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Darren New wrote:
> You can't get a forever continuous string of heads from a coin because
> as far as we know, there's no such thing as forever. :-) Fair nuff.
Well, maybe there _is_. I just know of no way of *testing* it.<G>
--
If a pig lost it's voice, would it become disgruntled?
/\ /\ /\ /
/ \/ \ u e e n / \/ a w a z
>>>>>>mue### [at] nawaz org<<<<<<
anl
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Mueen Nawaz wrote:
> Darren New wrote:
>> You can't get a forever continuous string of heads from a coin because
>> as far as we know, there's no such thing as forever. :-) Fair nuff.
>
> Well, maybe there _is_. I just know of no way of *testing* it.<G>
I know you're being funny, but even if there is a "forever", such is
unbounded, not infinite, so it still doesn't help.
It would be better to say "an infinite number of monkeys", because then you
don't have the confusion between unbounded and infinite.
Remember that "the limit as N approaches infinity" was originally designed
to calculate what happens *at* infinity, while avoiding the paradoxes.
It seems kind of silly to say "An infinite number of monkeys will hit upon
Shakespeare" and answer that with "no they won't, because there's no such
thing as an infinite number of monkeys." That's like me saying "If I was in
the WTC on the morning of 9/11/01, I'd be dead", and you answering "No you
wouldn't, because you were on the other side of the country that day."
I don't think anyone's arguing that there actually are or even could be an
infinite number of monkeys. :-)
--
Darren New, San Diego CA, USA (PST)
There's no CD like OCD, there's no CD I knoooow!
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> Nope. Probability theory gives zero.
OK, well I will have to bow down to your superior maths knowledge over mine.
I was just going by what I had read about 1/infinity not being strictly
speaking zero (only under certain circumstances) and the fact that the
infinite series seems to be defined as always equal to 1 in mathematics. I
was not aware of anything specific in probability theory that defined the
probability of choosing 1 item from an infinite number as zero. Writing the
probability as 1/infinity rather than zero explained the "never gets chosen"
paradox you mentioned quite nicely, but oh well...
>> Of course you can discuss how this relates to reality, but I'd rather
>> not get involved in that one, it could an infinitely long time :-)
>
> But that *is* what Warp and I are complaining about.
You should first discuss if there is enough time for an infinite number of
finite length events to ever happen :-) Personally I think you have to
treat this a strictly theoretical concept, otherwise you run into all sorts
of other technicalities about whether it can actually happen.
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The important thing to remember is that (as Warp has been trying to point
out), when the probability tends mathematically to 0 due to infinite
element, it neither *requires* nor *absolutely excludes* something from
happening. 0001020304...9596979899 is just as likely a random number as
7531963568...1739942680, and either *can* happen in an infinite sequence,
but neither *has to*.
--
Tim Cook
http://empyrean.freesitespace.net
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Tim Cook wrote:
> The important thing to remember is that (as Warp has been trying to
> point out), when the probability tends mathematically to 0 due to
> infinite element, it neither *requires* nor *absolutely excludes*
> something from happening.
I understand the assertion. Wikipedia (amongst other sources) disagrees.
> infinite sequence, but neither *has to*.
Do you have a citation for this assertion? Because the wikipedia entry on
Normal_numbers disagrees with you. If you have an equal probability for all
1-symbol elements to appear in an infinite string (i.e., if every symbol
appears an infinite number of times) then you have an equal probability of
every subsequence to show up - i.e., all possible combinations show up an
infinite number of times.
If you have an infinite number of trials and the letter 'a' never shows up,
it means it's impossible for the letter 'a' to show up. The probability a
priori of 'a' showing up is zero, because any non-zero number times infinity
is infinity.
--
Darren New, San Diego CA, USA (PST)
There's no CD like OCD, there's no CD I knoooow!
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"Darren New" <dne### [at] sanrrcom> wrote:
> Do you have a citation for this assertion? Because the wikipedia entry on
> Normal_numbers disagrees with you. If you have an equal probability for
> all 1-symbol elements to appear in an infinite string (i.e., if every
> symbol appears an infinite number of times) then you have an equal
> probability of every subsequence to show up - i.e., all possible
> combinations show up an infinite number of times.
While the probability is equal, it doesn't mean all possible combinations
necessarily show up an infinite number of times in an arbitrary sequence.
The aforementioned 0102030405..9596979899 is technically a possible random
number with each digit 0-9 having an equal probability of occurring. An
infinite sequence could be considered that fails to meet the 'all possible
combinations' feature.
> If you have an infinite number of trials and the letter 'a' never shows
> up, it means it's impossible for the letter 'a' to show up.
Not really. Randomly picking an infinite amount from the set {a,b} *could*
result in nothing but bbb..bbb. That doesn't mean it's impossible for the
letter 'a' to show up, only that the bbb..bbb sequence isn't very
likely...except it's exactly the same probability as any other sequence of
infinite length: 1/infinity.
--
Tim Cook
http://empyrean.freesitespace.net
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Tim Cook wrote:
> While the probability is equal, it doesn't mean all possible
> combinations necessarily show up an infinite number of times in an
> arbitrary sequence.
Why do you say that?
> The aforementioned 0102030405..9596979899 is
> technically a possible random number with each digit 0-9 having an equal
> probability of occurring. An infinite sequence could be considered that
> fails to meet the 'all possible combinations' feature.
I'm not sure I follow wht this example has to do.
>> If you have an infinite number of trials and the letter 'a' never
>> shows up, it means it's impossible for the letter 'a' to show up.
>
> Not really. Randomly picking an infinite amount from the set {a,b}
> *could* result in nothing but bbb..bbb.
Do you have a cite to support this contention?
> That doesn't mean it's
> impossible for the letter 'a' to show up, only that the bbb..bbb
> sequence isn't very likely...except it's exactly the same probability as
> any other sequence of infinite length: 1/infinity.
That's not my understanding of how the math works. Do you have any citation
as evidence for this? Because if the letter 'a' doesn't show up after an
*infinite* number of trials, you clearly don't have any probability for it
to show up at all, and indeed that's what the math pages I've cited already say.
--
Darren New, San Diego CA, USA (PST)
There's no CD like OCD, there's no CD I knoooow!
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"Darren New" <dne### [at] sanrrcom> wrote:
>> While the probability is equal, it doesn't mean all possible combinations
>> necessarily show up an infinite number of times in an arbitrary sequence.
> Why do you say that?
Because 0.50000000... is one possible arbitrary sequence, and nowhere in it
contains the sequence 123.
>>> If you have an infinite number of trials and the letter 'a' never shows
>>> up, it means it's impossible for the letter 'a' to show up.
>> Not really. Randomly picking an infinite amount from the set {a,b}
>> *could* result in nothing but bbb..bbb.
> Do you have a cite to support this contention?
Observe: the number set of decimals from 0 to 1 (inclusive) is infinitely
large, with each element containing an infinite amount of decimal places,
each having an equal probability of being a digit 0-9. This set contains
the number 0.000... QED.
>> That doesn't mean it's impossible for the letter 'a' to show up, only
>> that the bbb..bbb sequence isn't very likely...except it's exactly the
>> same probability as any other sequence of infinite length: 1/infinity.
> That's not my understanding of how the math works. Do you have any
> citation as evidence for this? Because if the letter 'a' doesn't show up
> after an *infinite* number of trials, you clearly don't have any
> probability for it to show up at all, and indeed that's what the math
> pages I've cited already say.
See above. The probability of selecting any particular number is
effectively zero, but that number exists and so *can* be chosen.
--
Tim Cook
http://empyrean.freesitespace.net
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Tim Cook wrote:
> "Darren New" <dne### [at] sanrrcom> wrote:
>>> While the probability is equal, it doesn't mean all possible
>>> combinations necessarily show up an infinite number of times in an
>>> arbitrary sequence.
>> Why do you say that?
>
> Because 0.50000000... is one possible arbitrary sequence, and nowhere in
> it contains the sequence 123.
The probability of a 1 showing up anywhere is zero, so yes, the probability
of 123 showing up anywhere is zero.
> Observe: the number set of decimals from 0 to 1 (inclusive) is
> infinitely large, with each element containing an infinite amount of
> decimal places, each having an equal probability of being a digit 0-9.
Yep.
> This set contains the number 0.000... QED.
And that's why the probability of picking a number like that is zero. :-)
>> That's not my understanding of how the math works. Do you have any
>> citation as evidence for this? Because if the letter 'a' doesn't show
>> up after an *infinite* number of trials, you clearly don't have any
>> probability for it to show up at all, and indeed that's what the math
>> pages I've cited already say.
>
> See above. The probability of selecting any particular number is
> effectively zero, but that number exists and so *can* be chosen.
Sorry, that's not a citation. I.e., I read experts who say you're wrong.
You're trying to convince me using intuitive reasoning that infinity works
in a way the experts say it doesn't work. I see that both arguments are
perfectly reasonable given certain assumptions about how infinity works.
Now, if you find something that explains *why* it's reasonable to pick an
infinite number of random digits and get all zeros, I'll look at it, but
right now we're both just asserting the truth of our own positions. :-)
--
Darren New, San Diego CA, USA (PST)
There's no CD like OCD, there's no CD I knoooow!
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"Darren New" <dne### [at] sanrrcom> wrote:
> Now, if you find something that explains *why* it's reasonable to pick an
> infinite number of random digits and get all zeros, I'll look at it, but
> right now we're both just asserting the truth of our own positions. :-)
I never said it was *reasonable* to get all zeroes, just that it's
technically possible. ;)
Besides, there's a third option: we're both right.
--
Tim Cook
http://empyrean.freesitespace.net
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