Re: Drawing Randomly from an Infinite Set
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It is impossible to "draw randomly" from an infinite set of numbers, if "randomly" means that each number has the same probability (uniform distribution). This is because the sum of the probabilities of all the numbers must equal 1, and there is no probability p that we can assign each number such that the sum of an infinite number of these probabilities is equal to 1. That is, p would have to be smaller than any positive real number, or else the sum of the probabilities would diverge to infinity. It could not be zero, because then the infinite sum of zeros would be zero, not 1 (by definition since the limit of partial sums is zero).
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It is possible to draw randomly from an infinite set. If you look at an atom (numbers) at the time it will have a probility of 0. But if you look at a group of atoms (number) they could have a probility greater then zero. Take for instance uniform measure on [0,1]. Their is infintly many real number but P(x>0.5)=0.5.
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