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Sphere Point Angle and Expected Value

Source:

August 8, 2024
geometry3D geometrysphereexpected valueprobability2022

Problem Statement

Let BB and DD be two points chosen independently and uniformly at random from the unit sphere in 3D space centered at a point AA (this unit sphere is the set of all points in R3\mathbb{R}^3 a distance of 11 away from AA). Compute the expected value of sin⁡2∠DAB\sin^2\angle DAB.
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