>>12088460This post inspired me to think about it myself before rewatching the video (not that anon BTW).
If you start by looking at a circle instead and 3 points, you see that when you have added 2 points you subdivide the circle into 4 region and only in one of them can the third point lie so that the center of circle lies in the triangle. Now instead of choosing the 2 points you could first fix one point and the choose 2 lines. For each line, a point lying at the end. Now in only 1/4 of the outcomes will the original point lie in the region opposite to the two points. By symmetry, the probability is 1/4.
Now in the 2-sphere case we do the same thing.
Fix one point, consider how the other three lie. Instead of distributing points, we distribute pairs of points (determined by lines through the center). For each such distribution of pairs, in only one of them does the center of the sphere lie in the tetrahedron. Thus since there are 8 equally likely ways to put the points when you have 3 pairs, the probability we want is 1/8.
It's been a while since I saw the video and it's been fun to think it through again :)
