probability on countable infinite sets My question relates to probabilities on countable infinite sets.  For example, what is the probability of choosing an even number from the positive integers.  Believe it or not I am interested in this question from a practical standpoint.  I am writing a paper on Boltzmann's brains (Brains that occur spontaneously from the vacuum in De Sitter space). Specifically the problem is, is it more likely that I am a Boltzmann brain or a regular brain if space is infinite.  There are of course no good calculations about what the odds of a Boltzmann brain identical to mine are vs. the odds of an identical brain evolving on a twin earth (my money would be that a copy on a twin earth is much more likely).  But would the specific odds matter if space is infinite and there are an infinite number of both, regardless of which is more common?  I know the answer would of course depend on your philosophy of probability (frequency, Bayesian, etc.) so I would be interested in the answer for each of the main theories in probability.
In conventional countably additive probability, in order to assign a probability to the set of even positive integers, you would need to have a probability assigned to each integer, and then you add up the ones for the even integers.

You can't assign equal probabilities to all positive integers, since then either they're all $0$ and their sum is $0$, whereas the sum should be $1$, or they're all some positive number as the sum is $\infty$, whereas it should be $1$.

However, there is also the idea of letting the probability of each set $A\subseteq\{1,2,3,\ldots\}$ be the "density" of $A$, which is
$$
\lim_{n\to\infty} \frac{|A\cap\{1,\ldots,n\}|} n.
$$
If you do that, then probabilities are finitely additive but not countably additive, and then not all of the theorems of conventional probability hold.
What do you think of the answer of Anubhav Balodhi here ? I have copied/pasted it in the following. He proposes a simple mathematical proof that leads to 0.5 probability to pick an even number among all natural numbers. His answer makes much more common sense than getting a 1/3 probability to pick an even number...