Question

If I choose two points completely at random from a sphere, exactly how far apart will they be (on average)?

Answer 1

on the surface of sphere?

Answer 2

I was thinking on the surface of the sphere. Another interesting problem is for inside the unit ball (and/or in other dimensions.)

Answer 3

well the min distance they could be apart would be 0 or very close to 0
the max distance would be pi*R, where R is radius of sphere

so assuming a uniform distribution of the random points i would say that on average the distance would be right in the middle at pi/2*R

Answer 4

if you are forced to travel overland on the surface of the sphere to measure distance, then @dumbcow's answer of pi/2 (for unit sphere) is correct. But what happens when I take the straight-line Euclidean distance through space ... distance(p,q) = |p-q|?

Answer 5

In that case, it should be equal to (0+diameter)/2 i.e. radius of sphere.

Answer 6

no, the straight-line "average" distance averaged over the sphere doesn't come out to exactly the radius.