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thebruce
Dances With Wikis

Joined: 16 Aug 2004
Posts: 6865
Location: Kitchener, Ontario

if there's an infinite number, there's an equal chance of a sequence definitely occurring as there is with it not possibly occurring at all... it's like division by 0... hehe

well, not exactly like division by 0, but as strange a concept to grasp...
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Posted: Sun Dec 19, 2004 7:54 pm
Olorin
Unfettered

Joined: 04 Nov 2004
Posts: 612
Location: Gainesville, FL

 tanner wrote: i take the first 3 out in one minute, the second 3 takes half minute the third in a quarter minute etc --- ill have em all done in 2 minutes But as has been said --- back to topic

Yeah,
back to the topic:
what did the number 3 did wrong to be left out?
F.O.R.

Posted: Sun Dec 19, 2004 8:10 pm
neon snake
Veteran

Joined: 18 Mar 2004
Posts: 70
Location: Chelmsford, UK

I was actually being relevant when I started this - as in, if you look at an infinite string, you'll always see what you are looking for at some point, so it's prudent to avoid wishful thinking!

thebruce - tentatively, and waiting to be proved wrong, I disagree. Over a long enough timeline, and barring removal of naughty number 3, all possible combinations must occur. Somewhere in my maths textbooks from a decade or so ago, I have the mathematical proof. I think.
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Posted: Mon Dec 20, 2004 2:15 am
Olorin
Unfettered

Joined: 04 Nov 2004
Posts: 612
Location: Gainesville, FL

 neon snake wrote: I was actually being relevant when I started this - as in, if you look at an infinite string, you'll always see what you are looking for at some point, so it's prudent to avoid wishful thinking! thebruce - tentatively, and waiting to be proved wrong, I disagree. Over a long enough timeline, and barring removal of naughty number 3, all possible combinations must occur. Somewhere in my maths textbooks from a decade or so ago, I have the mathematical proof. I think.

I think that over a long enough timeline all combinations have an equal chance of appearing.

As far as proving that all combinations *will* appear at some point,
I've never seen a proof. If you have one, I bet it uses the fact that e, pi, and such numbers are aperiodic.

F.O.R.

Posted: Mon Dec 20, 2004 6:39 am
SilentAvenger
Boot

Joined: 23 Oct 2004
Posts: 44

Assuming really each digit has an equal chance to appear, (say, not the infinite-digit number 2/3, which nothing but 6's), the chance of an exact string of digits with length N to appear, is 1/10^N.

This means that for a number with K digits, there is a 1/10^N chance for each substring of size N to be equal to the string we are trying to match, so, K-N+1 equalling the number of N sized substrings, there is chance of (K-N+1)/10^N to find a string of length N in a string of length K (actually, its slightly less, but in general this is it).

So, assuming K is infinite and N finite, this will come out as infinite, that means, you will find you string repeated infinite times, not only once

A common way to show this, is to picture a monkey on a typewriter, typing off stuff for all times. It is obvious (assuming he types letters randomly, and that the typwriter doesnt jam, as monkeys dont know how to fix jammed typewriters) that at one point he must type the full works of Shakespear, one after another, in alphabetical order.

*turns off mathmode*

Posted: Mon Dec 20, 2004 8:00 am
tanner
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Joined: 21 May 2003
Posts: 875
Location: (x,y,z,t,i, ...)+

 Quote: Assuming really each digit has an equal chance to appear

that wasnt mentioned in the original post which is why i pointed out that infinity does not necessarily contain all.
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Posted: Mon Dec 20, 2004 10:11 am
thebruce
Dances With Wikis

Joined: 16 Aug 2004
Posts: 6865
Location: Kitchener, Ontario

 SilentAvenger wrote: Assuming really each digit has an equal chance to appear, (say, not the infinite-digit number 2/3, which nothing but 6's), the chance of an exact string of digits with length N to appear, is 1/10^N. This means that for a number with K digits, there is a 1/10^N chance for each substring of size N to be equal to the string we are trying to match, so, K-N+1 equalling the number of N sized substrings, there is chance of (K-N+1)/10^N to find a string of length N in a string of length K (actually, its slightly less, but in general this is it). So, assuming K is infinite and N finite, this will come out as infinite, that means, you will find you string repeated infinite times, not only once A common way to show this, is to picture a monkey on a typewriter, typing off stuff for all times. It is obvious (assuming he types letters randomly, and that the typwriter doesnt jam, as monkeys dont know how to fix jammed typewriters) that at one point he must type the full works of Shakespear, one after another, in alphabetical order. *turns off mathmode*

However, correct me if I'm wrong, I still believe that there's an equal chance that a sequence will appear as there is it will never appear... theoretically, given a sequence of random numbers, and in infinite length of random numbers, I believe the chance would become a matter of approaching 100%, or approaching 0%... in other words, depending on how you look at it, there is still always going to be a chance that any specific sequence may never appear. just as there will never be a 100% certainty that it will appear... so technically, in dealing with infinite chances, math is pointless
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Posted: Mon Dec 20, 2004 10:14 am
thebruce
Dances With Wikis

Joined: 16 Aug 2004
Posts: 6865
Location: Kitchener, Ontario

I'll also add that the chance of e being a non-repeating infinite number in essence also becomes a matter of approaching 0%... but it will never be 0, because that would require a finite number to calculate the bounds... in essence, we'll never know if e is an infinite non-repeating number, if it is in fact just that.
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Posted: Mon Dec 20, 2004 10:18 am
tanner
Entrenched

Joined: 21 May 2003
Posts: 875
Location: (x,y,z,t,i, ...)+

if it were repeating it would be a rational number --- e is not only irrational but transcendental

see

for proof
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Posted: Mon Dec 20, 2004 10:25 am
darkblaze
Boot

Joined: 22 Oct 2004
Posts: 63
Location: US, North Carolina

all these numbers make me think about phi (its my favorite number!) theres an article in the guardian about it too but that was almost 2 years ago..
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Posted: Mon Dec 20, 2004 12:53 pm
am520dj
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Joined: 18 Oct 2004
Posts: 135
Location: United States

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Posted: Mon Dec 20, 2004 4:20 pm
neon snake
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Joined: 18 Mar 2004
Posts: 70
Location: Chelmsford, UK

 Quote: If you have one, I bet it uses the fact that e, pi, and such numbers are aperiodic.

Yep, exactly that.

 Quote: Assuming really each digit has an equal chance to appear

As long as each digit has a chance, regardless of how small the chance is, it still works - even if its one in a million, well, infinity is a hell of lot larger than that!
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Posted: Tue Dec 21, 2004 3:49 am
Shish
Boot

Joined: 24 Nov 2004
Posts: 51

I'm bored. Have some cents.

o) Any infinite random number will contain all finite number series
o) Any infinite non-repeating number will contain all finite number series
o) It's possible to have a non-repeating infinite number with a pattern like 123112233111222333 which will never contain the sequence "13"
o) It's possible for a random number generator to come up with the above pattern
o) perhaps the first two statements work if we also specify that it must be a patternless, unpredictable random number? Eg you can predict that the number after 1 will not be 3, and for it to be patternelss and unpredictable you must be wrong at some time.

One thinks this boils down to infinite force vs infinite wall: for a number to be random it must be unpredictable, and yet a RNG can come up with any series of numbers, including one with a pattern that has no 13. Using the first two unprovable statements, however, we can argue that for it to be random and infinite, it must at some point contain 13:

The chance of getting a one then a three is 1/9 * 1/9; 1/81. Given infinite chances and you get inf/81, aka infinite chance of it happening.

So which wins? the "all numbers must appear at some point" or the "if you can successfully predict that it'll happen, it isn't random"? Or, like the infinite force, are we thinking outside all the boxes, including that of reality?
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Posted: Tue Dec 21, 2004 7:50 am
thebruce
Dances With Wikis

Joined: 16 Aug 2004
Posts: 6865
Location: Kitchener, Ontario

I still choose to believe that if a chance ever approaches 100% (or 0%) but never reaches it, then you may as well throw the argument out the window... if there's never a definite chance a sequence will appear, then essentially, there's always a chance it won't appear, so as you march on towards infinity, there's always a chance it will never appear just as much as there's a chance it will appear... infinite chances aren't arguable, afaic

with a limit to the length of a random number, you can, and never will, know whether or not a sequence will appear or not. "Until you find the sequence" already assumes it will exist. If you never find it, it doesn't mean it will never appear, but it doesn't mean it will. So I agree, as soon as you say a sequence has a specific chance to appear, you claim to either know that the number is repeating (and have therefore uncovered the entire set of numbers) or know that it's not an infinite number - ie, you've reached the bounds in order to calculate a specific chance. An infinite number, by default has no bounds, therefore a chance can't be mathematically calculated... a theoretica chance could (whatever the term may be), but it's theoretical... like 'as x approaches 10' - for all intents and purposes, we know x will never pass 10 and it will pass all decimal values up to 10, but we know x will never actually be 10... making 10 the theoretical maximum of x, even though x will never be 10...

math & infinity is a strange thing
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Posted: Tue Dec 21, 2004 8:43 am
tanner
Entrenched

Joined: 21 May 2003
Posts: 875
Location: (x,y,z,t,i, ...)+

 Quote: I still choose to believe that if a chance ever approaches 100% (or 0%) but never reaches it

hmmm

99.999999... approaches 100 but never reaches it you say? (as an example)

well if:
x = 99.999999... (approaching 100)
10x = 999.999999...
10x - x = 900.000000...
10x - x = 900
x = 100

x has reached number 100

.but maybe this should be persued elsewhere --- OOG
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Posted: Tue Dec 21, 2004 9:57 am
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