Re: Calculus XOR Probability

Tony Orlow wrote:
cbrown@xxxxxxxxxxxxxxxxx said:
Han de Bruijn wrote:
cbrown@xxxxxxxxxxxxxxxxx wrote:

What does surprise me is that you don't see the obvious parallel
between my reasoning that since lim (n*(2/n))=2, therefore the diagonal
has length 2; and your reasoning that since the lim (n*(1/n))=1, that
therefore there must exist a uniform distribution on the naturals.

Since you seem to be in desperate need of a nonsense argument, I'll give
you one.

<snip amusing silliness>


End of nonsense argument.

But, according to Randy Poe and Robert Low, it is _valid_, because as
a physicist, I "meant" to say something different ...


What does your reply have to do with lim (n*(2/n)) = 2 implies the
length of the diagonal = 2, or that lim (n*1/n)=1 implies a uniform
distribution over the naturals?

Can you explain why the former reasoning is incorrect, and yet the
latter is correct?

Cheers - Chas



Hi Chas - Que Pasa?

It's an interesting example of faulty logic you gave. Whyever doesn't it work?

It's a great way to start understanding the vaue of analysis.


If you approximate with 1 step, you get a length of 2, but it's definitely not
the diagonal.

If you approximate a uniform distribution on the naturals with a 1
element set, you get a total probability of 1; but it's definitely not
a uniform distributionon the naturals.

If you use 2 steps, you still get 2.

If you use a 2 element set, you still get a total probability of 1.

As you use more and more
steps, it looks more and more like a diagonal line, but the length stays the
same as when it did not.

As you use more and more elements, it "looks" more and more like a
uniform distribution on the naturals, but the total probability stays
the same as when it did not (sic; i.e., whatever that means).

You are not approximating the length of the diagonal. That sum is never
equal nor getting closer to the answer.

You are not approximating a uniform distribution on the naturals. That
sum (1) is never equal to or getting closer to "the answer" (which is
either 0, since the sum of a countable number of 0's is 0; or infinite
by the archimedean property of the reals).

So, the question here
is, why would you think it gives a correct answer at oo, if it gives an equally
incorrect answer for all finite nnumber of steps? The limit of the error is not
0.

So, the question here is, why would you think it gives a correct answer
at oo, if it gives an equally incorrect answer for all finite number of
steps? The limit of the error is not 0.


If you have n possibilities all mutually exclusive and equally likely, and one
of them must occur, then the chances that any given one will occur is 1/n, so
that the probabilities of all will sum to 1, as expected.

If you have n steps in the diagonal, the length of each tread/riser is
2/n, so that the total length will sum to 2, as expected.

As n increases
without bund, this relationship is preserved and functions without issue, given
expected results.

As n increases without bund (sic), this relationship is preserved and
functions without issue, given expected results.

So, the question here is, why would you expect this
relationship between n 1/n's summing to 1 to break down at n=oo?

So the question here is, why would you expect this relationship between
n 2/n's summing to 2 to break down at n=oo?

Cheers - Chas

.

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