Re: Calculus XOR Probability

Tony Orlow wrote:

It's only zero in standard finite analysis, but where you are talking about an infinite set of possibilities, you are outside of standard territory, and the answer is nonstandard: each possiblity has an infinitesimal probability, and that infinity of infinitesimal values sums to 1 as expected. So, n*1/n=1 holds in the infinite case, and there is no reason to expect it to fail. Is there?

You did a whole lot of ground work for me, Tony. And I appreciate that
very much! When I was searching for Chas' staircase problem, yesterday
evening, I encountered an old little booklet, which is titled:

Riddles in Mathematics - A book of Paradoxes, by Eugene Purdy Northrop,
Penguin Books, 1944. [ http://isbn.nu/0442060785/ ]

The staircase problem is found on page 135. And, as _you_ have correctly
concluded, the limiting case _appears_ to be a straight line with length
equal sqrt(2). But, in fact, it still is a staircase (fractal) line with
length equal to 2. Thus there is no "jump" from (finite) 2 to (infinite)
sqrt(2), as has been suggested by Chas Brown. The argument is off-topic.

Yeah, sure. Like I said, it's not the diagonal line, even though it starts to look like it. You're talking about a fractal dimension on the line, basically. What does that have to do with probability? It's just an attempt to change the subject into something ridiculous and trying to say it's the same argument.

But, very much to my surprise, I found another chapter in the same book,
titled "Paradoxes in Probability". On page 169, we read the following:

At the risk of confusing the issue rather than clarifying it, let us
look at a different, but more concrete, example. Suppose that a box
contains 1 red marble and 9 white marbles, and that a single marble is
to be drawn. Then the probability of drawing the red marble is 1/10. If
we increase the number of white marbles to 99, the probability is 1/100.
If we increase the number of white marbles to 999, the probability is
1/1000. And so on. As we go on adding white marbles, the probability of
drawing the single red one becomes smaller and smaller, and we can make
it as small as we please by adding a sufficient number of white marbles.
But the probability of drawing the red marble is never zero - the red
marble is always there, and there is some chance, however small, that
it will be drawn.

Here comes (HdB):

In a word, we must distinguish in our minds between 'zero' on the one
hand and 'infinitely small', or 'infinitesimal', on the other. We can
say that although the desired probability is, for all practical purposes
, zero, it is, theoretical speaking, not zero but infinitesimal. This
same distinction must be made whenever the number of favourable cases is
finite and the number of possible cases is infinite.

Paradox 1. Since all even numbers are divisble by 2, the only even prime
number is 2 itself. That is to say, the number of even primes is 1. But
the total number of primes is infinite. Therefore the probability that
an arbitrary prime number is even is zero. This conclusion implies that
it is impossible for a prime number to be even. Consequently the prime
number 2 does not exist.

End of quotes.

Han de Bruijn

.

Relevant Pages

  • Re: Calculus XOR Probability
    ... If one of the elements is to be chosen from the set, then the probability of one being chosen is 1. ... then that 1 representing the fact that one of those will be chosen is defined to be the sum of the probabilities of each. ... If you can establish a uniform probability distribution, then you can say each possibility has the same probability. ... If we are discussing an infinite set of possiblities, and I define infinitesimal in terms of that infinity, it's not circular. ...
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  • Re: Calculus XOR Probability
    ... the sum of a countable number of 0's; he simply "wants" it to be 1. ... then the probability of one being chosen is 1. ... the question is why the standard system can't accomodate the infinite case, ... The probability of each is 0 in standard analysis, ...
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    ... therefore there must exist a uniform distribution on the naturals. ... If you use a 2 element set, you still get a total probability of 1. ... the sum of a countable number of 0's; he simply "wants" it to be 1. ... the question is why the standard system can't accomodate the infinite case, ...
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  • Re: Calculus XOR Probability
    ... in the infinite case, and there is no reason to expect it to fail. ... What does that have to do with probability? ... we increase the number of white marbles to 99, ... But the probability of drawing the red marble is never zero - the red ...
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  • Re: Calculus XOR Probability
    ... If one of the elements is to be chosen from the set, then the probability of one being chosen is 1. ... then that 1 representing the fact that one of those will be chosen is defined to be the sum of the probabilities of each. ... If you can establish a uniform probability distribution, then you can say each possibility has the same probability. ... If we are discussing an infinite set of possiblities, and I define infinitesimal in terms of that infinity, it's not circular. ...
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