04:52:00 -
[97]
Shush you, it's funnier when he stubs his toes on the meat of the matter rather than the definitions part _
CCP LEADERSHIP MENTALITY NEEDS TO CHANGE FAST ! "New junky features sell, old polished content doesn't" ? KILL IT WITH FIRE. |
Hieronimus Rex
Minmatar Infinitus Sapientia New Eden Research.
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Posted - 2011.04.17 06:34:00 -
[98]
Originally by: Katie Tanaka
Originally by: Whitehound No... The probability for an infinite number of movements (to reach a given point) is not the same thing as the probability for an infinite number of walks of infinite movements (to reach a given point). The later implies a set of walks, where each walk differs from another.
The nonsense continues; escalating now to a level that seeks to challenge the definition of "probability" itself.
I guess we need to abandon the Central Limit Theorem and go "back to basics".
Pretty much yeah. He also believes "everything must be either science or a shoe."
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Whitehound
The Whitehound Corporation Frontline Assembly Point
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Posted - 2011.04.17 08:05:00 -
[99]
Edited by: Whitehound on 17/04/2011 08:14:45
Originally by: Akita T Shush you, it's funnier when he stubs his toes on the meat of the matter rather than the definitions part
The meat that I stub upon sits between your ears. It is hard to believe that you find this a brain teaser and then f'ck it up. It really just means that you have not got a clue what you were really looking at.
It is as if you wanted to know what the probability is of a natural number to be dividable by 2 and then end up asking for the probability of a natural number to be dividable. The brain teaser as you call it has blown your brains out.
You have failed.
I also do not believe math is a shoe. I was asking if it was a shoe to you if it was not a science. What I do believe is that you are an idiot when you say math is not a science.
Please, any more math nerds who want to shed a tear over their beloved "brain teasers"? Next time just take off your shoes and sniff at them. It is going to tease your brains just as much. --
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Akita T
Caldari Navy Volunteer Task Force
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Posted - 2011.04.17 08:51:00 -
[100]
Originally by: Akita T I am curious though... How exactly do you reconcile saying that a) ANY *particular* infinite length walk WILL end up visiting the origin again (if the correct answer would be P=1) given the fact that b) ~66% of all the *possible* 3D walks of infinite lenghth do not go back to the origin (as per linked calculations)
You still haven't answered this. _
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Whitehound
The Whitehound Corporation Frontline Assembly Point
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Posted - 2011.04.17 09:01:00 -
[101]
Edited by: Whitehound on 17/04/2011 09:06:45
Originally by: Akita T
Originally by: Akita T I am curious though... How exactly do you reconcile saying that a) ANY *particular* infinite length walk WILL end up visiting the origin again (if the correct answer would be P=1) given the fact that b) ~66% of all the *possible* 3D walks of infinite lenghth do not go back to the origin (as per linked calculations)
You still haven't answered this.
Where is there a question, Akita? I understand you fail to comprehend the subject, but why should it bother me? You simply fail. You mix finite mediations with infinite ones as if it was all the same. You need to make a difference and keep it this way. --
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Akita T
Caldari Navy Volunteer Task Force
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Posted - 2011.04.17 09:16:00 -
[102]
Edited by: Akita T on 17/04/2011 09:27:04
Originally by: Whitehound
Originally by: Akita T I am curious though... how exactly do you reconcile saying that a) ANY *particular* infinite length walk WILL end up visiting the origin again (if the correct answer would be P=1) given the fact that b) ~66% of all the *possible* 3D walks of infinite lenghth do not go back to the origin (as per linked calculations)
Where is there a question, Akita?
Right there. TO ME, those statements look like they contradict eachother. Also, to me, it appears you simply made the first statement but you have not denied the validity of the second statement. So my question was, how do you personally reconcile those two conflicting pieces of information ?
Also, right here, black on white, it says "the probability that a random walk on a x-D lattice returns to the origin". It doesn't specify a finite length walk, so it is an infinite length walk. You say it is 1, the proof there says it is not 1 for 3-D or higher. Are you now disputing the accuracy of that, or maybe just the wording ?
Quote: I understand you fail to comprehend the subject, but why should it bother me? You simply fail.
Well, apparently, you care enough that you feel the need to keep spewing personal insults between your plumage displays of superiority without going into much detail as to why that is the case.
Quote: You mix finite mediations with infinite ones as if it was all the same. You need to make a difference and keep it this way.
That's not how I see it, so you need to explain it as if I know as little as you seem to think I know. Feel free to enlighten me with a detailed explanation of EXACTLY WHY I might be wrong.
Originally by: Whitehound I did give you a particular and infinite walk that did return to its origin if only for once.
And that is but one of the infinite number of possible infinite walks. What if I pick one of those infinite walks that DOES NOT return to the origin ? The proof which you did not deny its accuracy of states that such an INFINITE LENGTH walk does exist, moreso, that roughly 66% of them feature this peculiarity.
So, in other words, the probability of being in one of the walks that returns at least once is the same as the probability to EVER return, and the probability of being in one that does not is the same as the probability to never return. You say that this particular conclusion is wrong, and I am (paraphrasing) an idiot for not seeing that it's wrong. Feel free to explain in great detail WHY this is wrong.
_
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Scorpyn
Caldari Warp Ghosts Omega Spectres of the Deep
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Posted - 2011.04.17 09:25:00 -
[103]
Why is this discussion still going on?
Whitehound is correct.
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Katie Tanaka
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Posted - 2011.04.17 09:28:00 -
[104]
Originally by: Scorpyn Whitehound is correct.
Perhaps you can construct the proof that he is unable or unwilling to construct then? If not, GTFO.
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Mag's
the united Negative Ten.
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Posted - 2011.04.17 09:30:00 -
[105]
ITT Whitehound trolling.
Originally by: Allestin Villimar Also, if your bookmarks are too far out, they can and will ban you for it.
Originally by: Torothanax Low population in w systems makes afk cloaking unattractive.
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Whitehound
The Whitehound Corporation Frontline Assembly Point
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Posted - 2011.04.17 09:41:00 -
[106]
Originally by: Akita T Right there.
It is not a question to me. Like I said, why should I care?
Why do you think people say "as it approaches infinity", instead of just saying "in infinity" or to omit it and instead imply infinity for everything like you do?
There is no such thing as infinity and that is what you need to learn. If you are still stuck at looking at infinite walks of infinite movements then you are in your own personal hell of infinity. Seriously, I am fine with it. It does not bother me at all that it makes your brain go bonkers, if it teases it or makes parts of maths counter intuitive for you. --
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Akita T
Caldari Navy Volunteer Task Force
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Posted - 2011.04.17 09:45:00 -
[107]
Edited by: Akita T on 17/04/2011 09:47:19
So, basically, you are saying that even if the proof says quite clearly that ~66% of possible infinite-length walks NEVER, EVER return to the origin point, if you pick ANY SINGLE ONE of the possible walks, you ALWAYS get back to the origin at least once ? And your reasoning for that is solely that "infinity" is an abstract concept, and therefore EVERYTHING is possible "at infinity" ?
_
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Whitehound
The Whitehound Corporation Frontline Assembly Point
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Posted - 2011.04.17 10:05:00 -
[108]
Edited by: Whitehound on 17/04/2011 10:05:11
Originally by: Akita T So, basically, you are saying that even if the proof says quite clearly that ~66% of possible infinite-length walks NEVER, EVER return to the origin point, ...
It is not what the proof says.
There is no such thing as infinity. There is also no such thing as infinite movement and there is also no such things as infinite walks of infinite movements. These are abstracts.
You put a number like 0.34 on an infinite set of walks. But what do you get in return? You still get 34% of infinity. How can you there possibly say that in those 34% there is not a walk that does not return to its origin if only for once? It is just nonsense.
Polya only proofed that not all infinite walks in 3 dimensions return infinitely often. The proof does not imply that 34% of walks not return at all.
Only in infinite mediations is "one time" the same as "no time". It does however not allow you to say 1 = 0. --
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Akita T
Caldari Navy Volunteer Task Force
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Posted - 2011.04.17 10:18:00 -
[109]
Originally by: Whitehound Polya only proofed that not all infinite walks in 3 dimensions return infinitely often. The proof does not imply that 34% of walks not return at all.
He proved that there is a FINITE amount of times any infinite length random walk will pass through any specific point in the 3D (or higher) grid space, as opposed to an INFINITE amount of times for 1D and 2D for ANY of the points. Others later calculated that ~66% of those possible infinite length 3D walks NEVER return to the origin, while only ~34% of them do so at least once. It's right there in the text of the linked documents.
_
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Whitehound
The Whitehound Corporation Frontline Assembly Point
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Posted - 2011.04.17 10:42:00 -
[110]
Originally by: Akita T
Originally by: Whitehound Polya only proofed that not all infinite walks in 3 dimensions return infinitely often. The proof does not imply that 34% of walks not return at all.
He proved that there is a FINITE amount of times any infinite length random walk will pass through any specific point in the 3D (or higher) grid space, as opposed to an INFINITE amount of times for 1D and 2D for ANY of the points. Others later calculated that ~66% of those possible infinite length 3D walks NEVER return to the origin, while only ~34% of them do so at least once. It's right there in the text of the linked documents.
It is pointless. You keep mixing finite with infinite and still do not want to make a distinction. Have your 34% or 66% of infinity... Why do you even want to stretch this after you have already said that you failed to ask the right question? --
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Alpheias
Euphoria Released
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Posted - 2011.04.17 10:51:00 -
[111]
This is going to go on for an infinity....... *sigh*
♫ When your ship gets blown to bits ♫ And you lose your Faction fits \☻/ Don't worry ♪ ♫ ♪ ♫ ♫ ♪ ♫ ♪ Be Happy \☻/ |
Sitara
Minmatar Solar Flare Trade and Production
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Posted - 2011.04.17 10:55:00 -
[112]
Originally by: Mag's ITT Whitehound trolling.
This tbh - he's making vague general statements with no attempt to either answer the specific questions raised or prove what he's saying.
Drop it Akita - he's either just tolling or doesn't really understand the problem.
We could always do this one now if you've not seen it before :
"I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?"
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Whitehound
The Whitehound Corporation Frontline Assembly Point
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Posted - 2011.04.17 10:57:00 -
[113]
Edited by: Whitehound on 17/04/2011 10:58:39
Originally by: Sitara "I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?"
I like families with 2.3 children much better. It makes me think there is somewhere a child of 0.3 in size. --
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Akita T
Caldari Navy Volunteer Task Force
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Posted - 2011.04.17 11:16:00 -
[114]
Edited by: Akita T on 17/04/2011 11:25:18
Originally by: Whitehound Why do you even want to stretch this after you have already said that you failed to ask the right question?
No, what I actually said was...
Originally by: Akita T EDIT : YOU MIGHT (just might) sort of have a tiny bit of a point here. The absolutely and utterly inattackable formulation of the original problem should have read... "As the number of steps approaches infinity, what does the probability you will have landed on the origin point at least once more approach to, for each of the three cases ?" ...but... Isn't that pretty much the same thing anyway ?
Which is the normal, polite parlance translation of : "You're full of crap and splitting hairs, but for the sake of continuing the main argument I'll admit that the wording could have been formulated ever so slightly differently in order to comply to your nagging, even if it pretty much says the same thing".
Quote: You are still misinterpreting the proof.
No, I'm not misinterpreting the proof, you are.
Quote: Anyone can show that there is a finite number of walks that return to their origin.
No, there is not a finite amount of walks that return to the origin point. There is an INFINITE amount of walks that returns to the origin. In fact, 34% of all the possible infinite walks do return to the origin.
Quote: All you need to do is to give one example and you have your proof.
:sigh:
Quote: It is not what Polya did.
Apparently, you do not understand what Polya did, nor do you understand the ramifications of what he proved, and you certainly do not understand what the other guys mentioned there did either.
Quote: And not any walk will pass through any point ... not any walk will pass through all points *sigh*
Exactly.
ANY specific infinite length 3D walk will pass through ANY specific point a FINITE number of times (0, 1, 2, 3, etc), and NEVER an infinite number of times. However, for 1D and 2D, ANY specific walk will pass through ANY specific point an INFINITE number of times.
~66% of all possible infinite length 3D walks will NEVER AGAIN pass through the origin a second time, passing through the origin only ONCE, at the very start. Therefore, being right now at the start point at time zero, the probability to ever return to the origin without knowledge in advance about which particular walk you are in right now is ~34%.
And apparently, you can't accept that, because you insist on grasping to the conceptual straw that "but at infinity anything is possible". Newsflash : it's not. For 1D and 2D, it is, yes... but for 3D, it's not.
_
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Sturmwolke
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Posted - 2011.04.17 11:22:00 -
[115]
Originally by: Alpheias This is going to go on for an infinity
I heard infinity is a concept that can be often used to sweep a few untidy parts of an argument under the carpet. This thread might have a probability of extending into pages 7 and 8; oh! say, 15.9% ... given infinite time of course.
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Whitehound
The Whitehound Corporation Frontline Assembly Point
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Posted - 2011.04.17 12:01:00 -
[116]
Edited by: Whitehound on 17/04/2011 12:08:13 No need to emo-rage. You will only start to amuse the masses...
Originally by: Akita T No, there is not a finite amount of walks that return to the origin point. There is an INFINITE amount of walks that returns to the origin. In fact, 34% of all the possible infinite walks do return to the origin.
You have said:
"He proved that there is a FINITE amount of times any infinite length random walk will pass through any specific point in the 3D (or higher) grid space,"
Showing one example (one, as in the number 1), which passes through a specific point is a finite amount and counts as a proof. This is not what he did.
If you are trying to say any infinite walk will pass through all points at least once, then you are wrong and it can be proven with a counter example (a circling walk). This is also not what he did.
What he did was he showed that with each dimension you add after 2 dimensions the probability to return to the origin drops. His proof is important, because this could not be seen with only 1 and 2 dimensions. After 3 dimensions does it however match our understanding of the world around us and we can say things like "the more choices you have, the more easily you can get lost".
It does not mean that a infinite random walk is uncontrollable. Nor does it mean that a walk will or will not return. What defines the ability of a walk to return is not its probability.
If it was the probability that defines the outcome then two events could never be independent from one another. The probability for a series of coin tosses is however independent of the probability of a single coin toss.
Because of this independence can you not conclude that a single infinite and random walk in 3 dimensions returns, or not returns, to its origin by looking at the probability of an infinite set of walks. --
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Akita T
Caldari Navy Volunteer Task Force
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Posted - 2011.04.17 12:29:00 -
[117]
Edited by: Akita T on 17/04/2011 12:33:21 dang doublepost _
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Akita T
Caldari Navy Volunteer Task Force
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Posted - 2011.04.17 12:33:00 -
[118]
Edited by: Akita T on 17/04/2011 12:38:51
Originally by: Whitehound You have said: "He proved that there is a FINITE amount of times any infinite length random walk will pass through any specific point in the 3D (or higher) grid space," Showing one example (one, as in the number 1), which passes through a specific point is a finite amount and counts as a proof. This is not what he did.
The hell are you talking about ? This has absolutely nothing to do with what you quoted. What you quoted, expressed in other words, says "picking one of the infinite numbers of possible infinite 3D walks, for that particular walk, picking one specific point, that INFINITE random walk will pass through that point a FINITE number of times, NOT an infinite number of times (like in the case of 1D or 2D walks)".
Quote: If you are trying to say any infinite walk will pass through all points at least once, then you are wrong and it can be proven with a counter example (a circling walk). This is also not what he did.
This is also not what I said. Are you losing the capacity to read now too ? Coincidentally, yes, any infinite 1D and any infinite 2D walk will pass through all points, not just once, but an INFINITE number of times. However, for an infinite 3D walk, the number of times it passes through any arbitrary point can be zero times, one time, two times, three times, any FINITE number of times, but NOT an infinite number.
Quote: What he did was he showed that with each dimension you add after 2 dimensions the probability to return to the origin drops.
BINGO ! Given you at the start point, at time zero, the probability you will EVER return to the origin in an infinite length random 3D walk is ~34%. Q.E.D.
Quote: It does not mean that a infinite random walk is uncontrollable.
Nobody said anything about that.
Quote: Nor does it mean that a walk will or will not return.
Yes, it does.
Quote: What defines the ability of a walk to return is not its probability.
Yes, any particular walk has the ABILITY to return, indeed, all you need to do is reverse all steps made. Yet still, some NEVER WILL. In fact, ~66% of them never will.
Quote: If it was the probability that defines the outcome then two events could never be independent from one another. The probability for a series of coin tosses is however independent of the probability of a single coin toss.
Any infinite series of coin tosses can be emulated with the 1D random walk case. This however has no relevance for the 3D walk case anyway.
If you want to split hairs, here's one for you : if you have a "cheat" coin that lands 66% of the time on heads and 34% of the time on tails, what is the probability that the next coin toss will land heads ? What if you have an infinite number of coin tosses (yes, I know, you can't have an infinite number of cointosses, WHATEVER), what will the probability of the INFINITY+1 coin toss landing on heads be ? Just because the coin has the ABILITY to land tail side up 34% of the time does not mean that in step "infinity+1" it will land tails. Your whole argument so far seems to be predicated on the fact that JUST BECAUSE ANY PARTICULAR TRACK CAN BE BACKTRACKED to get to the origin again (ignoring the fact you would need "infinity times two" steps to get back), the probability to get back to the origin is P = 1.
Quote: Because of this independence can you not conclude that a single infinite and random walk in 3 dimensions returns, or not returns, to its origin by looking at the probability of an infinite set of walks.
That's the very definition of probability. What is the probability that event A will happen, if it happens X times out of Y possibilities ? Last time I checked, in this case, P = X/Y. Does your definition of probability say something different ? Or maybe for you 0.34 * x / x (when x goes to infinity) is somehow magically different from 0.34, turning into a 1 all of a sudden ? _
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Whitehound
The Whitehound Corporation Frontline Assembly Point
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Posted - 2011.04.17 12:56:00 -
[119]
Edited by: Whitehound on 17/04/2011 13:06:14
Originally by: Akita T Coincidentally, yes, any infinite 1D and any infinite 2D walk will pass through all points ..
No, wrong. A walk in 1 dimension that alternates only between left and right will not pass through all points. How can it?
Do you think it will stop alternating, because you say so? Stubbornness is not a good friend in an argument...
Originally by: Akita T That's the very definition of probability.
No. I can toss ten times "heads" in a row, but it does not change the probability for the eleventh coin toss to be either "heads" or "tails". It is unlikely to have ten times "heads", and even more unlikely to have eleven times "heads", but the probability for the coin to show "heads" or "tails" is still 1:1 and always will be 1:1. --
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Akita T
Caldari Navy Volunteer Task Force
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Posted - 2011.04.17 13:02:00 -
[120]
Originally by: Whitehound
Originally by: Akita T Coincidentally, yes, any infinite 1D and any infinite 2D walk will pass through all points ..
No, wrong. A walk in 1 dimension that alternates only between left and right will not pass through all points. How can it?
The probability of such a walk existing is P=0, which is not quite the same as saying that such a walk does not exist. I guess the correct way to express it in mathematical terms would be "the probability that an infinite 1D/2D walk will pass through any specific point an infinite number of times is P=1", which technically does not mean that ANY walk will, but that "almost any" walk will.
_
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