Planet Zog is a ball with centre . Three spaceships and 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 formed by the lines and .
Spaceships and can communicate directly by radio if , and similarly for spaceships and and spaceships and . Given angle , calculate the probability that can communicate directly with either or . Given angle , calculate the probability that can communicate directly with both and . Hence, or otherwise, show that the probability that all three spaceships can keep in in touch (with, for example, communicating with via if necessary) is .