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Mueen Nawaz wrote:
> Darren New wrote:
>> What is unintuitive to me is that if you draw a numberline on the wall
>> and toss a dart at it (figuratively speaking), your probability of
>> hitting a rational number is zero. That is, there are so many more
>> reals than rationals that the chance of picking a real that's rational
>> at random is literally zero. It would seem there's *some* epsilon
>> chance, but apparently not. :-)
>
> Well, your probability of hitting a given real number is also 0.
> Same amount of weirdness.
Yes. But I meant my probability of hitting *any* rational number is zero.
> Advanced probability theory involves measure theory - which I have
> yet to study properly. My guess is what you're saying holds true because
> the set of rationals is a set of measure 0 w.r.t. to the measure they
> use in probability.
I think that was the technical term I saw, yes. Just seemed strange. Not
wrong, but strange. :-)
--
Darren New / San Diego, CA, USA (PST)
Helpful housekeeping hints:
Check your feather pillows for holes
before putting them in the washing machine.
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Darren New <dne### [at] san rrcom> wrote:
> > Well, your probability of hitting a given real number is also 0.
> > Same amount of weirdness.
> Yes. But I meant my probability of hitting *any* rational number is zero.
I think that this confuses people because they think that if the
probability for something is 0, that means that it's *impossible* for
that something to happen.
This is indeed so in the discrete case. However, in this case the
probability is 0 because the total probability of 1 has been divided
among an *infinite* amount of numbers. Thus, mathematically, the resulting
probability of hitting a given number is 0, as 1/infinite = 0.
*After* you have thrown the dart, the probability to hit the number
the dart did hit grows to 1 (or 100%).
Or if we say the same thing in another way: Even though there is an
infinite amount of numbers, that doesn't mean you can't choose one.
(I really wonder if this has any relation whatsoever to the so-called
axiom of choice.)
--
- Warp
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Warp wrote:
> *After* you have thrown the dart, the probability to hit the number
> the dart did hit grows to 1 (or 100%).
But is the probability that it hit any rational number (vs any
irrational number) non-zero at that point? If you do it countably many
times, will you *ever* hit *any* rational number?
> (I really wonder if this has any relation whatsoever to the so-called
> axiom of choice.)
That's pretty much exactly what the axiom of choice is. Given an
infinite set of sets, can you create a set by picking one element from
each subset?
--
Darren New / San Diego, CA, USA (PST)
Helpful housekeeping hints:
Check your feather pillows for holes
before putting them in the washing machine.
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"Warp" <war### [at] tagpovrayorg> wrote in message
news:4877ba85@news.povray.org...
> It kind of "makes sense" that there are "more" reals than integers.
> What is more unintuitive is that the amount of rational numbers is the
> same as the amount of integers.
And now the really fun part: Is there a subset of real numbers that is "more
numerous" than intergers or rationals but less so than reals?
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somebody wrote:
> And now the really fun part: Is there a subset of real numbers that is "more
> numerous" than intergers or rationals but less so than reals?
That's actually a long-standing problem, but I remember hearing that
someone had figured out the answer must be "No." I believe they were
waiting on others verifying the proof. (Which, as you can imagine, must
be pretty hairy.)
--
Darren New / San Diego, CA, USA (PST)
Helpful housekeeping hints:
Check your feather pillows for holes
before putting them in the washing machine.
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somebody wrote:
> And now the really fun part: Is there a subset of real numbers that is "more
> numerous" than intergers or rationals but less so than reals?
I bet even Gail didn't know that Oracle has solved *that* problem.
http://laurentschneider.com/wordpress/2007/10/what-is-bigger-than-infinity.html
(Check out the comments, too.)
--
Darren New / San Diego, CA, USA (PST)
Helpful housekeeping hints:
Check your feather pillows for holes
before putting them in the washing machine.
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somebody wrote:
> And now the really fun part: Is there a subset of real numbers that is "more
> numerous" than intergers or rationals but less so than reals?
Apparently, in most maths, it's neither true nor false, but an axiom to
be adopted.
http://en.wikipedia.org/wiki/Continuum_hypothesis
(That's the *actual* answer to the question, methinks.)
--
Darren New / San Diego, CA, USA (PST)
Helpful housekeeping hints:
Check your feather pillows for holes
before putting them in the washing machine.
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Darren New wrote:
>> Well, your probability of hitting a given real number is also 0.
>> Same amount of weirdness.
>
> Yes. But I meant my probability of hitting *any* rational number is zero.
Because both are sets of measure 0 here. That's why it's the same
amount of weirdness.
--
Cut my pizza in six slices, please; I can't eat eight.
/\ /\ /\ /
/ \/ \ u e e n / \/ a w a z
>>>>>>mue### [at] nawazorg<<<<<<
anl
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Warp wrote:
> I think that this confuses people because they think that if the
> probability for something is 0, that means that it's *impossible* for
> that something to happen.
>
> This is indeed so in the discrete case. However, in this case the
> probability is 0 because the total probability of 1 has been divided
> among an *infinite* amount of numbers. Thus, mathematically, the resulting
> probability of hitting a given number is 0, as 1/infinite = 0.
Maybe so, but it's not as compelling as a formal proof. In math, you
can't divide by infinity.
I'm not saying it's "wrong", and your message addresses the dilemma in
another way. The question ultimately is the interpretation of
probability. I know the mathematical definition, but how do I give it
meaning in the real world?
--
Cut my pizza in six slices, please; I can't eat eight.
/\ /\ /\ /
/ \/ \ u e e n / \/ a w a z
>>>>>>mue### [at] nawazorg<<<<<<
anl
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Darren New wrote:
> somebody wrote:
>> And now the really fun part: Is there a subset of real numbers that is
>> "more
>> numerous" than intergers or rationals but less so than reals?
>
> Apparently, in most maths, it's neither true nor false, but an axiom to
> be adopted.
>
> http://en.wikipedia.org/wiki/Continuum_hypothesis
Yes. Haven't studied it, but I remember that the answer was either
"No", or "Can't be decided." Your message makes me think it's the latter.
--
Cut my pizza in six slices, please; I can't eat eight.
/\ /\ /\ /
/ \/ \ u e e n / \/ a w a z
>>>>>>mue### [at] nawazorg<<<<<<
anl
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