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There should be infinitely many "solutions" to this if you are allowed to define the sampling method arbitrarily. Intuitively, one would think that random poin

There should be infinitely many "solutions" to this if you are allowed to define the sampling method arbitrarily. Intuitively, one would think that random points are uniformly distributed, but that is not maintained in two of these solutions. If you randomly generate two points in [0,2pi) with a uniform distribution then you will quickly get the 1/3 probability empirically or provably by the limit. However, if you are allowed to derive the second point from the first (such as the cross section example) or create some other dependency, then you are given great liberty as to what the resulting "probability" will be. Sure you can even guarantee that every two points on the circle will be present "eventually" but infinity is very large and how you approach it will determine your eventual outcome. Here we could subsample preferred points first and actually derive any specific probability (0,1) and maybe even make it sound reasonable. This behaviour will even be maintained into the limit, but it is certainly not intuitive, at least to me.

xyzzyz 11y ago
The other two solutions are also results of generating points from uniform distribution: radius and point on radius result from uniformly drawing pair from [0, 2pi) x [0, 1) (radius, distance from center), while "random midpoint" draws from the uniform distribution on the open unit disc. There's no "dependency" at the drawing time. However, if one remembers the substitution formula for multiple variable integrals, this hints why the Jacobian multiplier is necessary in it: all three underlying spaces mentioned ([0,2pi)^2, [0,2pi)x[0,1], B^2\boundary) are diffeomorphic. They clearly don't have the same measure though -- the first one measures 4pi^2, second 2pi, and third pi. However, one can obtain one value from another by integrating Jacobian of appropriate diffeomorphism.
- andrew-lucker 11y ago
  On most discrete samples for the 1/4 or 1/2 case, the probability of choosing any two points in the subset may be uniform, but the location of the permitted points is definitely not. If you add the requirement that all discrete point samples should be equidistant, then you fix yourself on 1/3 and there is really no way to break that conclusion. Sorry, in this case "uniform" is ambiguous and I didn't mean the usual definition related to probability, but rather to point spacing.
tripzilch 11y ago
> infinity is very large and how you approach it will determine your eventual outcome. Here we could subsample preferred points first and actually derive any specific probability (0,1) and maybe even make it sound reasonable Your reasoning seems right to me, but can you come up with an example way of sampling that gives a probability very close to 0? What about very close to 1? I'm curious, but having a hard time coming up with a reasonable way of sampling that makes for "obviously you're almost never/always getting a chord longer than sqrt3 if you do it like this".
- andrew-lucker 11y ago
  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.
  - tripzilch 11y ago
    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?

xyzzyz 11y ago

The other two solutions are also results of generating points from uniform distribution: radius and point on radius result from uniformly drawing pair from [0, 2pi) x [0, 1) (radius, distance from center), while "random midpoint" draws from the uniform distribution on the open unit disc. There's no "dependency" at the drawing time. However, if one remembers the substitution formula for multiple variable integrals, this hints why the Jacobian multiplier is necessary in it: all three underlying spaces mentioned ([0,2pi)^2, [0,2pi)x[0,1], B^2\boundary) are diffeomorphic. They clearly don't have the same measure though -- the first one measures 4pi^2, second 2pi, and third pi. However, one can obtain one value from another by integrating Jacobian of appropriate diffeomorphism.

andrew-lucker 11y ago
On most discrete samples for the 1/4 or 1/2 case, the probability of choosing any two points in the subset may be uniform, but the location of the permitted points is definitely not. If you add the requirement that all discrete point samples should be equidistant, then you fix yourself on 1/3 and there is really no way to break that conclusion. Sorry, in this case "uniform" is ambiguous and I didn't mean the usual definition related to probability, but rather to point spacing.

tripzilch 11y ago

> infinity is very large and how you approach it will determine your eventual outcome. Here we could subsample preferred points first and actually derive any specific probability (0,1) and maybe even make it sound reasonable Your reasoning seems right to me, but can you come up with an example way of sampling that gives a probability very close to 0? What about very close to 1? I'm curious, but having a hard time coming up with a reasonable way of sampling that makes for "obviously you're almost never/always getting a chord longer than sqrt3 if you do it like this".

andrew-lucker 11y ago
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.
- tripzilch 11y ago
  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?