# probability involving distance between points on a sphere

Find the probability that given two random points on a sphere of radius $$k$$, their distance is at most $$d,$$ where $$0\leq d \leq 2k.$$

Obviously the probability function is increasing in $$d$$. By scaling, we may assume WLOG that the sphere has radius $$1.$$ So we want to find the probability of the distance between the two points being at most $$\frac{d}k.$$ But I have no idea how to compute it. Maybe considering the probability a point is of the form $$(x,0,0)$$ given that it is a distance $$r$$ from the center of the sphere might be useful? I know some integral will definitely be necessary here, perhaps involving some variable $$r$$ that could represent the distance from the first point to the center of the sphere, which ranges from $$0$$ to $$r$$.

• Just compute the area of the spherical cap as defined by the distance and divide it by the total area of the sphere. Sep 19 '20 at 1:20
• Do you have $k=r=x$ here? Sep 19 '20 at 1:21
• @Henry no. At the beginning, I already mentioned scaling the circle.
– user747916
Sep 19 '20 at 1:51
• Also, @DavidG.Stork what do you even mean? I don't even know what the "spherical cap" you're talking about is.
– user747916
Sep 19 '20 at 1:51
• en.wikipedia.org/wiki/Spherical_cap Sep 19 '20 at 2:03
