Accepted! Now let's derive almost any "probability" from almost any deterministic variable through arbitrarily bad sampling methods...
Choose a goal probability B in the range (0,1). It can't be at 0 or 1, but can be arbitrarily close.
Now let's start sampling, and we'll take this all the way to the limit. For any discrete sample size as sample size goes to infinity, let's choose some positive/negative outcomes such that our probability approaches B. Remember that because our goal probability is greater than 0 and less than 1, there are infinite positive/negative outcomes to choose from.
This distribution has some interesting properties. The derived probability is B. The sample will grow to include all points. There are infinitely many functions to generate this outcome.
Right! That does half answer my request :) Those crazy infinities,right? :)
The other half is where I said "reasonable", like the three approaches from the article are--in my subjective opinion--quite reasonable. Picking samples from an uncountable (right?) infinity of options explicitly to satisfy the desired probability, somewhat less subjectively, is not :)
So yes, your method allows for B arbitrarily close to 0 or 1 (or anywhere in between), and it's clever.
But let's not go for "arbitrarily close", but just "very close", or "closer". I wonder if there's (subjectively) reasonable ways of sampling chords from a circle, like you could explain with a realistic(ish) scenario for a ninja pizza slicerator or an erratic mermaid, that give a probability even less than 1/4th or more than a half?
So if we want to get pedantic, "random chord" is undefined here, which means that all the methods presented in the article as well as my sampling methods, and even the 0 and 1 case are all technically correct probabilities.
For example one of the most relatable alternative probabilities for me are, now thinking about it, actually the 0 or 1 cases: spinning a compass will generate a randomly oriented chord inside the face. What is the probability that the chord is of length > sqrt3? 1. Is this result informative or useful? no.
The result is that there are different generating functions with different biases which may or may not be useful. The three sources of randomness in the article correspond to physical stories that are possibly useful, and as such relatable. Otherwise they are no more legitimate than my absurdist examples.
Choose a goal probability B in the range (0,1). It can't be at 0 or 1, but can be arbitrarily close.
Now let's start sampling, and we'll take this all the way to the limit. For any discrete sample size as sample size goes to infinity, let's choose some positive/negative outcomes such that our probability approaches B. Remember that because our goal probability is greater than 0 and less than 1, there are infinite positive/negative outcomes to choose from.
This distribution has some interesting properties. The derived probability is B. The sample will grow to include all points. There are infinitely many functions to generate this outcome.