Re: Probability in an infinite sample space

Han de Bruijn wrote:
Gerry Myerson wrote:

Excellent. I take it we can generalize, and agree that the probability of picking 39 is also zero. So are the probabilities of picking 36, or 33, or ... or 3 - or 45, or 48, or 51, or ... well, or any other specified multiple of 3.
So how come when you add up all those zeros, you get one-third?
This is the standard, but quite vulnerable, argument of any mainstream
mathematician.

Standard, and quite correct.

The solution is that the probabilities of picking these numbers are not
zero, but infinitesimals. Given an infinity of them, they CAN therefore
add up to one-third. I've made some attempts in the thread Calculus XOR
Probability to establish this on a more rigorous footing.


And had it explained to you in great and hideous detail why
it doesn't work. You *cannot* assign uniform infinitesimal probabilities
to the integers in such a way that the probability of choosing
a multiple of three is one-third. You can't assign infinitesimal
probabilities to the integers in *any way* so that the probability
of choosing integers from any particular set is a finite number:
it is always either infinitesimal or undefined.
.

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