Can probabilities be calculated using cardinality? The probability that a positive natural number drawn at random is odd, is $\frac{1}{2}$.
Is there a way of calculating this fact using the cardinalities of the sets?  Or is there some measure of cardinality, or of the number of elements within each set, which captures this fact?
If we say the sample space is $\mathbb{N}$ and the successful outcomes are $O=\{2n-1:n\in\mathbb{N}\}$ then we might (obviously incorrectly) infer from the cardinalities of the sets that:
$(\lvert\mathbb{N}\rvert=\lvert O\rvert)\implies P(n$ is odd$)=1$
It would seem to me that the closest we can come is to say the densities of $O$ and $\mathbb{N}\cap O$ in $\mathbb{N}$ are equal to each other and make our deduction by that means.  Is that the normal approach within set theory?

No, there isn't any way to do it using the cardinality of the set. The cardinality of the set of odd numbers, the cardinality of the set of prime numbers, and the cardinality of the set of natural numbers are all the same: $\aleph_0$. But obviously the probability of choosing an odd number, a prime number, or a number at all are not the same - respectively, they should be $\frac{1}{2}$, $0$, and $1$. Within the field of set theory, then, there really isn't a way to calculate the density.
You can define probability using limiting density: take $\lim_{n \to \infty}\frac{|X \cap n|}{n}$, where $X \cap n$ is the set of members of $X$ less than $n$. This limit will give the correct probabilities for everything I noted above. However, I wouldn't call this a set-theoretic approach - this is more number-theoretic in character, because it depends heavily on the order of the natural numbers.
