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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] san rrcom> 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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