I'm a bit sheepish asking this since I know close to nil on uncomputability. I ask mainly due to Chaitin's constant, which to my knowledge, is the probability that any given program will halt or not, though I'm not sure if the logic behind the undecidability of the halting problem is similar to that of, for example, the continuum hypothesis (independence from ZFC implying unanswerability).
If an un(dis)provable statement could be written in the form:
$\text {There always exists some A such that B is satisfied}$
Is the probability that any given A satisfies B uncomputable
Small note: initially my thoughts were to let P be the probability that for any set S chosen from the set of all sets with cardinality less $\aleph_2$,
$\aleph_0 <|S|<\aleph_1$,
which is how I arrived at the above idea
edit: I'll narrow it down further to let A be a real in the interval $[0,1]$ and 'any given A' be a real from $[0,1]$
The Chaitin's Omega story is a rather special one, and not something associated with non-computability in general. It was already discussed in the comments that only few mathematical structures come with a natural probability measure on them, but even for $[0,1]$ we don't get lucky. As $[0,1]$ is connected, its only decidable subsets are $\emptyset$ and $[0,1]$ itself, so sets of measure other than $0$ and $1$ are always undecidable.
Fixing that issue, we can consider $2^\omega$ with the fair coin toss measure instead. But we can still easily built undecidable sets of any desired measure, as messing around with a measure 0 set suffices for undecidability.