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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