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http://news.bbc.co.uk/1/3013959.stm
WTF-O-Meter: 2.8
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Invisible <voi### [at] dev null> wrote:
> http://news.bbc.co.uk/1/3013959.stm
>
> WTF-O-Meter: 2.8
Ha!
"The work was interesting but had little scientific value, except to show that
the 'infinite monkey' theory is flawed."
I love it. Flawed, as in he thinks nobody had ever noticed that an infinite
number of monkeys would be difficult to arrange? Or does he really think that
infinity can be approximated by eight? And whoever said it was a theory in the
first place? :-D
Although to be honest, this wtf is probably as much lazy reporting by the BBC as
anything said or done by those supposedly involved...
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Bill Pragnell wrote:
> Ha!
>
> "The work was interesting but had little scientific value, except to show that
> the 'infinite monkey' theory is flawed."
>
> I love it. Flawed, as in he thinks nobody had ever noticed that an infinite
> number of monkeys would be difficult to arrange? Or does he really think that
> infinity can be approximated by eight? And whoever said it was a theory in the
> first place? :-D
Well, let's face it, the theory really states that the probability of a
string of random text containing the works of Shakespear approaches
unity as the length of the string approaches infinity. Which actually
has nothing to do with monkeys, really. Monkeys don't type statistically
uncorrelated letters. They bash keys. The text thus far produced is
anything but statistically random!
> Although to be honest, this wtf is probably as much lazy reporting by the BBC as
> anything said or done by those supposedly involved...
My personal favourit is the quote about this being cheaper and more
entertaining than reality TV. ;-)
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Bill Pragnell <bil### [at] hotmailcom> wrote:
> I love it. Flawed, as in he thinks nobody had ever noticed that an infinite
> number of monkeys would be difficult to arrange? Or does he really think that
> infinity can be approximated by eight? And whoever said it was a theory in the
> first place? :-D
Actually there's no law of mathematics or physics which says that an
infinite number of monkeys will produce the works of Shakespeare (especially
given that the problem is incomplete: The amount of time is unspecified).
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.
(Many people think that in this last case the works *will* eventually
appear with absolute certainty, but that's just the gambler's fallacy.)
--
- Warp
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Invisible <voi### [at] devnull> wrote:
> They bash keys.
If you're lucky. They're just as likely to sit on it, or try to mate with it.
:-)
> My personal favourit is the quote about this being cheaper and more
> entertaining than reality TV. ;-)
Cheaper? Certainly. More entertaining? Well, is there really any fundamental
difference between them? ;-)
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Warp <war### [at] tagpovrayorg> wrote:
> Bill Pragnell <bil### [at] hotmailcom> wrote:
> > And whoever said it was a theory in the
> > first place? :-D
>
> Actually there's no law of mathematics or physics which says that an
> infinite number of monkeys will produce the works of Shakespeare (especially
> given that the problem is incomplete: The amount of time is unspecified).
That's what I was getting at... it's not really a theory, more a
pop-thought-experiment. And a badly-formulated one, as you've pointed out.
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Also there was only one computer. The original scenario had a 1:1
monkey-typewriter ratio.
--
Tim Cook
http://empyrean.freesitespace.net
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>> My personal favourit is the quote about this being cheaper and more
>> entertaining than reality TV. ;-)
>
> Cheaper? Certainly. More entertaining? Well, is there really any fundamental
> difference between them? ;-)
...which is kind of my point...
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>>> And whoever said it was a theory in the
>>> first place? :-D
>> Actually there's no law of mathematics or physics which says that an
>> infinite number of monkeys will produce the works of Shakespeare (especially
>> given that the problem is incomplete: The amount of time is unspecified).
>
> That's what I was getting at... it's not really a theory, more a
> pop-thought-experiment. And a badly-formulated one, as you've pointed out.
More like, it's a precise mathematical statement, rendered casually as
"if you gave a monkey infinite time to randomly type stuff"... which is
rather less precise. And quite probably wrong in a number of ways. (For
starters, monkeys don't type statistically random text...)
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Tim Cook <z99### [at] gmailcom> wrote:
> Also there was only one computer. The original scenario had a 1:1
> monkey-typewriter ratio.
What is the "original scenario"?
As far as I know, the compeltely original statement is: "One million
monkeys hammering on one million typewriters for one million years will
eventually produce the entire works of Shakespeare."
Of course that statement is patently false in all possible interpretations.
Later the falsity was attempted to be fixed by changing the "one million
years" to "an infinite amount of time".
Of course if you change it to an infinite amount of time, then the one
million number becomes superfluous. One monkey and one typewriter would
essentially say the exact same thing.
The spirit of the metaphor is, of course, that the "monkey" is really
a true evenly-distributed random number generator. While in this situation
the probability of the works popping up approaches 1, and the works will
thus almost surely pop up at some point, it's still not a guarantee. The
probability *approaches* 1, it never becomes it.
--
- Warp
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Warp wrote:
> probability *approaches* 1, it never becomes it.
I think if you mean "the probability as the time approaches infinity" then
yes. I think if you look at the actual result with infinite time provided,
you get the probability 1. It's not like you have different probabilities;
there's only one situation under consideration: an infinite amount of time.
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.
--
Darren New, San Diego CA, USA (PST)
There's no CD like OCD, there's no CD I knoooow!
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Invisible escreveu:
> http://news.bbc.co.uk/1/3013959.stm
>
> WTF-O-Meter: 2.8
Yep, completely missing the point: it's just a thought experiment of
sorts. They should get more luck from a computer simulation with
cellular automata and let it run for a few millenia on common alphabet.
Eventually scanning through the generated texts we could see out of
order streams resembling the complete works, mangled among gibberish.
--
a game sig: http://tinyurl.com/d3rxz9
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Darren New <dne### [at] sanrrcom> wrote:
> Warp wrote:
> > probability *approaches* 1, it never becomes it.
> I think if you mean "the probability as the time approaches infinity" then
> yes. I think if you look at the actual result with infinite time provided,
> you get the probability 1. It's not like you have different probabilities;
> there's only one situation under consideration: an infinite amount of time.
Just because the limit of the probability is 1 doesn't say anything about
whether the works will surely appear with complete certainty at some point.
It only says that the probability for them to appear is larger and larger
as the amount of time is increased.
To understand what I'm talking about, consider the opposite:
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.
Since the probability for them appearing is zero, does that mean that
they will never appear?
No, it doesn't mean that. The zero is just a consequence of calculating
the limit of the probability against infinity. It says nothing about what
will *surely* happen or not happen.
> 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.
There is an infinite amount of sequences *not* containing the works of
Shakespeare. What stops the RNG from generating *all* of those other
sequences before generating Shakespeare's works?
Also, even if the RNG has already produced all the possible letter
combinations with as many letters as there are in Shakespeare's works,
that doesn't mean that the works are next. When we pop out the next set
of letters, there's still a really small probability for them being the
works of Shakespeare. Even if the RNG would have already produced all the
possible other combinations a billion times, it still doesn't make the
probability of Shakespeares' works popping out any more probable. Past
events do not affect future randomness.
Yes, it's extremely unlikely that popping up all the possible distinct
combinations of letters (amounting to the total number of letters in the
works) a billion times will not produce Shakespeare's works, but this
probability is still larger than zero. There is no guarantee.
--
- Warp
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nemesis wrote:
> Eventually scanning through the generated texts we could see out of
> order streams resembling the complete works, mangled among gibberish.
No, all the text is already there. It's your *brain* that's running in the
wrong order.
--
Darren New, San Diego CA, USA (PST)
There's no CD like OCD, there's no CD I knoooow!
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nemesis <nam### [at] gmailcom> wrote:
> Yep, completely missing the point: it's just a thought experiment of
> sorts. They should get more luck from a computer simulation with
> cellular automata and let it run for a few millenia on common alphabet.
> Eventually scanning through the generated texts we could see out of
> order streams resembling the complete works, mangled among gibberish.
I think sampling the noise produced by a resistor would be a better
way of simulating the "monkeys".
--
- Warp
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Warp wrote:
> It only says that the probability for them to appear is larger and larger
> as the amount of time is increased.
Right. And I'm talking about when you actually consider the end result, and
not the limit.
I don't think you can look at a limit as n -> infinity and say what the
answer is at infinity by analogy.
> 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.
Err, no. There are an infinite number of sequences that are as long as
Shakespeare, and only a finite number of *possible* sequences. If all
sequences are equally likely, they'll all appear an equal number of times.
> Since the probability for them appearing is zero, does that mean that
> they will never appear?
I think you're mistaken that the probability is zero.
>> 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.
>
> There is an infinite amount of sequences *not* containing the works of
> Shakespeare. What stops the RNG from generating *all* of those other
> sequences before generating Shakespeare's works?
Nothing. Nor does doing so keep Shakespeare's works from being generated.
You need to address the theorems about normal numbers. There's actual math
here not relying on intuition about infinitites.
> Also, even if the RNG has already produced all the possible letter
> combinations with as many letters as there are in Shakespeare's works,
> that doesn't mean that the works are next. When we pop out the next set
> of letters, there's still a really small probability for them being the
> works of Shakespeare. Even if the RNG would have already produced all the
> possible other combinations a billion times, it still doesn't make the
> probability of Shakespeares' works popping out any more probable. Past
> events do not affect future randomness.
COrrect. I think you're failing to account for infinity.
> Yes, it's extremely unlikely that popping up all the possible distinct
> combinations of letters (amounting to the total number of letters in the
> works) a billion times will not produce Shakespeare's works, but this
> probability is still larger than zero. There is no guarantee.
A billion? Sure. Infinity? I don't think so. Anything less than infinity? Sure.
--
Darren New, San Diego CA, USA (PST)
There's no CD like OCD, there's no CD I knoooow!
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Warp escreveu:
> nemesis <nam### [at] gmailcom> wrote:
>> Yep, completely missing the point: it's just a thought experiment of
>> sorts. They should get more luck from a computer simulation with
>> cellular automata and let it run for a few millenia on common alphabet.
>> Eventually scanning through the generated texts we could see out of
>> order streams resembling the complete works, mangled among gibberish.
>
> I think sampling the noise produced by a resistor would be a better
> way of simulating the "monkeys".
BTW, I think there's the complete works of Shakespeare among the
billions of usenet posts since its inception. Google Groups might index
it. :)
--
a game sig: http://tinyurl.com/d3rxz9
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nemesis <nam### [at] gmailcom> wrote:
> BTW, I think there's the complete works of Shakespeare among the
> billions of usenet posts since its inception. Google Groups might index
> it. :)
But wouldn't that be more akin to Intelligent Design? ;)
(Ok, I gave the average internet poster too much credit by implying that
it has a higher intelligence than a monkey... ;) )
--
- Warp
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Warp <war### [at] tagpovrayorg> wrote:
> nemesis <nam### [at] gmailcom> wrote:
> > Yep, completely missing the point: it's just a thought experiment of
> > sorts. They should get more luck from a computer simulation with
> > cellular automata and let it run for a few millenia on common alphabet.
> > Eventually scanning through the generated texts we could see out of
> > order streams resembling the complete works, mangled among gibberish.
>
> I think sampling the noise produced by a resistor would be a better
> way of simulating the "monkeys".
OMG, I misread that as 'stimulating' o_O
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Warp wrote:
> (Ok, I gave the average internet poster too much credit by implying that
> it has a higher intelligence than a monkey... ;) )
WIN!
--
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*
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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] nawazorg<<<<<<
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
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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
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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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Mueen Nawaz wrote:
> OK. Not how I was interpreting it...
That's how the people in cryptography define it. :-) Basically, you can't
have a random sequence. You can only have a random process. It's impossible
to look at a sequence that has already been generated and decide if it's
random or not.
> I know - just thought I'd point out it could be confusing.
Fair enough.
> 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.
I didn't define the term "normal sequence", except by reference to "normal
numbers", via Wikipedia.
> 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).
Correct.
> However, for this sequence "every block of a particular
> length occurs with equal probability" does not hold.
I think there's confusion conflating "random" and "normal" in our
conversation about distributions here. Not every random sequence is normal,
and not all normal sequences are random. Sorry if I confused "the kind of
random you assume idealized monkeys will generate on keyboards" with "any
sort of random distribution".
To have a "normal number", you need linear distribution, basically. I.e.,
all substrings of a given length appear with equal asymptotic probability as
the number of symbols you look at gets large. *Given* that, every sequence
of any given length will appear if you let N be infinity.
In other words, in order to avoid Shakespeare appearing, you would have to
have a non-random distribution *because* the sequence is infinite and not
just unbounded. In exactly the same way that 0*X is zero no matter how big X
gets, until X is actually infinite.
(lim(x->inf) 0*X) =/= (0*inf)
You can't reason about it based on "no matter how big it gets, ..."
--
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 wrote:
>> 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).
>
> Correct.
>
>> However, for this sequence "every block of a particular
>> length occurs with equal probability" does not hold.
>
> I think there's confusion conflating "random" and "normal" in our
> conversation about distributions here. Not every random sequence is
> normal, and not all normal sequences are random. Sorry if I confused
> "the kind of random you assume idealized monkeys will generate on
> keyboards" with "any sort of random distribution".
Not sure I'm getting it. Just to clarify, I'm using the word normal in
the way you originally meant it (normal numbers). I'm using normal
sequence in the same sense.
I was merely pointing out that you originally said:
"Given that truly random sequences are normal, and in a normal sequence
every block of a particular length occurs with equal probability"
I was giving you an example of a truly random sequence that was _not_
normal.
> To have a "normal number", you need linear distribution, basically.
> I.e., all substrings of a given length appear with equal asymptotic
> probability as the number of symbols you look at gets large. *Given*
> that, every sequence of any given length will appear if you let N be
> infinity.
Agree - other than with the use of the word "will".<G> What I mean is
that what you say is how I understand what probability theory says.
> In other words, in order to avoid Shakespeare appearing, you would have
> to have a non-random distribution *because* the sequence is infinite and
I think I get the general idea, but to nitpick, you can avoid it even
with a random distribution. If I have a distribution where the
probability of typing the letter 'e' is forbidden, it's still random
(not "truly" random).
I suppose you may object to my referring to it as random, but it is
consistent with probability theory: A uniform distribution from 0 to 1
is a valid random distribution - even though you've excluded all numbers
greater than 1.
> not just unbounded. In exactly the same way that 0*X is zero no matter
> how big X gets, until X is actually infinite.
>
> (lim(x->inf) 0*X) =/= (0*inf)
That's really not helpful. I get quite fussy when people use infinity,
and insist on a lot of rigor (which I may not understand<G>) before I
accept anything said by it. The LHS is _always_ equal to 0 (assuming x
is a real number). It's not equal to the RHS because the RHS is
meaningless. It's undefined.
--
"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 wrote:
> Not sure I'm getting it. Just to clarify, I'm using the word normal in
> the way you originally meant it (normal numbers). I'm using normal
> sequence in the same sense.
>
> I was merely pointing out that you originally said:
>
> "Given that truly random sequences are normal, and in a normal sequence
> every block of a particular length occurs with equal probability"
>
> I was giving you an example of a truly random sequence that was _not_
> normal.
Yes. I was phrasing it sloppily there, since we hadn't talked about other
distributions at that point in the conversation. To clarify, a "normal
sequence" in the way I'm using it means all possible subsequences have the
same probability distribution in the asymptote. Hence, you can't take a
gausian distribution of symbols and expect to get a normal sequence from them.
>
>> To have a "normal number", you need linear distribution, basically.
>> I.e., all substrings of a given length appear with equal asymptotic
>> probability as the number of symbols you look at gets large. *Given*
>> that, every sequence of any given length will appear if you let N be
>> infinity.
>
> Agree - other than with the use of the word "will".<G> What I mean is
> that what you say is how I understand what probability theory says.
OK. I don't think I know enough math to convince someone else of it. I'm
just taking it on authority. It would seem to be the sort of thing amenable
to proof, so when a math text says it's so, I'll take their word for it. :-)
Especially when they point out a bunch of other hypotheses where they say
"We think so but haven't proven it."
>> In other words, in order to avoid Shakespeare appearing, you would have
>> to have a non-random distribution *because* the sequence is infinite and
>
> I think I get the general idea, but to nitpick, you can avoid it even
> with a random distribution. If I have a distribution where the
> probability of typing the letter 'e' is forbidden, it's still random
> (not "truly" random).
No, it's not random. If you go an infinite number of symbols and "e" doesn't
appear, then "e" doesn't have the same frequency in the asymptote as
something that *does* appear an infinite number of times.
What you do in that case is remap the output to encode fewer letters of
english per bit, and interpret it that way. That's the equivalent of reading
the number in some other base (binary/hex/etc). I'm dealing with English as
base-26 or base-127 or whatever character set you want to use.
> I suppose you may object to my referring to it as random, but it is
> consistent with probability theory: A uniform distribution from 0 to 1
> is a valid random distribution - even though you've excluded all numbers
> greater than 1.
Sure. But we're starting with the assumption that all the letters of the
alphabet can be typed. If you give the monkeys a typewriter with no "e" key,
of course they're not going to type out shakespeare.
And you can't say "they might never type the word 'the'", because if they
typed "th" and you knew they never typed "the", then you again wouldn't have
a random sequence, because you could predict with 1/25 assurance what the
next letter would be instead of 1/26.
> It's not equal to the RHS because the RHS is
> meaningless. It's undefined.
It's not meaningless. It's merely not a number. That doesn't mean you can't
work with it. Just like you can work with math where you have functions that
are uncalculable, corresponding to functions that get stuck in infinite
loops in a programming language.
In any case, therefore, the statement still holds, because 0 is defined and
0*inf is not.
But I think I've exhausted my ability to convince you that Shakespeare
necessarily appears in the output, if the output is infinite and making the
other normal assumption that each possible character is typed with equal
probability. :-)
--
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 wrote:
> Yes. I was phrasing it sloppily there, since we hadn't talked about
> other distributions at that point in the conversation. To clarify, a
> "normal sequence" in the way I'm using it means all possible
> subsequences have the same probability distribution in the asymptote.
To clarify more, this includes sequences of length 1. So the mathematical
proof I read a long time ago (and which I don't think I really followed at
the time) basically said "if you have an equal probability of picking each
symbol at random, and you string together an infinite number of those
symbols, then you have an equal probability of picking any particular
substring of any particular length." Hence, since there are an infinite
number of substrings of length (size of shakespeare) and *something* in
there must appear an infinite number of times, shakespeare too must appear
an infinite number of times.
The heights of people aren't "truly random" even tho they might be
statistically distributed. It depends on what you ignore when you do the
measurements.
The difference between arbitrary math (unrelated to the universe) and
statistics and science (related to the universe) that you described can be
attributed to what you ignore when you take your measurements. It's actually
kind of fascinating to think on. Someone once convinced me that subatomic
particles like electrons aren't just fungible but actually identical, but I
don't recall what the proof was. It was just logic and relativity and stuff
like that when you got down to it.
--
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 wrote:
> Yes. I was phrasing it sloppily there, since we hadn't talked about
> other distributions at that point in the conversation. To clarify, a
> "normal sequence" in the way I'm using it means all possible
> subsequences have the same probability distribution in the asymptote.
> Hence, you can't take a gausian distribution of symbols and expect to
> get a normal sequence from them.
OK - Agreed.
>> Agree - other than with the use of the word "will".<G> What I mean is
>> that what you say is how I understand what probability theory says.
>
> OK. I don't think I know enough math to convince someone else of it. I'm
> just taking it on authority. It would seem to be the sort of thing
There's really nothing to convince. I'm agreeing with you that
probability theory states the same thing that you state. I have no
_mathematical_ argument against what you're saying (other than the
nitpicking - I agree with your math). I'm just uncomfortable with this
application of probability theory to the "real" world.
See, in mathematics, everywhere I've seen infinity used with rigor (and
where I understood it), behind all the formalism is basically a
definition of what they mean. Sometimes it is intuitive, but there's no
guarantee that it will conform to reality. Indeed, it is often
impossible to compare it to reality, given the absence of infinities (or
at least our ability to measure them). We can say the cardinality of the
set of natural numbers is that of the rationals, but that's not a
universal truth in the "real" world - it's just a consequence of how we
define cardinality (and numbers).
It's because of stuff like this that you get the Banach Tarski paradox
(among a bunch of paradoxes). It's due to the Axiom of Choice, which is
quite intuitive to most people. If the BT paradox seems impossible in
the real world, you'll have to give up on the A of C, which seems
counterintuitive.
Now in probability theory, it's really just mathematics where we're
manipulating numbers and symbols. For cases where we deal with finite
and discrete quantities, probability theory seems to agree with reality.
If it states the probability of something is 0, we interpret that to
mean it will definitely _not_ happen (unless our modeling was off).
Likewise, if it states it is 1, we interpret that to mean it _will_ happen.
When you now go to continuous distributions, picking a point is, in a
sense, meaningless to the theory. It can only give nonzero probabilities
to intervals, or collections of intervals, etc.** Likewise, probability
theory gives you a 0 for flipping a coin every second forever and never
getting a tails.
The (usual) interpretation when applied to the real world is that
because it gave me a 0, such a thing can never happen.
Can it? Can't it? I don't know. I can never know, because I can never
test it. Because of that, I'm wary of results probability theory gives
me for stuff like infinite sequences. I _can_ test it with the real
world for finite events. I'm not saying probability theory is
inconsistent or the math behind it is bad. I'm just saying I shouldn't
blindly accept it as being consistent with the real world just because
it works for finite cases.
I won't accept that I can't get a forever continuous string of heads
from a coin unless someone can give me a _physical_ reason. There isn't
any - there's only a mathematical one.
<snipped a whole bunch of stuff because I'd just nitpick further>
>> It's not equal to the RHS because the RHS is
>> meaningless. It's undefined.
>
> It's not meaningless. It's merely not a number. That doesn't mean you
Well, I guess "meaningless" is vague. It _is_ undefined. About as
useful as dividing by 0.
> But I think I've exhausted my ability to convince you that Shakespeare
> necessarily appears in the output, if the output is infinite and making
Well, perhaps we're just playing semantic games.
As I said, I agree that probability theory agrees with you.
I'm simply wary of using a _purely_ mathematical argument to make
statements about the real world. It invokes no aspect of the real world,
and no laws of physics that I'm aware of. It's just like saying that
yes, you _can_ execute the BT paradox in the real world...
(As you can guess by now, I'm one of those who feel that the theorems
in mathematics are not necessarily tied to the physical laws of the
universe - other than in the manner that we can think them and
presumably our brains conform to the physical laws...).
** The reason, I suspect, is that theories of integration generally
don't have the integral changing if you happen to remove a point or any
set of measure 0 from the domain of integration.
--
"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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Invisible <voi### [at] devnull> wrote:
> http://news.bbc.co.uk/1/3013959.stm
> "...and mostly typing the letter "s"."
For a second, I thought maybe they had discovered radio communication, but I
guess it was the humans that set up the radio link. Too bad.
- Ricky
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Invisible <voi### [at] devnull> wrote:
> http://news.bbc.co.uk/1/3013959.stm
What amazes me is that the Arts Council doled out £2,000 for this idiocy. Hmm...
getting grant money seems to be easier than I thought!
KW
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> http://news.bbc.co.uk/1/3013959.stm
>
> WTF-O-Meter: 2.8
Hehe, I guess for some people it doesn't take much to "learn an awful lot"
:-)
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> 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.
>
> (Many people think that in this last case the works *will* eventually
> appear with absolute certainty, but that's just the gambler's fallacy.)
Isn't it mathematical fact that the probability of the works not appearing
is zero in the limit condition?
Like if you ask what is the probability of getting no heads when a coin is
tossed N times, it is 2^-N, which is *equal* to zero in the limit as N tends
to infinity. That's what I was taught at school/university anyway.
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> 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.
Sorry Warp, but that's just bad math.
One way to explain why you are wrong is that yes, there are an infinite
number of finite length sequences of letters, but then there are also an
infinite number of finite length sequences that *contain* the works (plus
other junk). So the probability calculation is infinity/infinity, not
1/infinity.
If you restrict the problem down to finite sequences of the same length as
"the works", then now there are only a finite number of possiblities, and
exactly 1 of them will be "the works".
Just to assume "the works" are W letter long, then the probability of a
random sequnce R of length W exactly matching is 64^-W (assuming 64
characters here). Then, the probability of R *not* matching is (1-64^-W).
If we take N sequences of random letters, then the probability of finding
"the works" is given by 1-(1-64^-W)^N, which *equals* 1 in the limit of N
tending to infinity.
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scott wrote:
> Hehe, I guess for some people it doesn't take much to "learn an awful
> lot" :-)
Uh, yeah...
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Kenneth wrote:
> What amazes me is that the Arts Council doled out £2,000 for this idiocy. Hmm...
> getting grant money seems to be easier than I thought!
After the study to determine whether a duck's quack echos, nothing
surprises me any more. ;-)
Hmm, maybe we could get some kind of a grant? Maybe to determine whether
people can tell the difference between POV-Ray and the Real World?
...so basically, get paid to play with POV-Ray all day! :-D
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Mueen Nawaz <m.n### [at] ieeeorg> wrote:
> Which was my whole long winded point.
To make it clear: I was not saying anything *against* your arguments.
I was trying to complement them. :)
--
- Warp
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scott <sco### [at] scottcom> wrote:
> Isn't it mathematical fact that the probability of the works not appearing
> is zero in the limit condition?
In the exact same way as the probability of getting a specific value in
a continuous range is zero (for the sole reason that a continuous range has
an infinite amount of values).
Just because the probability is mathematically zero does not mean you
will not get any specific value.
--
- Warp
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scott <sco### [at] scottcom> wrote:
> Just to assume "the works" are W letter long, then the probability of a
> random sequnce R of length W exactly matching is 64^-W (assuming 64
> characters here). Then, the probability of R *not* matching is (1-64^-W).
> If we take N sequences of random letters, then the probability of finding
> "the works" is given by 1-(1-64^-W)^N, which *equals* 1 in the limit of N
> tending to infinity.
So exactly at which point are the works forced to appear, to fulfill
the probability of 1?
The answer is: They are never forced to appear. And that is not a
contradiction of the probability being 1 when dealing with infinity
(any more than a value in a continuous range having a probability of
zero is a contradiction that that value might be chosen at random).
--
- Warp
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> So exactly at which point are the works forced to appear, to fulfill
> the probability of 1?
In the limit of N --> infinity. If N is a finite number then the works are
not forced to appear, by saying N is infinite you are forcing them to
appear. Just think of "infinite" to mean "repeat until it does appear".
Saying "it might never appear" is not a valid argument, because an infinite
list of random sequences does not have a concept of "never", there are
always infinitely more sequences to come no matter how many you go through.
> The answer is: They are never forced to appear.
...unless N is allowed to be infinite, which the original problem states
quite clearly.
Or do you also disagree with 0.99999... with infinitely many 9's equals 1?
Perhaps you should read this page:
http://en.wikipedia.org/wiki/Limit_(mathematics)
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scott <sco### [at] scottcom> wrote:
> > So exactly at which point are the works forced to appear, to fulfill
> > the probability of 1?
> In the limit of N --> infinity.
That's, in fact, the exact same thing as saying "never".
And that's how it is: Each *individual* round of popping up values from
the RNG has a smaller-than-1 probability for the works to appear. Thus at
no point are the works *forced* to appear.
> > The answer is: They are never forced to appear.
> ...unless N is allowed to be infinite, which the original problem states
> quite clearly.
So at which point are the works forced to appear?
> Or do you also disagree with 0.99999... with infinitely many 9's equals 1?
Do you disagree that a probability of zero does not mean that the event
will never happen?
--
- Warp
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http://everything2.com/title/If%2520you%2520have%2520enough%2520monkeys%2520banging%2520randomly%2520on%2520typewriters%252C%2520they%2520will%2520eventually%2520type%2520the%2520works%2520of%2520William%2520Shakespeare
http://everything2.com/title/Monkey%2520Shakespeare%2520Simulator mentions a
site that I'd come across a while back that simulated this, but it's since
vanished.
--
Tim Cook
http://empyrean.freesitespace.net
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>> In the limit of N --> infinity.
>
> That's, in fact, the exact same thing as saying "never".
Ermm, no, it's saying that the probability will get as close to 1 as you
want, and in the limit it will *equal* 1. Exactly the same as saying
0.999999... can get as close to 1 as you want, and in the limit is actually
*equal* to 1.
> And that's how it is: Each *individual* round of popping up values from
> the RNG has a smaller-than-1 probability for the works to appear. Thus at
> no point are the works *forced* to appear.
For a finite number of tries, no.
>> Or do you also disagree with 0.99999... with infinitely many 9's equals
>> 1?
No go on, please answer this one, because it's the same thing. At which
point does 9/10 + 9/100 + 9/1000 suddenly equal 1? Answer: when you extend
the series to infinity.
> Do you disagree that a probability of zero does not mean that the event
> will never happen?
A probability of zero means an event does not happen by definition, however
if you try an infinite number of times it might not necessarily never
happen.
A good example has already been mentioned, of choosing an exact number
between 0 and 1. The probability is zero for any specific number, but if
you sum up the infinite number of probabilities between say 0.2 and 0.3, you
will get a non-zero probability.
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scott <sco### [at] scottcom> wrote:
> >> Or do you also disagree with 0.99999... with infinitely many 9's equals
> >> 1?
> No go on, please answer this one, because it's the same thing. At which
> point does 9/10 + 9/100 + 9/1000 suddenly equal 1? Answer: when you extend
> the series to infinity.
I really can't understand why you are so fixated with that question.
I never doubted or denied its veracity. My reply clearly implied that it
is indeed so.
> > Do you disagree that a probability of zero does not mean that the event
> > will never happen?
> A probability of zero means an event does not happen by definition, however
> if you try an infinite number of times it might not necessarily never
> happen.
You don't have to try an infinite number of times to get a value from
a continuous range. You only have to try once. And the value you get had
a probability of zero of being chosen. Yet it was chosen.
> A good example has already been mentioned, of choosing an exact number
> between 0 and 1. The probability is zero for any specific number, but if
> you sum up the infinite number of probabilities between say 0.2 and 0.3, you
> will get a non-zero probability.
You are not choosing a range of values. You are choosing *one* value at
random.
--
- Warp
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