Suppose $K/\mathbb{Q}$ is a finite extension (degree $n$ say). Choose a prime $\Lambda$ of $K$ lying above $p$ and suppose $K_{\Lambda}/\mathbb{Q}_p$ has ramification index $e$.
How "likely" is it that $p>e+1$ holds?
Eventually I am going to want to choose $p$ to satisfy this condition as well as not lying in certain arithmetic progressions.