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Invisible wrote:
> http://news.bbc.co.uk/1/3013959.stm
"...using it as a lavatory..."
Now if they aimed at the keyboard, there's _some_ hope!
--
"Now we all know map companies hire guys who specialize in making map
folding a physical impossibility" - Adult Kevin Arnold in "Wonder Years"
/\ /\ /\ /
/ \/ \ u e e n / \/ a w a z
>>>>>>mue### [at] nawaz org<<<<<<
anl
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Darren New wrote:
> Given that truly random sequences are normal, and in a normal sequence
> every block of a particular length occurs with equal probability, and
> we're talking an infinite sequence, it follows that the bard is in there
> somewhere. If I'm not mistaken about the math of it.
What's a "truly random" sequence?
I think I know what you mean, because I used to think the same, but
that's just my bias - I don't think there's any mathematical backing. Or
rather, the "truly random" is just yet another distribution like the
Gaussian, etc.
(And be careful when you say Normal. The distribution called "normal"
in probability is the Gaussian distribution, which I think is not what
you meant).
--
"Now we all know map companies hire guys who specialize in making map
folding a physical impossibility" - Adult Kevin Arnold in "Wonder Years"
/\ /\ /\ /
/ \/ \ u e e n / \/ a w a z
>>>>>>mue### [at] nawazorg<<<<<<
anl
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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"
/\ /\ /\ /
/ \/ \ u e e n / \/ a w a z
>>>>>>mue### [at] nawazorg<<<<<<
anl
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Warp wrote:
> Since there is an infinite amount of different finite sequences of
> letters, the probability of one specific sequence (in this case the works
> of Shakespeare) to appear is, mathematically speaking, zero.
If you _specify_ exactly some sequence containing Shakespeare's works,
then yes - it is 0 to get that particular sequence.
However, there are _infinitely_ many infinite sequences that contain
the works of Shakespeare...
--
"Now we all know map companies hire guys who specialize in making map
folding a physical impossibility" - Adult Kevin Arnold in "Wonder Years"
/\ /\ /\ /
/ \/ \ u e e n / \/ a w a z
>>>>>>mue### [at] nawazorg<<<<<<
anl
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Mueen Nawaz <m.n### [at] ieeeorg> wrote:
> Invisible wrote:
> > http://news.bbc.co.uk/1/3013959.stm
>
> "...using it as a lavatory..."
>
> Now if they aimed at the keyboard, there's _some_ hope!
Only if they have very dense number twos.
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Mueen Nawaz wrote:
> What's a "truly random" sequence?
This is a well-defined concept. It means that (basically) the probability
of you being able to predict the next item in the sequence is unchanged by
any knowledge you might have.
> I think I know what you mean, because I used to think the same, but
> that's just my bias - I don't think there's any mathematical backing.
That's what I'm saying - I think there is.
> rather, the "truly random" is just yet another distribution like the
> Gaussian, etc.
No, you can have any distribution you want and not be random. You can have
any distribution you want and be random, too. Random is about predictability
with better-than-the-distribution-suggests accuracy.
If I say "the first two characters my generator output were 'th'", you'd
have no idea what the third letter is if it's random, but a pretty good idea
what it might be if I said "it's an english sentence." That sort of
predictability.
No matter what you know, and no matter how often you've rolled the dice,
you're not going to guess the next side with >1/6 probability (assuming
perfect dice, of course). Any computer PRNG dice rolling can be easily
predicted simply by looking at the PRNG and doing the math.
> (And be careful when you say Normal. The distribution called "normal"
> in probability is the Gaussian distribution, which I think is not what
> you meant).
I meant what are called "normal numbers" in math, not normal statistical
distributions. Check wikipedia.
--
Darren New, San Diego CA, USA (PST)
There's no CD like OCD, there's no CD I knoooow!
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Mueen Nawaz <m.n### [at] ieeeorg> wrote:
> 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.
I'd say that the probability being 1 with infinite throws is exactly as
meaningful as the probability of getting a certain value of a continuous
range being 0. Yes, it's mathematically zero, but that doesn't mean that
the value will never be chosen (if it meant that, it would mean that no
value would *ever* be chosen because all the individual values have a
probability of zero).
Likewise the probability of getting tails being 1 with infinite throws
doesn't really tell us that tails must appear at some point. After all,
you *can't* throw a coin an infinite amount of times, and even the idea
has no correspondence to any physical event. It's a purely mathematical
construct. What the 1 tells as is that it's the upper limit for the
probability.
I'd think about it this way: No matter how many times you throw, there's
*always* a non-zero probability that it will be all heads. At no point is
tails *forced* to appear by any law of nature or mathematics. (This is
because past tosses do not affect future ones.)
--
- Warp
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Darren New wrote:
> Mueen Nawaz wrote:
>> What's a "truly random" sequence?
>
> This is a well-defined concept. It means that (basically) the
> probability of you being able to predict the next item in the sequence
> is unchanged by any knowledge you might have.
OK. Not how I was interpreting it...
>> (And be careful when you say Normal. The distribution called "normal"
>> in probability is the Gaussian distribution, which I think is not what
>> you meant).
>
> I meant what are called "normal numbers" in math, not normal statistical
> distributions. Check wikipedia.
I know - just thought I'd point out it could be confusing.
Anyway, your original assertion was that a truly random sequence is
necessarily a normal sequence. That doesn't match up with your
definition of normal.
If I have a sequence derived from a Gaussian distribution, then it is
truly random by the way you defined it. Given the whole sequence, up to
a point, it doesn't tell me anything about what the next element could
be (other than what is obvious - that it follows a Gaussian
distribution). However, for this sequence "every block of a particular
length occurs with equal probability" does not hold.
--
"Now we all know map companies hire guys who specialize in making map
folding a physical impossibility" - Adult Kevin Arnold in "Wonder Years"
/\ /\ /\ /
/ \/ \ u e e n / \/ a w a z
>>>>>>mue### [at] nawazorg<<<<<<
anl
Post a reply to this message
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Warp wrote:
> I'd say that the probability being 1 with infinite throws is exactly as
> meaningful as the probability of getting a certain value of a continuous
> range being 0. Yes, it's mathematically zero, but that doesn't mean that
Which was my whole long winded point. The monkey problem is identical
to this one.
> the value will never be chosen (if it meant that, it would mean that no
> value would *ever* be chosen because all the individual values have a
> probability of zero).
In classical probability, that's kind of the case. Perhaps it's better
to say that the model is only valid if you're talking of an interval or
collection of intervals (my interpretation - not necessarily shared by
probabilists). I was pointing out that if it's flawed to use this model
to calculate the probability of getting a single point, it's equally
flawed in trying to calculate the probability of getting all heads, or
of monkeys producing the works of Shakespeare.
--
"Now we all know map companies hire guys who specialize in making map
folding a physical impossibility" - Adult Kevin Arnold in "Wonder Years"
/\ /\ /\ /
/ \/ \ u e e n / \/ a w a z
>>>>>>mue### [at] nawazorg<<<<<<
anl
Post a reply to this message
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Warp wrote:
> I'd think about it this way: No matter how many times you throw, there's
> *always* a non-zero probability that it will be all heads. At no point is
> tails *forced* to appear by any law of nature or mathematics. (This is
> because past tosses do not affect future ones.)
That's the difference between "unbounded" and "infinite". You're thinking
"no matter how big it gets..." but "infinite" doesn't mean that. Unbounded
means that.
Can you write a turing machine that uses up an infinite amount of space on
the tape? No, only an unbounded amount. Can you have a countably infinite
set? Yes. Hence, infinity is not computable by a turing machine.
As long as you keep trying to do the math in terms of "you compute
successive values", you'll not get the answer.
--
Darren New, San Diego CA, USA (PST)
There's no CD like OCD, there's no CD I knoooow!
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