(Note that this topic has been covered in detail by several well-known websites. While their material is sound, I feel that some of the interesting details in this problem are obscured by multivariate calculus. In this series of posts, I will attempt to explain this problem assuming limited prior mathematical knowledge. Readers with a good grasp of probability will want to skip past the preliminaries.)
In this series of posts we will answer the question: how do we choose a point uniformly at random on the surface of a sphere? (Think: how do we pick a random place on a globe without bias to any particular location(s)?)
Before we get to the solution, we need to address a few basics. First consider the phrase “uniformly at random.” What does this mean?