I’ve never seen “asymptomatic density” referred to as “asymptomatic probability.”

Anyways, if anyone is interested is a more complete survey of similarly flavored computability/density results, this paper is good: https://faculty.math.illinois.edu/~jockusch/DensityCE_JML.pd...

There is even some pure combinatorics involved, for example I remember lemma 6.7 from my combinatorics class back in undergrad