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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&gt;0.5)=0.5.

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From an uncountable infinite set (like [0,1]), yes. But, from an countable infinite set (like the set of itnegers, which the original question was about):

The Kolmogorov probability axioms state that the probability of a union of countably many disjoint sets is the sum of the probabilities of those sets. In a countable infinite set the entire set is a countable union of all singletons. Thus the probability of the entire set is zero (or infinite if the single events have probability a&gt;0), which is a contradiction as the probability of the entire set must be one.


{This assumes using Kolmogorov's probability axiomatization, maybe probability can be defined in some other way to make uniform distributions in countable infinite sets possible, but I'm not familiar with any other definitions. }