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

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



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

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


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