When responding to Mathematician in Triangular Circles (a little play on words, I know circles can't actually be triangular), I wrote this:
On parameterisation, I did some thinking about this along the lines of saying that if you have a 1/3 answer, then it seems (to me) that your selection method must simply have missed some of the chords. In my way of thinking (standard caveat about the possibility of being wrong), if we are asked to select a chord "at random" then it follows that we would be selecting from a set of ALL chords, rather than from a specific subset, unless advised otherwise. Thought from this perspective, our first concern is making sure that we have ALL chords available to select from. The question then is how to express this properly. I'm probably going to mess this up in some obscure way, but if you can at least try to understand what I am saying (and criticise the best formulation of my argument, rather than the worst), it would be appreciated.
I suggest that an expression for ALL chords in a circle defined by x2+y2=1 (in units of R where R is the radius of the circle) goes something like this:
The infinite set S of all unique sets Si of points that fulfil the following criteria:
-1 > c > 1 (defining the y axis intercept of the chord)
0 > θ > 2π (defining the gradient of the chord)
-√((-cosθ)2+(c-sinθ)2) > r > √((cosθ)2+(c+sinθ)2)
(x,y) = (r.cosθ,r.sinθ+c)
Note: the combined effect of these two conditions is (or is intended) to include all and only points between intercepts of the line defined by (x,y) = (r.cosθ,r.sinθ+c) and the circle defined by x2 + y2 = 1, thus defining a chord. In other words a unique set Si is intended to define a unique chord.
When corrected in terms of mathematical terminology, etc, is this a parameterisation and, if so, does it establish or define a structure (per u/Vietoris) for which there is a defined probability measure (per u/Vietoris) or probability distribution (per u/overconvergent)? And, if so, what Bertrand Paradox related answer would be expected from this parameterisation and associated probability measure/distribution?
Well, I was certainly right. I did mess it up.
First, I've doubled the number of chords by using the intervals c:[-1,1] and θ:[0,2π] (hopefully this terminology is clear, it's slightly more convenient than using the -1 > c > 1 and 0 > θ > 2π structure.
I should have used either c:[0,1] and θ:[0,2π] OR c:[-1,1] and θ:[0,π]. Mathematician, in his response, went with the latter, so I'll use that to explain the second, more egregious stuff up.
Note that I said that "our first concern is making sure that we have ALL chords available to select from". The whole purpose my sets was to achieve this and they don't.
For any value of θ<>0 (assuming that θ=0 is standard and aligns with the positive x-axis, the notional horizontal axis and that c is the point at which the resultant chord intersects the y-axis or notional vertical axis (I did use the word "intercept" before, which is apparently right in some cases but there may be some subtlety that I am missing - or perhaps my wording was just clumsy), there are chords are missed in my schema.
We've agreed that the interval [-1,1] is (or can be) uniform, so imagine 11 equally spaced points on the y-axis in that range and say we look at θ=π/4:
This cannot produce the set of ALL chords. Additionally, there is a "skewing" of chords towards those that are longer, so it should come as no surprise that (as Mathematician intuited) there would be substantially more chords of length greater than √3.R. For this reason, I don't think the following comment was nearly as silly as Mathematician later thought it was:
Ok, actually I'm not sure that I am computing the correct probability here. Tell me if this is your idea :
First you pick a number between -1 and 1, uniformly on the interval [-1,1]. And then you pick an angle between 0 and pi, uniformly on the interval [0,pi]. The chord corresponding to the couple (r,θ) is the unique chord that has slope θ and that cuts the horizontal axis at r. Is that okay ?
So this defines a probability on the set of chords. And with this, the probability that a random chord is longer than sqrt(3) is given by the following formula :
P= 1/3 + ln(7+4*sqrt(3))/2pi = 0.7525...
I might be wrong here, but it seems reasonable.
To Mathematician, in answer to the embedded question " … Is that okay?" Yes, I am reasonably happy with that, once I get over my confusion about the use of r (which I normally think of as the length of a vector from (0,0) to some other point). If forced to pick something similar, I'd have gone with (c,θ) since we already have c defined - this would be the unique chord with slope θ that is offset from the x-axis by c when x=0. But I get what you mean,
I'll try to define a set of ALL chords again (this requires more than just a minor shuffle, I suspect).
An expression for ALL chords on a disc defined by x2+y2=1 (in units of R where R is the radius of the disc) goes something like this:
The infinite set S of all unique sets Si of all unique sets Sj of points that fulfil the following criteria:
0 > θ > π (defining the gradient of the chord)
locus defined as (0,0)
-1/cosθ > c > 1/cosθ (defining the y-intercept of the chord)
-√((-cosθ)2+(c-sinθ)2) > r > √((cosθ)2+(c+sinθ)2)
(x,y) = (r.cosθ,r.sinθ+c)
Note: the combined effect of these two conditions is (or is intended) to include all and only points on the intersection of the lines (x,y) = (r.cosθ,r.sinθ+c) and the disc defined by x2+y2=1, thus defining a chord.
Defining the locus as (0,0) removes some complications to the equations that would otherwise be required to achieve invariance in terms of translation (by which I mean movement of the circle to another location). Setting the radius of the disc to R and making R the units of length in all considerations addresses the question of invariance in terms of scale. Defining the set of y-intercepts such that c:[-1/cosθ,1/cosθ] goes only part of the way to addressing invariance in terms of rotation.
If we revisit the image above but extend out the range of c, we will get:
The gap has gone, but we've now got more chords at θ=π/2 than we had at θ=0, so we no longer have rotational invariance. To get it back, we need to introduce a concept that probably has another proper term to it, but I call "granularity".
Say we select an arbitrarily large number (N+1) of evenly spaced samples over the interval from which we take c. If c:[-1,1] because θ=0, then there would be N/2 samples above the locus and N/2 below the locus and one on the locus. If we generalise this, for c:[-ci,ci], then there would be still N/2 samples above the locus and N/2 below the locus, but with a different separation - rather than the samples being 2/N apart, they would be 2/N/ci apart. I refer to this figure, 2/N/ci, as the "granularity".
In order to maintain invariance in terms of rotation, we need to set the granularity of the sets to 2.cosθ/N with N->∞. If there is a better way to word this, please let me know.
If there is an iron-clad rule that says that I cannot parameterise my chord selection with anything akin to this concept of granularity, then I guess I have to graciously concede defeat, albeit with the residue of the itchy feeling that maths shouldn't be like this. But if it is possible, without necessarily being conventional, then I think my selection of chords makes sense, is invariant in terms of scale, translation and rotation and results in the 1/2 answer. And while it does not seem quite as elegant as my first (incorrect) version, it is more general and I don’t know that an attempt to do something similar with the 1/3 and 1/4 methods can be done as elegantly. Perhaps it can be done, perhaps there are even more elegant ways to do it, I'm in absolutely no way certain of this.