Follow up I wanted to post this because I fall into what seems to be the unusual camp of intuitively feeling that 1/4 is the correct answer. The first option doesn’t necessarily feel wrong, since...

I wanted to post this because I fall into what seems to be the unusual camp of intuitively feeling that 1/4 is the correct answer. The first option doesn’t necessarily feel wrong, since it’s simply generating two points in {(cos(t), sin(t)) such that t is uniformly sampled from [-pi, pi)}, but option three does feel wrong to me. Option two and option three both use the same mapping from point to cord, and the core difference between the two methods is the method used to generate the point. Method two simply generates a point uniformly in two dimensions and then maps it to a cord, whereas method three generates a rotation and radius separately, then maps it to a cord. Anyone who has studied this before will know that the second method of generating points results in a higher density of points closer to the origin - worth simulating if you are curious and ever need to write a Monte Carlo simulation for a problem with data involving angles.

Follow up

I wanted to post this because I fall into what seems to be the unusual camp of intuitively feeling that 1/4 is the correct answer. The first option doesn’t necessarily feel wrong, since it’s simply generating two points in {(cos(t), sin(t)) such that t is uniformly sampled from [-pi, pi)}, but option three does feel wrong to me. Option two and option three both use the same mapping from point to cord, and the core difference between the two methods is the method used to generate the point. Method two simply generates a point uniformly in two dimensions and then maps it to a cord, whereas method three generates a rotation and radius separately, then maps it to a cord. Anyone who has studied this before will know that the second method of generating points results in a higher density of points closer to the origin - worth simulating if you are curious and ever need to write a Monte Carlo simulation for a problem with data involving angles.