Warp wrote:
>   One monkey and an infinite amount of time is closer, but still not a
> guarantee.
> 
>   A true evenly-distributed random number generator and an infinite amount
> of time is a lot, lot closer to fulfilling the claim, and the probability
> of the works coming up is unlimitedly high, but there's still no absolute
> guarantee.

	Heh heh. I felt this would create an argument.

	First, don't say "unlimitedly high" The upper limit is 1. ;-) I think
you mean that it gets arbitrarily close to 1, but only hits 1 in the
infinite limit.

	Personally, I've always had two problems with probability:

	1) Given a continuous distribution, the probability of each individual
point is 0. I mean, if someone asked me to come up with a number between
0 and 1, and I said pi/4, then there was "some" chance that I'd pick it,
right? If it were 0, then I couldn't have said that.

	2) The monkey problem is isomorphic to this one: If I have an unbiased
coin (and assume the outcome _is_ random, and not related to chaos -
such as the force I hit it with, etc), and I keep flipping it, is it
possible that I can get a continuous string of heads indefinitely?

	Intuition tells me yes. I just can't see any physical reason why I
_have_ to get a tails at some point.

	Yet _every_ single person - mathematician, probabilist, or otherwise
says the answer is no because the probability of such an event happening
is 0 in the limit to infinity. If you do 1 minus that, you get  that the
probability of a tails appearing at _some_ point is 1. In fact, you're
the _first_ person I've seen who has the same dilemma with this problem
as I do.

	They're mathematics, as far as probability theory goes, is valid.

	The question is then, "Is their theory valid with respect to this
universe?"

	For whatever reason, I've always viewed math to be "independent" of the
universe, but probability and statistics should conform to it. In a
sense, I view statistics to be more like a "science" than math. The
reason is that whenever someone comes to me with a probability scenario
and his result is different from mine, I always appeal to a computer
simulation as the ultimate judge (ignoring that the RNG is not random
enough...). I don't care how valid his logic may sound, but if the
computer (or the world) gives a different answer, and if my simulation
is sound, then he's wrong. The ultimate criterion is experiment - not
logic. Just as in science.

	That criterion doesn't work here. I can't simulate anything to infinity.

	I think at the end of the day it's a philosophical question. One of
those things that whichever stance you take won't affect anything in the
real world. I very strongly suspect, though, that probability theory
states that given infinite time, a single monkey on a single keyboard
hitting the keys with, say, uniform randomness will somewhere in that
string produce the works of Shakespeare. The real question is how you
interpret "will".

	Oh, and BTW, I was discussing this very same "coin" problem just a few
days ago. And I realized that in fact, problems 1) and 2) are identical.
It's quite simple to see:

Let 0 represent heads, and 1 represent tails. Flip a coin indefinitely,
and write the sequence in order:

001011100011

	What you get is essentially a binary expansion of a number between 0
and 1, inclusive. It's a 1-1 correspondence. Thus, the probability of
getting _any_ fixed infinite sequence (be it all heads or HTHTHTHTHT...
or whatever) is the probability of getting its representation as a
number in a uniformly continuous distribution from 0 to 1 - which is 0.

	So in summary, if the whole monkey thing bothers you, then the fact
that probabilities of points in continuous distributions being zero
should bother you as well.

	

-- 
"Now we all know map companies hire guys who specialize in making map
folding a physical impossibility" - Adult Kevin Arnold in "Wonder Years"


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