More than one "zero"?
Phil Freedenberg
THe same is true if I ask you to pick a real number at random, or a point in N-dimensional space, etc.
The reason that all these probabilities are zero is that there is an infinite sample space in each case. But the various infinities are not equal, not even comparable, so why don't the a priori probabilities reflect this?
Phil
pubkeybreaker
> IF I ask you to pick an integer at random, the probability that I can guess your selection is zero.
False. It depends on the density function that you
specify. There is no uniform density function for
the entire set of integers. It does not exist.
It is impossible to 'pick an integer at random'
in the sense that you mean.
Dave L. Renfro
>> IF I ask you to pick an integer at random, the probability
>> that I can guess your selection is zero.
pubkeybreaker wrote:
> False. It depends on the density function that you
> specify. There is no uniform density function for
> the entire set of integers. It does not exist.
> It is impossible to 'pick an integer at random'
> in the sense that you mean.
Perhaps, but you can also model the situation Phil
is talking about in a limiting frequency manner.
The "probability" [if it exists] of a subset B
of positive integers is the limit [if it exists]
as n --> infinity of card(B intersect {1, 2, ..., n})
divided by n. In this sense, the probability of
{7} is 0. An appropriate 2-variable version of
this interpretation is what lies behind the often
seen result that the probability two randomly chosen
positive integers are relatively prime is 6 / pi^2.
Dave L. Renfro
Tim Little
> IF I ask you to pick an integer at random, the probability that I
> can guess your selection is zero.
Actually there is no way at all that you can pick an integer at random
for which my probability of guessing it is zero - even if I tell you
my guessing strategy. The same is true of rationals or algebraic
numbers, or even computable reals. Any countable set. You can make
the probability as small as you like, but not zero.
> The same is true if I ask you to pick a real number at random, or a
> point in N-dimensional space, etc.
Yes, it can be zero in the case of real numbers and tuples of real
numbers.
- Tim
William Elliot
On Tue, 10 Mar 2009, Phil Freedenberg wrote:
> IF I ask you to pick an integer at random, the probability that I can
> guess your selection is zero.
> THe same is true if I ask you to pick a real number at random,
Except that it's more zero than before.
Or do I mean it's much less zero than before? :-)
sri
wrote:
Multiple zeros are as meaningless (or meaningful) as multiple
infinities. I'm sure the math community will soon decide to kill some
more time in the joy of mysticism and building the artificial math
rigor around multiple zeros as well.
-venkat
David C. Ullrich
<renf...@cmich.edu> wrote:
>Phil Freedenberg wrote:
>
>>> IF I ask you to pick an integer at random, the probability
>>> that I can guess your selection is zero.
>
>pubkeybreaker wrote:
>
>> False. It depends on the density function that you
>> specify. There is no uniform density function for
>> the entire set of integers. It does not exist.
>> It is impossible to 'pick an integer at random'
>> in the sense that you mean.
>
>Perhaps, but you can also model the situation Phil
>is talking about in a limiting frequency manner.
>The "probability" [if it exists] of a subset B
>of positive integers is the limit [if it exists]
>as n --> infinity of card(B intersect {1, 2, ..., n})
>divided by n.
That's a reasonable and common substitute for the
undefined notion of "probability that a random
integer lies in B".
But I don't see how it makes sense of the phrase
"pick an integer at random". Exactly what are
you saying I should do to pick that integer?
>In this sense, the probability of
>{7} is 0. An appropriate 2-variable version of
>this interpretation is what lies behind the often
>seen result that the probability two randomly chosen
>positive integers are relatively prime is 6 / pi^2.
>
>Dave L. Renfro
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
amy666
well actually no.
thats the irony.
they use different kinds of infinities.
but only one inverse : 0.
i agree with the OP.
the probability of guessing a real is smaller than that of guessing an integer.
because P ( guess integer ) = 1 / aleph_0
and P ( guess real ) = 1 / aleph_1
i adressed this too.
the idea is not new , but not accepted.
basicly it is said probability 0 does not mean it cannot happen.
perhaps a better way - in some contexts , not in general ! ( for pratical reasons ) - is to just leave the limit :
P = 1/N or P = 1/R
however 2N * 1/N =/= 2 etc
so its a cardinal like number anyway ...
what i have suggested in the past was :
aleph_-1
aleph_-2
regards
tommy1729
Dave L. Renfro
>> Perhaps, but you can also model the situation Phil
>> is talking about in a limiting frequency manner.
>> The "probability" [if it exists] of a subset B
>> of positive integers is the limit [if it exists]
>> as n --> infinity of card(B intersect {1, 2, ..., n})
>> divided by n.
David C. Ullrich wrote:
> That's a reasonable and common substitute for the
> undefined notion of "probability that a random
> integer lies in B".
>
> But I don't see how it makes sense of the phrase
> "pick an integer at random". Exactly what are
> you saying I should do to pick that integer?
Yep, there's a difference between these two ideas
that I overlooked.
Dave L. Renfro
Denis Feldmann
Have a look at surreal numbers (where those exists) to see what is a
real *suggestion*...
Then, have a look at proba theory, to see why this particullat
suggestion would have no real interest...
>
> regards
>
> tommy1729
Richard Tobin
Tim Little <t...@little-possums.net> wrote:
>Actually there is no way at all that you can pick an integer at random
>for which my probability of guessing it is zero - even if I tell you
>my guessing strategy.
Well, that depends on your guessing strategy. If your strategy is
to guess 2, then choosing 3 will have the desired effect.
-- Richard
--
Please remember to mention me / in tapes you leave behind.
Tim Little
> Well, that depends on your guessing strategy. If your strategy is
> to guess 2, then choosing 3 will have the desired effect.
Any guessing stategy that assigns a nonzero probability to each
integer will suffice. E.g. guess zero with probability 1/2 and guess
z != 0 with probability 2^-(|z|+2).
- Tim
Phil Freedenberg
William Elliot
> What if you pick a real number and then truncate it to an integer?
>
You get an integer.
Dave Seaman
> What if you pick a real number and then truncate it to an integer?
What probability distribution do you have in mind? There is no uniform
probability distribution on R, but there is a uniform distribution on
[0,1], for example.
--
Dave Seaman
Third Circuit ignores precedent in Mumia Abu-Jamal ruling.
<http://www.indybay.org/newsitems/2008/03/29/18489281.php>
amy666
i am aware of that.
in fact , you know i am.
your just pretending im not here.
> >
> > regards
> >
> > tommy1729
Tim Little
> What if you pick a real number and then truncate it to an integer?
Then at least some integers have a nonzero probability of being
picked. After all, for each one there's a whole (uncountably
infinite) interval of reals all giving the same integer.
- Tim