2.II.12F

Planet Zog is a ball with centre $O$. Three spaceships $A, B$ and $C$ land at random on its surface, their positions being independent and each uniformly distributed on its surface. Calculate the probability density function of the angle $\angle A O B$ formed by the lines $O A$ and $O B$.

Spaceships $A$ and $B$ can communicate directly by radio if $\angle A O B<\pi / 2$, and similarly for spaceships $B$ and $C$ and spaceships $A$ and $C$. Given angle $\angle A O B=\gamma<\pi / 2$, calculate the probability that $C$ can communicate directly with either $A$ or $B$. Given angle $\angle A O B=\gamma>\pi / 2$, calculate the probability that $C$ can communicate directly with both $A$ and $B$. Hence, or otherwise, show that the probability that all three spaceships can keep in in touch (with, for example, $A$ communicating with $B$ via $C$ if necessary) is $(\pi+2) /(4 \pi)$.

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