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From: Kory Heath <kory.heath.domain.name.hidden>

Date: Wed, 21 Jan 2004 07:30:17 -0500

At 1/21/04, David Barrett-Lennard wrote:

*>Saying that the probability that a given integer is even is 0.5 seems
*

*>intuitively to me and can be made precise (see my last post).
*

We can say with precision that a certain sequence of rational numbers

(generated by looking at larger and larger finite sets of integers from 0 -

n) converges to 0.5. What we can't say with precision is that this result

means that "the probability that a given integer is even is 0.5". I don't

think it's even coherent to talk about "the probability of a given

integer". What could that mean? "Pick a random integer between 0 and

infinity"? As Jesse recently pointed out, it's not clear that this idea is

even coherent.

*>For me, there *is* an intuitive reason why the probability that an
*

*>integer is a perfect square is zero. It simply relates to the fact that
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*>the squares become ever more sparse, and in the limit they become so
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*>sparse that the chance of finding a perfect square approaches zero.
*

Once again, I fully agree that, given the natural ordering of the integers,

the perfect squares become ever more sparse. What isn't clear to me is that

this sparseness has any affect on "the probability that a given integer is

a perfect square". Your conclusion implies: "Pick a random integer between

0 and infinity. The probability that it's a perfect square is zero." That

seems flatly paradoxical to me. If the probability of choosing "25" is

zero, then surely the probability of choosing "24", or any other specified

integer, is also zero. A more intuitive answer would be that the

probability of choosing any pre-specified integer is "infinitesimal" (also

a notoriously knotty concept), but that's not the result your method is

providing. Your method is saying that the chances of choosing *any* perfect

square is exactly zero. Maybe there are other possible diagnoses for this

problem, but my diagnosis is that there's something wrong with the idea of

picking a random integer from the set of all possible integers.

Here's another angle on it. Consider the following sequence of integers:

0, 1, 2, 4, 3, 9, 5, 16, 6, 25 ...

Here we have the perfect squares interleaved with the non perfect-squares.

In the limit, this represents the exact same set of integers that we've

been talking about all along - every integer appears once and only once in

this sequence. Yet, following your logic, we can prove that the probability

that a given integer from this set is a perfect square is 0.5. Can't we?

-- Kory

Received on Wed Jan 21 2004 - 07:35:50 PST

Date: Wed, 21 Jan 2004 07:30:17 -0500

At 1/21/04, David Barrett-Lennard wrote:

We can say with precision that a certain sequence of rational numbers

(generated by looking at larger and larger finite sets of integers from 0 -

n) converges to 0.5. What we can't say with precision is that this result

means that "the probability that a given integer is even is 0.5". I don't

think it's even coherent to talk about "the probability of a given

integer". What could that mean? "Pick a random integer between 0 and

infinity"? As Jesse recently pointed out, it's not clear that this idea is

even coherent.

Once again, I fully agree that, given the natural ordering of the integers,

the perfect squares become ever more sparse. What isn't clear to me is that

this sparseness has any affect on "the probability that a given integer is

a perfect square". Your conclusion implies: "Pick a random integer between

0 and infinity. The probability that it's a perfect square is zero." That

seems flatly paradoxical to me. If the probability of choosing "25" is

zero, then surely the probability of choosing "24", or any other specified

integer, is also zero. A more intuitive answer would be that the

probability of choosing any pre-specified integer is "infinitesimal" (also

a notoriously knotty concept), but that's not the result your method is

providing. Your method is saying that the chances of choosing *any* perfect

square is exactly zero. Maybe there are other possible diagnoses for this

problem, but my diagnosis is that there's something wrong with the idea of

picking a random integer from the set of all possible integers.

Here's another angle on it. Consider the following sequence of integers:

0, 1, 2, 4, 3, 9, 5, 16, 6, 25 ...

Here we have the perfect squares interleaved with the non perfect-squares.

In the limit, this represents the exact same set of integers that we've

been talking about all along - every integer appears once and only once in

this sequence. Yet, following your logic, we can prove that the probability

that a given integer from this set is a perfect square is 0.5. Can't we?

-- Kory

Received on Wed Jan 21 2004 - 07:35:50 PST

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