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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] san rrcom> 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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