Paper 2, Section II, F

Let $A, B$ and $C$ be three random points on a sphere with centre $O$. The positions of $A, B$ and $C$ are independent, and each is uniformly distributed over the surface of the sphere. Calculate the probability density function of the angle $\angle A O B$ formed by the lines $O A$ and $O B$.

Calculate the probability that all three of the angles $\angle A O B, \angle A O C$ and $\angle B O C$ are acute. [Hint: Condition on the value of the angle $\angle A O B$.]

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