Let $p$ be a fixed prime number. Roughly speaking, I am interested in the following ratio

$$ \frac{ |\{\text{ all CM number fields of degree }2g}|} { |\{\text{CM fields of degree 2g, such that p splits completely in K}\}|} $$

A possible definition could be the following: let $d_{K}$ be the discriminant of K, then we can define this ratio as

$$ \lim_{d \to \infty} \frac {|\{ \text{all CM fields of degree 2g and}\ d_{K} \le d\}|} {|\{ \text{CM fields of degree 2g such that p splits completely and}\ d_{K} \le d \}|}.$$

Was it studied by anyone? I would appreciate any reference.