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.