A probability question on infinite sets

Question

Suppose all members of a countably infinite set of people each pick an integer "at random" (I want to avoid uniform distributions for the obvious reasons). (I hope this setting makes enough sense for my question to have meaning; I would certainly appreciate constructive comments on this matter.)

Then what is the probability that at least one integer has been picked an infinite number of times ?

EDIT: my sentence "I want to avoid uniform distributions for the obvious reasons" was not understood by some, so I will formulate a more precise version as well which is less confusing (I hope). For this question, I will only assume finite additivity for probabilities (and not Kolgomorov's countable additivity), so that one can work with uniform distributions on countable infinite sets, which is what we do here. (I will call these generalized probabilities "f-probabilities.")

QUESTION: What is the f-probability that at least one integer has been picked an infinite number of times ?

Answer 1

If the probabilities are finitely additive then the uniform distribution on a countably infinite set $X$ is trivial. Namely, suppose that each $x\in X$ has a probability $p(x)=p$ to be chosen. Since for each finite subset $F$ of $X$ we have $1\ge \sum_{x\in F} p(x)= \sum_{x\in F} p=p|F|$, we have $p=0$.

Answer 2

In a first step, let us assume that each person selects a number $i$ with same probability $p_i$ ($\sum p_i = 1$).

A simple example:

$$p_i=2^{-i}$$

considering here only strictly positive integers.

Then for $N$ persons, the number $i$ will as an average be selected $N\,p_i$ times, assuming selections are done independently.

If $p_i>0$, the average number of times $i$ is selected tends to infinity, when $N$ tends to infinity.

With the above example, all numbers are selected an infinite number of times.

If we assume a different selection probability law for each person, then we cannot conclude. For example, if $i^{th}$ person selects number $i$, no number is selected an infinity number of times.