Beautiful little probability/geometry problem here: what's the probability that four points chosen uniformly from the sphere lie in some hemisphere?.

A solution by J. G. Wendel does it by solving the much more general problem where you've got N random points in Rn that obey any joint probability distribution (not even necessarily IID!) that is reflection-invariant and guarantees that all N-point marginal distributions are linearly independent with probability 1. Out of the linear-independence assumption he notes that N hyperplanes in general position chop Rn up into a certain constant number of connected components (for which it's easy to get a nice recurrence), and from reflection invariance he gets that the number of components determines the probability the problem is asking for.

So cute!
Tags:
• #### (no subject)

A paper on describing circuits in an agda DSL: http://www.staff.science.uu.nl/~swier004/publications/2015-types-draft.pdf

• #### (no subject)

Going more carefully now through this little tutorial on fpga programming with the iCEstick. It's in spanish, which makes it slightly more…

• #### (no subject)

Some further progress cleaning up the https://xkcd.com/1360/ -esque augean stables that is my hard drive. Tomato chicken I made a couple days ago…

• #### Error

Anonymous comments are disabled in this journal

default userpic