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

$
\frac{ \#\{all CM number fields of degree 2g}}
{\#\{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 

$lim_{d \to \infty}( \frac
{\#{all CM fields of degree 2g and d_{K} \le d\}}
{\#\{CM fields of degree 2g, such that p splits completely  and d_{K} \le d \}})$


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