

Tested in the field: Will monkeys type shakespeare? - huangm
http://bakadesuyo.com/will-monkeys-really-type-shakespeare-if-given

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RiderOfGiraffes
People really, really don't understand the concept of infinity, nor the
concept of "with probability 1." The most interesting point about the idea
that people actually go out and try these things, is to prove that people just
don't understand these things.

The conclusion that "the theory is flawed" shows they don't understand the
theory.

The question I have is - should we care? Should we try to explain these
things? Or should we just shrug and give up. How will it actually make their
lives better? For some people it might, but then again ...

~~~
astine
If the probability of producing the desired result in any given attempt is
zero, then the probability of producing the desired result given infinite
attempts is still zero.

If monkeys are incapable of typing, then the probability is zero.

~~~
RiderOfGiraffes

      > If the probability of producing the desired result
      > in any given attempt is zero, then the probability
      > of producing the desired result given infinite
      > attempts is still zero.
    

This is a common misconception. Pick a point uniformly at random in the
interval from 0 to 1. The probability of any given point is zero, but the
probability of being in the interval [a,b) is b-a (assuming 0<=a<=b<=1). This
is a case of infinitely many zeroes giving a non-zero sum, and why you need to
deal carefully with infinities.

Yes, I know about countable and uncountable infinities. This is a big subject,
and small comments here won't cover it. There are rules and exceptions.

~~~
astine
One over infinity is a different kind of zero than we're talking about. An
infinitesimal value is not a nil value.

~~~
RiderOfGiraffes
One over infinity is not a zero at all, because it's an ill-defined operation.
You need to say exactly what you mean, and which number system you're working
in. Are you in the surreals? Non-standard analysis? The hyper-reals?

If you work with standard analysis, your comment seems to be a non-sequiteur.
If not, you need to say which system you're in. Personally, I think the other
systems are largely unhelpful, although independently interesting.

And of course an infinitesimal is not zero (nil being a different technical
term).

If I've missed the point(s) you were tying to make, you'll need to be more
explicit. Probably further "discussion" here is not going to be fruitful, but
I'll be interested to read anything you care to add.

~~~
astine
I'm sorry, I'm not a statistician (or any kind of mathematician) so I don't
know the proper terminology. I'm just noting that the only way your previous
example seems work from my perspective is if you take one over infinity to be
zero.

I'll explain: The set of all of the points between two points is an infinite
set, and the probability of selecting any single member of a set is
approximately one over the size of the set, so it would seem that the
probability of selecting any individual member of an infinite set would be one
over 'infinity' that is, an undefined value.

Granted, the limit of one over x as x approaches infinity is zero, but the
probability of selecting any single member of an infinite set isn't zero in
the same sense that the probability of selecting a member outside the set is
zero. That is, if I pick a point between 0 and 1, and I select point x, there
was obviously not _impossible_ to select point x. So while it may be
'mathematically convenient'* to say that the probability of selecting any
given point is zero I think that it might be better to say that it might be
better to say that the chance is infinitesimally small, rather than zero per
se.

So while I may certainly be wrong, I think that you are univocating when you
say that a range is an example of an infinite number of zeros being non-zero.
When I was talking about the chance of a individual monkey typing out
Shakespeare was zero, I meant that, if it was impossible for any monkey to
type (imagine that we are using exclusively dead monkeys) no monkey could ever
type out anything, much less Shakespeare.

~~~
RiderOfGiraffes
The problem is that if you assign a probably to a point, then the probability
you assign has to be strictly smaller that every positive number, so it has to
be zero. The system that works with finite sets of taking the number of things
you're interested in and dividing by the number of things that might happen
doesn't apply. Contradictions and/or inconsistencies arise from that sort of
thinking, seductive though it is.

My example about the ranges was an attempt to show that the otherwise
excellent ideas from the finite world simply don't apply, so you have to do
something else. You suggested that ...

    
    
      > If the probability of producing the desired result
      > in any given attempt is zero, then the probability
      > of producing the desired result given infinite
      > attempts is still zero.
    

... and that's what I was refuting.

And perhaps the things to take away from this is that just because the
probability of an event is zero, that doesn't mean the event can't happen.
Talking about the probability being "infinitesimally small" requires that you
make a whole pile of other definitions precise, and that way lies madness. Or
at least, non-standard analysis, which is pretty close to the same thing.

In my opinion it doesn't help the understanding of the "interested layman."

All this, of course, is only valid in the theoretical musings of
mathematicians (or those doing math at the time) In the "real world" it's all
different again.

Doing this sort of thing correctly and consistently is hard.

~~~
astine
_And perhaps the things to take away from this is that just because the
probability of an event is zero, that doesn't mean the event can't happen._

Ah! That does seem to be the take-away here. In that case, however, how would
you measure the probability of an impossible event?

I still maintain that an invinite number of dead monkeys would never type
Shakespeare, no matter the resources alloted to them. :)

