Is the probability of observing a specific event in a countably infinite set of events over countably infinte samples 1? I saw a question on Reddit that was along the lines of this:
"If I randomly generate an infinite number of integers, what's the probability I see some specific integer $n$"
An answer there said the question did not make sense, since there is no uniform distribution for all the integers (makes sense to me).
Which has me wondering, if you define a distribution over the integers, whatever it may be such that the probabilities over all integers sum to one as required, and take a countably infinite number of samples from that distribution, is the probability of observing a specific integer in that countably infinite sample then 1?
Does asking such a question even make sense, and if does and the answer is not 1, what might it be?
 A: Certainly not. Take the distribution given by $\delta(x)$, where $\delta$ is the Kronecker delta function: 1 when the input is zero and 0 otherwise. Then the probability of choosing any positive number is exactly zero.
However, we can say something more interesting: the probability of seeing any particular number is either 0 or 1, no matter what distribution we choose. This can be computed via elementary means (as in Carmeister's answer), or we can bash the question open with a sledgehammer by noting that the event you describe is a tail event, and therefore the Kolmogorov Zero-One Law kicks in and provides the answer.
A: As other answers have mentioned, it's possible that your distribution has a $0$ chance of picking some number, in which case of course you will never get that number no matter how many numbers you pick.
But if every number has a positive probability in the distribution, then every number will eventually be chosen, if you take enough samples:
For any number $m$, let $p_m$ be the probability of choosing that number. Then after $N$ samples, the chance that you have never chosen $m$ is equal to $(1-p_m)^N$. Since $p_m>0$, this quantity goes to $0$ as $N$ gets large - so eventually, you'll find an $m$.
A: For clarity's sake, note that we can still have a distribution on $\mathbb{N}$ that is non-zero for each $n \in \mathbb{N}$ -- that is, our distribution doesn't have to be finite. For example, the distribution where $P(n) = \frac{1}{2^n}$ ($n = 1, 2, \dots$) works. We just can't define a uniform distribution over $\mathbb{N}$.
A: In general, this may not be true. As a trivial example, consider the distribution on Natural numbers such that probability of getting $100$ is $1$ and all other numbers having zero probability. Now, its clear that the probability of getting any number apart from $100$ is zero if we randomly generate infinite number of integers with this distribution.
A: I'm not clear what you are asking.  You first note that there cannot be a uniform probability distribution over the counting numbers so I thought your question would be whether it is possible to have a probability distribution where every integer has non-zero probability of being chosen. It is not too difficult to show that $\sum_{n=1}^\infty \frac{1}{n^2}= \frac{\pi^2}{6}$ so $\sum_{n=1}^\infty \frac{6}{\pi^2n^2}= 1$.  But then you ask about the probability of a specific number being 1.  I would think it obvious that, if the probability of a specific number is 1, the probability of any other number must be 0!  That is, such a probability would be, say, $P(3)= 1, P(n)= 0$ if $n$ is not 3.
