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Sitara
Minmatar
Solar Flare Trade and Production
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Posted - 2011.04.16 03:57:00 -
[1]
Ok - here goes.
With an infinite amount of time you will visit each point that can be reached an infinite number of times. So all we need to show is that the origin *can* be reached again from any point and the probability is 100%
In case 1D - trivial - 2 possible directions to move at each step and you can take either - say you're at position x where x!=0 then if x<0 move x+1 else move x-1 ; repeat until back at origin : 100%
In case 2D - extension of above. 4 possible directions. if at any point (x,y) apply the rule above to each of x and y till you reach the origin : 100%
In case 3D - extension of above. 6 possible directions. if at point (x, y, z) apply the rule to each of x, y and z till you reach the origin : 100%
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Sitara
Minmatar
Solar Flare Trade and Production
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Posted - 2011.04.16 11:01:00 -
[2]
hehe kk - I've seen the proof online now and given the sources I have little reason to disbelieve (even if I would need to invest several days to fully follow the proof - its been like 20 years since I studied advanced probability theory and not an area of maths I really use in practice heh).
It does appear bonkers counter-intuitive but then that's infinities and probability theory for you 
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Sitara
Minmatar
Solar Flare Trade and Production
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Posted - 2011.04.16 11:24:00 -
[3]
Yeah, you're kind of talking at cross purposes - Akita is (correctly) talking about the probability of a *single* random walk returning to O, wheras you are (also correctly) saying that the with an infinite number of random walks will get one that does if the probability of a single one is non-zero.
To be fair the way the original question was posed does not limit you to a single random walk.
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Sitara
Minmatar
Solar Flare Trade and Production
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Posted - 2011.04.16 11:40:00 -
[4]
Edited by: Sitara on 16/04/2011 11:41:09
 Originally by: Whitehound
Such a limit will only lead to ~34% of all people saying that it does return, and ~64% of all saying that it does not. Both will be correct, but the majority of 64% will win, because they are in the majority. 
Heh - got to love majority rule 
Edit: I think I'll join the other 2% who must have gone down the pub 
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Sitara
Minmatar
Solar Flare Trade and Production
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Posted - 2011.04.16 11:51:00 -
[5]
Originally by: Whitehound
Originally by: Sitara
Edit: I think I'll join the other 2% who must have gone down the pub
I will simply go straight home, not stagger left and right, no matter how drunk I am.
That would be a deterministic walk then 
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Sitara
Minmatar
Solar Flare Trade and Production
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Posted - 2011.04.17 10:55:00 -
[6]
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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Sitara
Minmatar
Solar Flare Trade and Production
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Posted - 2011.04.17 13:09:00 -
[7]
Originally by: Whitehound
No, wrong. A walk in 1 dimension that alternates only between left and right will not pass through all points. How can it?
That's not a random walk 
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Sitara
Minmatar
Solar Flare Trade and Production
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Posted - 2011.04.17 22:00:00 -
[8]
I think the main point is that if you specify the behaviour of a walk into infinity its clearly not a random walk but a deterministic one (as you've said what the values will be all the way into infinity, where's the randomness?)
There is a key difference here, that if you specify a finite length series then it is a possible outcome from a random walk, just as Akita says, the probability of such an outcome reducing as its length increases. If you attempt to extrapolate this into infinity you've crossed the boundary from a possible random walk outcome to a deterministic walk and whatever argument you were using that worked on the finite series can no longer be said to be applying to random walks.
This is where the argument 'what about the series that just goes left then right forever' breaks down - as a truly random walk cannot do this into infinity. (as we've already shown above - in one dimension a true infinite random walk will go through every point an infinite number of times).
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Sitara
Minmatar
Solar Flare Trade and Production
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Posted - 2011.04.18 13:01:00 -
[9]
Originally by: Scorpyn
Since no infinity is bigger than infinity
Actually incorrect. You do get 'orders of infinity' (think density) eg: The set of Irrational numbers is said to possess a higher order of infinity than the set of whole numbers. - they are both infinite sets but as irrational numbers have an infinite number of values between every two whole numbers it is said to have a higher order of infinity.
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Sitara
Minmatar
Solar Flare Trade and Production
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Posted - 2011.04.18 13:13:00 -
[10]
Originally by: Whitehound
randomness.... includes the freedom to be deterministic.
Originally by: Katie Tanaka
You've said all kinds of totally meaningless froth in this thread.......I'm afraid "proof by vigorous assertion and name calling" is not accepted as a legitimate technique in mathematics.
this
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Sitara
Minmatar
Solar Flare Trade and Production
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Posted - 2011.04.18 15:04:00 -
[11]
Originally by: Pr1ncess Alia
A very well laid out statement of the 'un-intuitiveness'
I'm conceptualising it in terms of 'degrees of infinity' : in one or two dimensions the available degrees of infinity in the possible random walks is such that such a walk will visit all points - add a 3rd (and a whole new order of infinity) and its entirely possible for a true (none of this 'what if it just goes left then right' rubbish) random walk to go off on a path that does not cover all points in 3 dimensions yet still be both entirely random and itself infinite.
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Sitara
Minmatar
Solar Flare Trade and Production
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Posted - 2011.04.18 15:23:00 -
[12]
Akita - I'm developing a theory here that Whitehound is CCP's revenge for the 'discussions' on the new forums - a bot designed to post words in an *almost* meaningful manner pitched to keep you busy thinking if you just get the right explanation he'll finally see it
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Sitara
Minmatar
Solar Flare Trade and Production
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Posted - 2011.04.19 10:48:00 -
[13]
Originally by: Whitehound
Edited by: Whitehound on 19/04/2011 06:53:17
Originally by: Scorpyn
Originally by: Whitehound
...I can take each of these infinite, never-returning walks and insert a finite walk into them to make them return. For each infinite, never-returning walk would I get an infinite, returning walk...
That isn't relevant.
Of course it is. There is nothing limiting infinity, because that is what it is - limitless. I can add many more walks to infinity and it stays infinity.
I suggest you study the concepts of cardinality of infinite sets and the aleph numbers :
Pretty good basic explanation
Quote:
Can one possibly say, then, that the number of aleph-one numbers are greater than aleph-null numbers? Not exactly. They are both infinity. One can be considered greater than the other, however, and that is what makes this idea of cardinality of infinite sets so mathematically interesting. The cardinality of infinities does not end here, of course. This is just one example of a conclusion that might be drawn from this idea. It should be able, at the very least, to allow one to realize that perhaps not all infinities are equal, and that mathematicians have their work cut out for them in making sense of these things
Linkage
Linkage
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Sitara
Minmatar
Solar Flare Trade and Production
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Posted - 2011.04.21 18:33:00 -
[14]
Originally by: Whitehound
Do you never get tired of trolling
The irony
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