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Mueen Nawaz <m.n### [at] ieeeorg> wrote:
> http://en.wikipedia.org/wiki/Almost_surely
Seems like my claims about the monkeys and Shakespeare were not, after all,
completely ridiculous.
--
- Warp
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"Mueen Nawaz" <m.n### [at] ieeeorg> wrote in:
> http://en.wikipedia.org/wiki/Almost_surely
So, a probability of 1 doesn't mean 'always', and a probability of 0 doesn't
mean 'never' when there's infinity involved. Was that so hard?
--
Tim Cook
http://empyrean.freesitespace.net
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Warp wrote:
> Mueen Nawaz <m.n### [at] ieeeorg> wrote:
>> http://en.wikipedia.org/wiki/Almost_surely
>
> Seems like my claims about the monkeys and Shakespeare were not, after all,
> completely ridiculous.
I never thought they were completely ridiculous. Just unconvincing. :-)
It's still up in the air for me as to who is right. :-)
--
Darren New, San Diego CA, USA (PST)
There's no CD like OCD, there's no CD I knoooow!
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From: Nicolas Alvarez
Subject: Re: Infinite sequences and probability
Date: 28 Apr 2009 13:22:43
Message: <49f73b62@news.povray.org>
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Mueen Nawaz wrote:
> Let's say that instead of a 2 sided coin, I have a three state
> random generator, which generates each state with equal probability.
>
> Now I had already shown that the likelihood of getting a
> _particular_ infinite sequence was the same as getting a _particular_
> value between 0 and 1. (That is, the likelihood is zero - I'm arguing
> purely mathematically - let's put the "real" world aside).
>
> Intuitively, that's because a point is infinitely smaller than
> the interval [0,1].
>
> In fact, if I were to ask the probability of getting a rational
> number, the answer is still 0. While the set of rationals is infinite,
> it's still infinitely smaller than the unit interval - the latter is of
> a higher infinity.
I think this is relevant:
http://en.wikipedia.org/wiki/Almost_surely
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From: Nicolas Alvarez
Subject: Re: Infinite sequences and probability
Date: 28 Apr 2009 13:25:00
Message: <49f73beb@news.povray.org>
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Nicolas Alvarez wrote:
> I think this is relevant:
> http://en.wikipedia.org/wiki/Almost_surely
And I should learn to read the rest of the thread before posting a reply.
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John VanSickle wrote:
> Mueen Nawaz wrote:
>
>> It turns out this set has "measure" 0. And any integral of a
>> function (the uniform distribution in this case) over a set of measure
>> zero is 0.
>
> I understand that the Dirac delta function,
>
> http://en.wikipedia.org/wiki/Dirac_delta_function
>
> is an exception to this.
Well, it's not exactly a function.
--
Every hard drive I've ever bought has been larger than all my previous
hard drives combined. And this is without even trying.
--Seen on Slashdot.org
/\ /\ /\ /
/ \/ \ u e e n / \/ a w a z
>>>>>>mue### [at] nawazorg<<<<<<
anl
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Mueen Nawaz wrote:
> Put another way, I showed that the even if I allow some sequences with
> a single 1 to sneak in, the probability is still 0.
Oh yeah, I certainly wasn't debating the accuracy of the result. It
appears that you already considered this detail anyway, so my
nit-picking was entirely unnecessary.
I did like the Cantor set analogy by the way. I didn't forsee it when I
was reading your post and was delighted when you pointed it out.
>
>> I think represent the same number on the real line, but one would be in
>> the set and one wouldn't. Nevertheless I do think that the Cantor set
>
> One would be in the set of sequences we're interested in, and the other
> wouldn't. In terms of the Cantor set, they're both in it as they are the
> same number.
Yeah, that's what I meant.
>> lim_{n->inf} a^n = 0 if 0 <= a < 1
>
> (Incidentally, I'm not sure I understand your limit. What is 'a' and
> why is it fixed?
For base m a would be (m-1)/m -- the probability that a single given
digit of the random sequence doesn't contain a 1 in base m.
>> 1 = sum_{0:inf} p
>>
>> Since no real number has this property, p cannot exist. Thus there is
>> no probability (zero or otherwise) that a random sequence is in S. Said
>> in more standard terminology: S has no measure.
>
> I didn't read the whole thing - xors make my head hurt. But I got the
> general idea, having seen similar arguments before. When I was learning
> integration theory using measures, I noticed that integrals were always
> defined over _measurable_ sets.
In that case I'll spare you the trouble of being tempted to read it
later. I just adapted the construction of a Vitali set from the real
line to infinite binary sequences by replacing addition with xor. A few
small things changed as a result, but nothing major.
I'm enveious you had an integration theory class that got into measure
theory. I don't recall such a course being available back when I was
taking math classes, but it would have been pretty fun had there been.
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From: Kevin Wampler
Subject: Re: Infinite sequences and probability
Date: 28 Apr 2009 14:46:14
Message: <49f74ef6@news.povray.org>
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Invisible wrote:
> (For example, if you have a group of size X, it has subgroups of every
> size that is a factor of X - including 1 and X itself.
I assume by "group" you actually mean to say "cyclic group"? If not
that might explain why it's proving hard to understand.
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>> (For example, if you have a group of size X, it has subgroups of every
>> size that is a factor of X - including 1 and X itself.
>
> I assume by "group" you actually mean to say "cyclic group"? If not
> that might explain why it's proving hard to understand.
I was under the impression the result applies to all groups - which is
why the groups of prime order are always cyclic.
--
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*
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Nicolas Alvarez wrote:
> Nicolas Alvarez wrote:
>> I think this is relevant:
>> http://en.wikipedia.org/wiki/Almost_surely
>
> And I should learn to read the rest of the thread before posting a reply.
That is almost surely correct. ;-)
--
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*
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