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scott <sco### [at] scott com> wrote:
> >> 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 :-)
Great out of context quote.
The "never" was not referring to "will it appear?". It referred to
"is it *forced* to appear"?
No, at no point is it forced to appear. Thus it's *never* forced to
appear. That's not the same thing as "it will never appear".
--
- Warp
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> The "never" was not referring to "will it appear?". It referred to
> "is it *forced* to appear"?
If the probability is exactly 1, which you agree it is, then of course it is
forced to appear at some point in the sequence. If it wasn't then the
probability would be less than 1.
Or are you going to argue now about the meaning of "forced"? ;-)
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scott <sco### [at] scottcom> wrote:
> > The "never" was not referring to "will it appear?". It referred to
> > "is it *forced* to appear"?
> If the probability is exactly 1, which you agree it is, then of course it is
> forced to appear at some point in the sequence. If it wasn't then the
> probability would be less than 1.
And if the probability of getting 0.5 is zero, then of course it means
that it will never be chosen.
Explain to me the exact mechanics which force the works to appear.
--
- Warp
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>> If the probability is exactly 1, which you agree it is, then of course it
>> is
>> forced to appear at some point in the sequence. If it wasn't then the
>> probability would be less than 1.
>
> And if the probability of getting 0.5 is zero, then of course it means
> that it will never be chosen.
Of course.
But the probability of 0.5 turning up is not zero, it is 1/infinity.
Sometimes they can be used interchangably, but in this situation they
cannot. You won't find a mathematician that claims 1/infinity universally
is equal to zero under all circumstances in normal number systems.
On the other hand, writing an infinite sum like 1/2+1/4+1/8+.. does
*exactly* equal one, always, never with any doubt, it is universally
accepted in mathematics.
> Explain to me the exact mechanics which force the works to appear.
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.
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scott <sco### [at] scottcom> wrote:
> > Explain to me the exact mechanics which force the works to appear.
> 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.
Thanks for not answering the question.
--
- Warp
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>> 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.
>
> Thanks for not answering the question.
Well you can read a few posts back to clear up the detail.
Glad to have helped :-)
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scott wrote:
> But the probability of 0.5 turning up is not zero, it is 1/infinity.
> Sometimes they can be used interchangably, but in this situation they
> cannot. You won't find a mathematician that claims 1/infinity
> universally is equal to zero under all circumstances in normal number
> systems.
>
> On the other hand, writing an infinite sum like 1/2+1/4+1/8+.. does
> *exactly* equal one, always, never with any doubt, it is universally
> accepted in mathematics.
Sorry, but no - the explanation doesn't work. If you read my original
long post, you'd see that the two cases are *identical*.
Getting a sequence of all heads forever is identical to picking a point
from 0 to 1. Both have probability 0. There's a 1-1 correspondence
(which I showed earlier) between the two. They're not just similar -
they're the same.
Conversely, saying that a tails _must_ appear in a sequence is just
doing 1 - (1/2)^n, and taking the limit to infinity, giving 1. Which is
the same as saying that the probability that I _won't_ pick 0.5 when I
choose a number between 0 and 1 is 1.
I even forget how the 1/2 + 1/4 + 1/8 sequence enters the picture - was
that the result of some probability calculation?
>> Explain to me the exact mechanics which force the works to appear.
>
> 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.
--
"The security of the Enterprise is of Paramount importance.
/\ /\ /\ /
/ \/ \ u e e n / \/ a w a z
>>>>>>mue### [at] nawazorg<<<<<<
anl
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John VanSickle wrote:
> The Internet has already proven that a multitude of monkeys would not
> duplicate the works of Shakespeare, regardless of the amount of time
> allowed.
For a very narrow definition of "never".
--
"The security of the Enterprise is of Paramount importance.
/\ /\ /\ /
/ \/ \ u e e n / \/ a w a z
>>>>>>mue### [at] nawazorg<<<<<<
anl
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Warp wrote:
> So exactly at which point are the works forced to appear, to fulfill
> the probability of 1?
Infinity. 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.
--
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:
> 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.
--
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:
> 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] ieeeorg> 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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Warp wrote:
> True. The event happens with *one* trial. You don't even need an infinite
> amount of them.
Selection bias. :-)
--
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:
> 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.
No. You're getting wrapped up in Zeno's paradox here.
> The analogy is that the monkey will type Shakespeare's work for sure.
> Exactly when is this 100% certainty reached? Never.
Yes. After infinite time. But you're refusing to look at that case, trying
to get to "after infinite time has passed" by looking one second at a time.
Assume you have an infinite number of monkeys, each typing one block of text
the size of shakespeare, and they all finish within one day. Is Shakespeare
in there?
--
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:
> > So at which point exactly will it happen with 100% certainty?
> When an infinite number of flips are flipped, and not before.
In other words, never.
> > "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.
Thus the monkey analogy is technically flawed. Which is my original point.
> >> 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.
But it happens with only one attempt, not an infinite amount of them.
> Do you understand what I'm saying when I say you're confusing unbounded with
> infinite?
The monkey analogy is trying to make a connection between theoretical
math dealing with infinites and physical reality, which are at completely
different conceptual levels. I do not fully understand the math, but I'm
pretty sure I understand the physical process depicted by the analogy,
and I'm pretty sure the claim is flawed.
--
- Warp
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Darren New <dne### [at] sanrrcom> wrote:
> Assume you have an infinite number of monkeys, each typing one block of text
> the size of shakespeare, and they all finish within one day. Is Shakespeare
> in there?
If all the monkeys press the key 'a', then obviously not.
What forces any of the monkeys to press some other key?
--
- Warp
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On Tue, 21 Apr 2009 09:15:29 -0700, Darren New wrote:
>> So at which point exactly will it happen with 100% certainty?
>
> When an infinite number of flips are flipped, and not before.
Disagree. The point at which it will happen with 100% certainty is the
point at which it happens, and not before. ;-)
Jim
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Warp wrote:
>> 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.
>
> Thus the monkey analogy is technically flawed. Which is my original point.
Well, yes. You're saying "No, an infinite number of monkeys won't type out
the works of shakespeare *because* there aren't an infinite number of
monkeys in the universe."
Certainly the argument falls down if you don't accept the premise. So does
the proof that 1+1=2, so I'm not sure what your point is.
>>>> 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.
>
> But it happens with only one attempt, not an infinite amount of them.
Because you're incapable of picking one number from an infinite number of
possible numbers, in precisely the same way it's impossible to have one
real-world monkey typing for an infinite length of time.
>> Do you understand what I'm saying when I say you're confusing unbounded with
>> infinite?
>
> The monkey analogy is trying to make a connection between theoretical
> math dealing with infinites and physical reality, which are at completely
> different conceptual levels.
You're denying that the premise could possibly be true in the real world.
That isn't how math works.
> I do not fully understand the math, but I'm
> pretty sure I understand the physical process depicted by the analogy,
> and I'm pretty sure the claim is flawed.
No, I'm pretty sure you don't understand "after an infinite amount of time"
in the physical process sense of the word. I don't think anyone does, except
maybe perhaps Hawking.
--
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:
>> Do you understand what I'm saying when I say you're confusing unbounded with
>> infinite?
>
> The monkey analogy is trying to make a connection between theoretical
> math dealing with infinites and physical reality,
So what you're saying is there aren't *really* an infinite number of
positive integers, because if you count "one, two, three, ..." you'll never
get to all the integers.
Yeah, OK, that's not what infinite means.
I'll agree that it's impossible to have an infinite number of monkeys. But
if you *did*, they'd hit shakespeare with 100% certainty.
--
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:
> If all the monkeys press the key 'a', then obviously not.
> What forces any of the monkeys to press some other key?
Randomness. If you do an infinite number of trials and only get the 'a'
key, then your process isn't random, by definition.
--
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:
> > If all the monkeys press the key 'a', then obviously not.
> > What forces any of the monkeys to press some other key?
> Randomness. If you do an infinite number of trials and only get the 'a'
> key, then your process isn't random, by definition.
Then there must be some kind of law of the universe which forces at
least some of the monkeys to type something else than 'a'.
The probability that all of them will press 'a' is zero, but is this
a "different zero" from the one related to choosing one value from a
continuous range?
--
- Warp
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Warp wrote:
> Then there must be some kind of law of the universe which forces at
> least some of the monkeys to type something else than 'a'.
No. It's part of the definition of infinity and random.
--
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:
> True. The event happens with *one* trial. You don't even need an infinite
> amount of them.
>
> Now explain that.
Actually, it means you didn't pick at random from an infinite number of
possibilities, in exactly the same way you didn't *really* have an infinite
number of monkeys doing the typing.
--
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:
> > True. The event happens with *one* trial. You don't even need an infinite
> > amount of them.
> >
> > Now explain that.
> Actually, it means you didn't pick at random from an infinite number of
> possibilities, in exactly the same way you didn't *really* have an infinite
> number of monkeys doing the typing.
Do you mean that it's not possible to choose a value from an infinite
set of values?
(Isn't that the axiom of choice?)
--
- Warp
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Warp wrote:
> Do you mean that it's not possible to choose a value from an infinite
> set of values?
No. I mean it's impossible for you, Warp, to choose a value from an infinite
set of values, in *exactly* the same way it's impossible to have an infinite
number of monkeys typing Shakespeare. Your brain is physically incapable of
picking any one of those values with equal probability because there are
infinite numbers of choices which you are capable of expressing or thinking
about, due to being finite yourself.
Hence, when you pick "0.5", you haven't picked it from the infinite number
of choices available, but from the finite subset you have ever in y our life
ever happened to think about beforehand.
> (Isn't that the axiom of choice?)
Not exactly, no. The axiom of choice says that if you have a set of infinite
cardinality where each element is a set of infinite cardinality, it's
possible to create an infinite set by picking one element of each of the
subsets.
--
Darren New, San Diego CA, USA (PST)
There's no CD like OCD, there's no CD I knoooow!
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scott wrote:
> 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
Nope. Probability theory gives zero. Not "arbitrarily small". The very
same mathematics that gives you a 1 for the monkey scenario gives you a
0 here. Which is why I keep saying they are identical.
Rather, the "equivalent" argument to what you're saying is that the
probability of getting at least one tails in an infinite number of flips
is 1 - 1/infinity, and then me claiming that it isn't 1.
As far as the math goes, there's no such thing as 1/infinity, nor is
there 1 - 1/infinity. The process to calculate the first gives you a 0,
the process to calculate the latter gives you 1.
> 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.
I don't see the logic. The only reason it is 1 is that after n flips,
the probability of getting a tails is 1 - (1/2)^n (which is the same as
your sum, which I could loosely (and meaninglessly) write as 1 -
1/infinity). Then they just take the limit to infinity to get 1. The
probability of getting all heads after n flips is (1/2)^n, which goes to
0 as you take the limit.
Likewise, there is a 1-1 correspondence between each number from 0 to 1
and an infinite sequence of heads and tails. In other words, if you
state some number between 0 and 1, I can give you a unique sequence that
corresponds to that number (and vice versa). Hence, the probability of
picking that number is the same as the probability of getting that exact
sequence.
Thus, if you state that the probability of getting all heads is 0, then
so is the probability of picking 0.5. Likewise, if you state that you
must get at least one tails, then I can likewise state that you must
pick some number _other_ than 0.5.
When I say a 1-1 correspondence exists, I'm not saying it loosely, or
intuitively. I can rigorously show it to you: Just take the binary
expansion of any number from 0 to 1, and replace all 0's with heads and
all 1's with tails.
> 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 :-)
But that *is* what Warp and I are complaining about. I don't disagree
with the math, and neither does he.
--
If a pig lost it's voice, would it become disgruntled?
/\ /\ /\ /
/ \/ \ u e e n / \/ a w a z
>>>>>>mue### [at] nawazorg<<<<<<
anl
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Darren New wrote:
> 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.
Well, maybe there _is_. I just know of no way of *testing* it.<G>
--
If a pig lost it's voice, would it become disgruntled?
/\ /\ /\ /
/ \/ \ u e e n / \/ a w a z
>>>>>>mue### [at] nawazorg<<<<<<
anl
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Mueen Nawaz wrote:
> Darren New wrote:
>> 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.
>
> Well, maybe there _is_. I just know of no way of *testing* it.<G>
I know you're being funny, but even if there is a "forever", such is
unbounded, not infinite, so it still doesn't help.
It would be better to say "an infinite number of monkeys", because then you
don't have the confusion between unbounded and infinite.
Remember that "the limit as N approaches infinity" was originally designed
to calculate what happens *at* infinity, while avoiding the paradoxes.
It seems kind of silly to say "An infinite number of monkeys will hit upon
Shakespeare" and answer that with "no they won't, because there's no such
thing as an infinite number of monkeys." That's like me saying "If I was in
the WTC on the morning of 9/11/01, I'd be dead", and you answering "No you
wouldn't, because you were on the other side of the country that day."
I don't think anyone's arguing that there actually are or even could be an
infinite number of monkeys. :-)
--
Darren New, San Diego CA, USA (PST)
There's no CD like OCD, there's no CD I knoooow!
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> Nope. Probability theory gives zero.
OK, well I will have to bow down to your superior maths knowledge over mine.
I was just going by what I had read about 1/infinity not being strictly
speaking zero (only under certain circumstances) and the fact that the
infinite series seems to be defined as always equal to 1 in mathematics. I
was not aware of anything specific in probability theory that defined the
probability of choosing 1 item from an infinite number as zero. Writing the
probability as 1/infinity rather than zero explained the "never gets chosen"
paradox you mentioned quite nicely, but oh well...
>> 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 :-)
>
> But that *is* what Warp and I are complaining about.
You should first discuss if there is enough time for an infinite number of
finite length events to ever happen :-) Personally I think you have to
treat this a strictly theoretical concept, otherwise you run into all sorts
of other technicalities about whether it can actually happen.
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The important thing to remember is that (as Warp has been trying to point
out), when the probability tends mathematically to 0 due to infinite
element, it neither *requires* nor *absolutely excludes* something from
happening. 0001020304...9596979899 is just as likely a random number as
7531963568...1739942680, and either *can* happen in an infinite sequence,
but neither *has to*.
--
Tim Cook
http://empyrean.freesitespace.net
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Tim Cook wrote:
> The important thing to remember is that (as Warp has been trying to
> point out), when the probability tends mathematically to 0 due to
> infinite element, it neither *requires* nor *absolutely excludes*
> something from happening.
I understand the assertion. Wikipedia (amongst other sources) disagrees.
> infinite sequence, but neither *has to*.
Do you have a citation for this assertion? Because the wikipedia entry on
Normal_numbers disagrees with you. If you have an equal probability for all
1-symbol elements to appear in an infinite string (i.e., if every symbol
appears an infinite number of times) then you have an equal probability of
every subsequence to show up - i.e., all possible combinations show up an
infinite number of times.
If you have an infinite number of trials and the letter 'a' never shows up,
it means it's impossible for the letter 'a' to show up. The probability a
priori of 'a' showing up is zero, because any non-zero number times infinity
is infinity.
--
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:
> Do you have a citation for this assertion? Because the wikipedia entry on
> Normal_numbers disagrees with you. If you have an equal probability for
> all 1-symbol elements to appear in an infinite string (i.e., if every
> symbol appears an infinite number of times) then you have an equal
> probability of every subsequence to show up - i.e., all possible
> combinations show up an infinite number of times.
While the probability is equal, it doesn't mean all possible combinations
necessarily show up an infinite number of times in an arbitrary sequence.
The aforementioned 0102030405..9596979899 is technically a possible random
number with each digit 0-9 having an equal probability of occurring. An
infinite sequence could be considered that fails to meet the 'all possible
combinations' feature.
> If you have an infinite number of trials and the letter 'a' never shows
> up, it means it's impossible for the letter 'a' to show up.
Not really. Randomly picking an infinite amount from the set {a,b} *could*
result in nothing but bbb..bbb. That doesn't mean it's impossible for the
letter 'a' to show up, only that the bbb..bbb sequence isn't very
likely...except it's exactly the same probability as any other sequence of
infinite length: 1/infinity.
--
Tim Cook
http://empyrean.freesitespace.net
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Tim Cook wrote:
> While the probability is equal, it doesn't mean all possible
> combinations necessarily show up an infinite number of times in an
> arbitrary sequence.
Why do you say that?
> The aforementioned 0102030405..9596979899 is
> technically a possible random number with each digit 0-9 having an equal
> probability of occurring. An infinite sequence could be considered that
> fails to meet the 'all possible combinations' feature.
I'm not sure I follow wht this example has to do.
>> If you have an infinite number of trials and the letter 'a' never
>> shows up, it means it's impossible for the letter 'a' to show up.
>
> Not really. Randomly picking an infinite amount from the set {a,b}
> *could* result in nothing but bbb..bbb.
Do you have a cite to support this contention?
> That doesn't mean it's
> impossible for the letter 'a' to show up, only that the bbb..bbb
> sequence isn't very likely...except it's exactly the same probability as
> any other sequence of infinite length: 1/infinity.
That's not my understanding of how the math works. Do you have any citation
as evidence for this? Because if the letter 'a' doesn't show up after an
*infinite* number of trials, you clearly don't have any probability for it
to show up at all, and indeed that's what the math pages I've cited already say.
--
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:
>> While the probability is equal, it doesn't mean all possible combinations
>> necessarily show up an infinite number of times in an arbitrary sequence.
> Why do you say that?
Because 0.50000000... is one possible arbitrary sequence, and nowhere in it
contains the sequence 123.
>>> If you have an infinite number of trials and the letter 'a' never shows
>>> up, it means it's impossible for the letter 'a' to show up.
>> Not really. Randomly picking an infinite amount from the set {a,b}
>> *could* result in nothing but bbb..bbb.
> Do you have a cite to support this contention?
Observe: the number set of decimals from 0 to 1 (inclusive) is infinitely
large, with each element containing an infinite amount of decimal places,
each having an equal probability of being a digit 0-9. This set contains
the number 0.000... QED.
>> That doesn't mean it's impossible for the letter 'a' to show up, only
>> that the bbb..bbb sequence isn't very likely...except it's exactly the
>> same probability as any other sequence of infinite length: 1/infinity.
> That's not my understanding of how the math works. Do you have any
> citation as evidence for this? Because if the letter 'a' doesn't show up
> after an *infinite* number of trials, you clearly don't have any
> probability for it to show up at all, and indeed that's what the math
> pages I've cited already say.
See above. The probability of selecting any particular number is
effectively zero, but that number exists and so *can* be chosen.
--
Tim Cook
http://empyrean.freesitespace.net
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Tim Cook wrote:
> "Darren New" <dne### [at] sanrrcom> wrote:
>>> While the probability is equal, it doesn't mean all possible
>>> combinations necessarily show up an infinite number of times in an
>>> arbitrary sequence.
>> Why do you say that?
>
> Because 0.50000000... is one possible arbitrary sequence, and nowhere in
> it contains the sequence 123.
The probability of a 1 showing up anywhere is zero, so yes, the probability
of 123 showing up anywhere is zero.
> Observe: the number set of decimals from 0 to 1 (inclusive) is
> infinitely large, with each element containing an infinite amount of
> decimal places, each having an equal probability of being a digit 0-9.
Yep.
> This set contains the number 0.000... QED.
And that's why the probability of picking a number like that is zero. :-)
>> That's not my understanding of how the math works. Do you have any
>> citation as evidence for this? Because if the letter 'a' doesn't show
>> up after an *infinite* number of trials, you clearly don't have any
>> probability for it to show up at all, and indeed that's what the math
>> pages I've cited already say.
>
> See above. The probability of selecting any particular number is
> effectively zero, but that number exists and so *can* be chosen.
Sorry, that's not a citation. I.e., I read experts who say you're wrong.
You're trying to convince me using intuitive reasoning that infinity works
in a way the experts say it doesn't work. I see that both arguments are
perfectly reasonable given certain assumptions about how infinity works.
Now, if you find something that explains *why* it's reasonable to pick an
infinite number of random digits and get all zeros, I'll look at it, but
right now we're both just asserting the truth of our own positions. :-)
--
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:
> Now, if you find something that explains *why* it's reasonable to pick an
> infinite number of random digits and get all zeros, I'll look at it, but
> right now we're both just asserting the truth of our own positions. :-)
I never said it was *reasonable* to get all zeroes, just that it's
technically possible. ;)
Besides, there's a third option: we're both right.
--
Tim Cook
http://empyrean.freesitespace.net
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Tim Cook wrote:
> "Darren New" <dne### [at] sanrrcom> wrote:
>> Now, if you find something that explains *why* it's reasonable to pick
>> an infinite number of random digits and get all zeros, I'll look at
>> it, but right now we're both just asserting the truth of our own
>> positions. :-)
>
> I never said it was *reasonable* to get all zeroes, just that it's
> technically possible. ;)
And I'm saying no, it isn't technically possible, unless the probability of
getting any digit other than zero is literally impossible. As I said, I
understand what you're asserting - each number is equally possible, so all
zeros is possible. I'm saying if each number is equally possible and you do
an *infinite* number of trials and don't get an *infinite* number of each
digit, then they're not all equally possible. The only way to make an
infinite number of choices and not get an infinite number of 9's is if 9's
cannot appear, because any non-zero number times infinity is infinity.
I don't personally know which is right, but from what I understand the
experts to be saying, my assertions are right. (I.e., I think I'm saying
what I read the experts to be saying.) That's why I'm asking you if you have
any actual expert descriptions that might shed light on why *you* are right,
beyond your personal logic.
> Besides, there's a third option: we're both right.
I'm not sure I follow how that can be. :-)
--
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:
> And I'm saying no, it isn't technically possible, unless the probability
> of getting any digit other than zero is literally impossible. As I said,
> I understand what you're asserting - each number is equally possible, so
> all zeros is possible. I'm saying if each number is equally possible and
> you do an *infinite* number of trials and don't get an *infinite* number
> of each digit, then they're not all equally possible. The only way to make
> an infinite number of choices and not get an infinite number of 9's is if
> 9's cannot appear, because any non-zero number times infinity is infinity.
That assertion would mean that randomly choosing any terminating decimal or
a decimal that doesn't include all ten digits is impossible. But it's not,
so...yeah.
> I don't personally know which is right, but from what I understand the
> experts to be saying, my assertions are right. (I.e., I think I'm saying
> what I read the experts to be saying.) That's why I'm asking you if you
> have any actual expert descriptions that might shed light on why *you* are
> right, beyond your personal logic.
If my personal logic is insufficiently transparent to need an outside expert
to be determined valid, it's obviously not very good.
>> Besides, there's a third option: we're both right.
> I'm not sure I follow how that can be. :-)
Math is strange that way.
--
Tim Cook
http://empyrean.freesitespace.net
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Tim Cook wrote:
> That assertion would mean that randomly choosing any terminating decimal
> or a decimal that doesn't include all ten digits is impossible. But
> it's not, so...yeah.
At random? Are you sure it's not? Remember, the irrational numbers are
infinitely more populous in any given segment than the rationals. There are
more irrational numbers between any two rational numbers than there are
rational numbers period.
>> I don't personally know which is right, but from what I understand the
>> experts to be saying, my assertions are right. (I.e., I think I'm
>> saying what I read the experts to be saying.) That's why I'm asking
>> you if you have any actual expert descriptions that might shed light
>> on why *you* are right, beyond your personal logic.
>
> If my personal logic is insufficiently transparent to need an outside
> expert to be determined valid, it's obviously not very good.
Well, your personal logic seems just as valid as my personal logic, but they
can't both be right. :-) Therefore, I turn to experts.
--
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:
> At random? Are you sure it's not? Remember, the irrational numbers are
> infinitely more populous in any given segment than the rationals. There
> are more irrational numbers between any two rational numbers than there
> are rational numbers period.
Except there's the conundrum: the amount of rational numbers in any given
segment is *also* infinite!
Boggles the mind, eh?
--
Tim Cook
http://empyrean.freesitespace.net
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Tim Cook wrote:
> That assertion would mean that randomly choosing any terminating decimal
> or a decimal that doesn't include all ten digits is impossible. But
> it's not, so...yeah.
Actually, probability theory states it **is** zero if some digit is
missing. I was going to write a separate post about it providing *some*
rigor, but am quite busy - hopefully some time this week.
(Essentially, if you take the interval from 0 to 1, write the decimal
expansion, and ask "what is the probability to get *any* number that
doesn't have the digit 4 in it", the answer is 0. The set of all numbers
in [0,1] that don't have the digit 4 anywhere in their expansion is a
set of measure zero, and thus the integral is 0).
--
Wear short sleeves! Support your right to bare arms!
/\ /\ /\ /
/ \/ \ u e e n / \/ a w a z
>>>>>>mue### [at] nawazorg<<<<<<
anl
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Tim Cook wrote:
> "Darren New" <dne### [at] sanrrcom> wrote:
>> At random? Are you sure it's not? Remember, the irrational numbers are
>> infinitely more populous in any given segment than the rationals.
>> There are more irrational numbers between any two rational numbers
>> than there are rational numbers period.
>
> Except there's the conundrum: the amount of rational numbers in any
> given segment is *also* infinite!
'Tis a bigger infinity.<G>
--
Wear short sleeves! Support your right to bare arms!
/\ /\ /\ /
/ \/ \ u e e n / \/ a w a z
>>>>>>mue### [at] nawazorg<<<<<<
anl
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