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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>  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.

So you would also agree then that 1/2 + 1/4 + 1/8 + ... equals 1 with 
infinitely many terms?  You see where this is going?

>  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.

You are trying to equate the probabilities of 1/infinity with 0/infinity, 
they cannot always be treated as the same (in some situations they can be).

Imagine choosing numbers between 0 and 1.  Getting exactly 0.5 (or any other 
number between 0 and 1) has a probability of 1/infinity, getting exactly 1.5 
has a probability of 0/infinity.  "Normally" they would be the same, but if 
you say something like "what is the probability of getting *any* value 
between 0 and 1" or "what is the probability of getting exactly 0.5 after 
infinitely many tries" then they are not the same.

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scott <sco### [at] scottcom> wrote:
> >  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.

> So you would also agree then that 1/2 + 1/4 + 1/8 + ... equals 1 with 
> infinitely many terms?  You see where this is going?

  I agree with it, and no, I don't see your point.

> You are trying to equate the probabilities of 1/infinity with 0/infinity, 
> they cannot always be treated as the same (in some situations they can be).

> Imagine choosing numbers between 0 and 1.  Getting exactly 0.5 (or any other 
> number between 0 and 1) has a probability of 1/infinity, getting exactly 1.5 
> has a probability of 0/infinity.  "Normally" they would be the same, but if 
> you say something like "what is the probability of getting *any* value 
> between 0 and 1" or "what is the probability of getting exactly 0.5 after 
> infinitely many tries" then they are not the same.

  I honestly don't understand.

-- 
                                                          - Warp

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>> So you would also agree then that 1/2 + 1/4 + 1/8 + ... equals 1 with
>> infinitely many terms?  You see where this is going?
>
>  I agree with it, and no, I don't see your point.

That the sequence is also the probability of a head being tossed after N 
tries of throwing a coin.  After 1 try it's 1/2, after 2 tries it's 1/2 + 
1/4, etc.  So after infinitely many tries the probability is *equal* to 1.

Now simply replace "head being tossed" with "this sequence of characters 
being the works".

>  I honestly don't understand.

If you are choosing numbers in the range 0...1, 0.5 has an infinitesimally 
small probability of being chosen, 1.5 has a zero probability of being 
chosen, they are not the same probability.  (In some situations you can call 
the probability of 0.5 being chosen "zero", but you must remember that the 
zero came from 1/infinity and not "real" zero incase you use it in later 
calculations).

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The Internet has already proven that a multitude of monkeys would not 
duplicate the works of Shakespeare, regardless of the amount of time 
allowed.

Regards,
John

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scott <sco### [at] scottcom> wrote:
> >> So you would also agree then that 1/2 + 1/4 + 1/8 + ... equals 1 with
> >> infinitely many terms?  You see where this is going?
> >
> >  I agree with it, and no, I don't see your point.

> That the sequence is also the probability of a head being tossed after N 
> tries of throwing a coin.  After 1 try it's 1/2, after 2 tries it's 1/2 + 
> 1/4, etc.  So after infinitely many tries the probability is *equal* to 1.

> Now simply replace "head being tossed" with "this sequence of characters 
> being the works".

  You talk as if I had said somewhere that in the infinite case the
probability does *not* equal 1. I don't remember saying such a thing.

> >  I honestly don't understand.

> If you are choosing numbers in the range 0...1, 0.5 has an infinitesimally 
> small probability of being chosen, 1.5 has a zero probability of being 
> chosen, they are not the same probability.  (In some situations you can call 
> the probability of 0.5 being chosen "zero", but you must remember that the 
> zero came from 1/infinity and not "real" zero incase you use it in later 
> calculations).

  Now it's you who sounds like saying that 1/2+1/4+... does not equal 1.

-- 
                                                          - Warp

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>> Now simply replace "head being tossed" with "this sequence of characters
>> being the works".
>
>  You talk as if I had said somewhere that in the infinite case the
> probability does *not* equal 1. I don't remember saying such a thing.

You said:

> That's, in fact, the exact same thing as saying "never".

Never is a probability of exactly 0, yet you've just agreed that the 
probability is actually exactly 1.  Make your mind up :-)

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