I was thinking about how probability is used in heuristic arguments, an example being the argument that there are an infinite number of twin primes: the probability that $n$ is the first of two twin primes is about $\frac{1}{(\log n)^2}$, and $\prod{(1-\frac{1}{(\log n)^2}}) \rightarrow 0$, so the probability that there are an infinite number of twin primes is $1$. (Another example provided by @joriki is the heuristic for the Collatz conjecture.) I then wondered if there are any heuristic arguments yielding probabilities strictly between $0$ and $1$ and I considered this example predicate:
$A(x,n)$ := "the $n^{th}$ bit of the binary expansion of $x$ is $1$".
Given certain assumptions there is a sense in which $A(\pi, n)$ is "true with probability $\frac{1}{2}$". Now this isn't a very useful notion, since it is either true or not, and we can find out by computing the $n^{th}$ bit of $\pi$, although perhaps there is some utility if $n$ is very large. However it is a second-class sort of heuristic because it only gives a hint to an answer we could find definitively with more work. But next I thought of this example:
$B(x,n)$ := "the binary expansion of $x$ starting from offset $n$, when interpreted as an Iota program, halts".
In a similar way we can lazily argue that $B(\pi, n)$ is "true with probability $\Omega_{\mathrm{Iota}}$". But in this case, under similar assumptions, there are values of $n$ for which $B(\pi, n)$ is not even decidable: we can construct a Gödel-sentence in our working theory, write an Iota program which searches for its proof, find that program in the binary expansion of $\pi$, and then construct $B(\pi, n)$ using its offset $n$. For that reason, it seems to be a first-class sort of heuristic (despite giving us a probability strictly between $0$ and $1$), in the same league as those that give us almost-certain conclusions to open problems.
One of the unstated assumptions (along with the 2-normality of $\pi$ and such) is that there is no "conspiracy" between the bits of $\pi$ and the semantics of Iota. It is equally believable that no such conspiracy exists between $\pi$ and $e$, or between $e$ and Iota, or amongst all three. And since the probability of the conjunction of independent events is equal to the product of their respective probabilities, it is just as reasonable to say that $B(\pi,n) \wedge B(e,n)$ is "true with probability $\Omega_{\mathrm{Iota}}^2$". On the other hand, the same doesn't apply to $B(\pi,n) \wedge B(\pi,n+1)$: since the programs substantially overlap we should expect some correlation in their halting status and therefore a different "truth probability" for the conjunction.
So here are my questions: Are there other (hopefully more natural) examples of problems (open or not) with simple heuristic arguments suggesting a particular probability of truth (other than $0$ or $1$)? Are my (admittedly vague) arguments above at least vaguely correct or are there major mistakes and conceptual problems? What is a good way of describing the dependencies between different heuristics? Can the idea of quantum probability amplitude be applied here? What are some other examples of heuristics that are not independent (e.g. two statements that are both heuristically true but contradict each other)? Any references to related topics that might help me further develop or discard this idea? I have read overviews of fuzzy logic and probabilistic logic but I don't know how to apply either here.