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Warp wrote:
> Thanks for not answering the question.
He's saying that the probability of getting heads while tossing a coin is
1/2 + 1/4 + 1/8 + ....
Nothing forces heads to appear.
Yet if you toss an *infinite* number of times, the total probability is
exactly equal to one, which means it is 100% guaranteed.
Resolve this discrepency, in your mind, and you'll understand why
shakespeare must appear.
--
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] ieee org> wrote:
> > Calculate the probability, you will find it is equal to one. Exactly
> > one. That, by definition, means it is guaranteed, or "forced" if you
> > like, to happen. QED.
> Yes, by *definition*.
> Warp asked for mechanics. Invoking mathematics is not invoking any
> physical reasons. The mathematicians have merely defined it that way.
> Which does not mean it will happen.
As far as I can see, the "mechanics" go like this:
Q: When popping up random values, where is the first point at which the
works of Shapespeare are forced to appear?
A: At infinity.
In other words, you would have to pop up an infinite amount of numbers
before you reach the situation where the works must appear. However, no
matter how many values you pop up, they will never be infinite. By definition
you can *not* pop up an infinite amount. You will never reach infinity.
Thus the works are *never* forced to appear.
Asking when the works must appear is like asking what is the smallest
real number larger than zero. There's no such a thing. There is no point
at which the works are forced to appear.
The monkey analogy is thus flawed. No matter how long the monkey hammers
the typewriter, at no point are the works of Shakespeare appearing with
absolute certainty. They may appear, but there's no absolute guarantee.
There may be mathematical background for the infinity and the certain
probability, but the *monkey analogy* is inherently flawed.
--
- Warp
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Mueen Nawaz wrote:
> 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.
Given that, as far as we know, the universe isn't infinite, I'd go so far as
to say there's a decent guarantee that it *doesn't* match the universe. :-)
> universal truth in the "real" world - it's just a consequence of how we
> define cardinality (and numbers).
Sure. Understood. The monkey thing is obviously not a "real world" thing.
> 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.
You can't get a forever continuous string of heads from a coin because as
far as we know, there's no such thing as forever. :-) Fair nuff.
>> 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.
I hadn't realized you were trying to talk about "really" outside of
mathematics. I think we're in complete agreement.
> I'm simply wary of using a _purely_ mathematical argument to make
> statements about the real world.
I agree. Math is only useful to the extent it's isomorphic to reality, which
is impossible to prove mathematically. It's pretty amazing that so much of
reality seems to conform to simple rules once you ignore the right stuff.
--
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 <dne### [at] sanrrcom> wrote:
> Warp wrote:
> > Thanks for not answering the question.
> He's saying that the probability of getting heads while tossing a coin is
> 1/2 + 1/4 + 1/8 + ....
> Nothing forces heads to appear.
> Yet if you toss an *infinite* number of times, the total probability is
> exactly equal to one, which means it is 100% guaranteed.
So at which point exactly will it happen with 100% certainty?
What are the exact mechanics which cause it to happen?
"At infinity" is the same thing as "never" in this case.
> Resolve this discrepency, in your mind, and you'll understand why
> shakespeare must appear.
Once you explain to me the discrepancy that an event having zero
probability can happen.
--
- Warp
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Darren New <dne### [at] sanrrcom> wrote:
> Warp wrote:
> > Do you disagree that a probability of zero does not mean that the event
> > will never happen?
> A probability of zero does not mean the event won't happen with an infinite
> number of trials. Zero times infinity is not zero.
True. The event happens with *one* trial. You don't even need an infinite
amount of them.
Now explain that.
--
- Warp
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> Sorry, but no - the explanation doesn't work. If you read my original
> long post, you'd see that the two cases are *identical*.
The mistake in your logic is assuming the probability of choosing 1 item
from an infinite set is zero, it isn't, it is 1/infinity or "infinitely
small". In many cases this can be treated as zero, but when you start
summing over an infinite number of items (ie what is the probability that I
chose any of these items, or if I try an infinite times will I get this
one?) there is an important difference.
The probability of you choosing 1.847 when asked to choose a number between
0 and 1 is really zero. Even if you try an infinite number of times, it's
still zero probability.
> Getting a sequence of all heads forever is identical to picking a point
> from 0 to 1. Both have probability 0.
Mathematicians seem to disagree with you on that one. 1/infinity is not
defined as zero (because it can often lead to problems like the above),
however the infinite sum of 1/2+1/4+1/8+... (ie probability of getting no
tails after infinite throws) is defined as 1.
Of course you can discuss how this relates to reality, but I'd rather not
get involved in that one, it could an infinitely long time :-)
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Darren New <dne### [at] sanrrcom> wrote:
> Warp wrote:
> > So exactly at which point are the works forced to appear, to fulfill
> > the probability of 1?
> Infinity. That's the point.
In other words: Never. At no point is it forced to happen.
That's the point.
> The limit does not equal the value at infinity. Unbounded is not the same as
> infinite. You're still confusing the two.
The analogy is that the monkey will type Shakespeare's work for sure.
Exactly when is this 100% certainty reached? Never.
--
- Warp
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> In other words, you would have to pop up an infinite amount of numbers
> before you reach the situation where the works must appear. However, no
> matter how many values you pop up, they will never be infinite. By
> definition
> you can *not* pop up an infinite amount. You will never reach infinity.
>
> Thus the works are *never* forced to appear.
It seems to me you are concerned about an event taking place an infinite
number of times, rather than anything to do with probability.
Yes I know the text says that "with infinite time", and in reality this can
never happen, so I guess that's the explanation you are looking for?
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Warp wrote:
> So at which point exactly will it happen with 100% certainty?
When an infinite number of flips are flipped, and not before.
> What are the exact mechanics which cause it to happen?
The mathematics of infinity.
> "At infinity" is the same thing as "never" in this case.
It has nothing to do with "time", so you are technically correct in saying
that it's physically impossible to do an infinite number of flips in a
non-infinite length of time.
>> Resolve this discrepency, in your mind, and you'll understand why
>> shakespeare must appear.
>
> Once you explain to me the discrepancy that an event having zero
> probability can happen.
When you do it an infinite number of times.
Do you understand what I'm saying when I say you're confusing unbounded with
infinite?
Is it possible to have an infinitely long string of zeros? Yes. Is it
possible to program a turing machine to write an infinitely long string of
zeros on its tape? No. A turing machine can only write an unbounded number
of zeros on the tape.
--
Darren New, San Diego CA, USA (PST)
There's no CD like OCD, there's no CD I knoooow!
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Warp wrote:
> In other words, you would have to pop up an infinite amount of numbers
> before you reach the situation where the works must appear. However, no
> matter how many values you pop up, they will never be infinite.
The problem statement presumes an infinite number of monkeys, or an infinite
duration of typing. Hence, yes, the number of values will be infinite.
If you have an infinite number of monkeys, and each types one page, you will
have an infinite number of pages.
You're arguing "but you can't *get* to infinity by counting one at a time."
Of course. But the premise is that you're already at infinity.
> The monkey analogy is thus flawed. No matter how long the monkey hammers
> the typewriter, at no point are the works of Shakespeare appearing with
> absolute certainty. They may appear, but there's no absolute guarantee.
Because you can't get to an infinite number of keypresses by looking at them
one at a time.
Would you agree that an infinite number of monkeys each pressing one key is
identical to one monkey pressing an infinite number of keys?
> There may be mathematical background for the infinity and the certain
> probability, but the *monkey analogy* is inherently flawed.
Well, yes. Because there aren't an infinite number of monkeys, nor is there
any way a single monkey will live an infinite number of seconds.
However, if you *presume* either of those has come to pass, you 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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