Planet Zog problems

Here are two challenging tripos questions from recent years. The first is a problem in geometric probability (as in Lecture 18). To do this question it will help to know that the surface area of a spherical cap is $2\pi R h$, where $R$ is the radius of the sphere and $h$ is the height of the cap. The surface area of the entire sphere is $4\pi R^2$.

2003/II/12
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 $AOB$ formed by the lines $OA$ and $OB$.

Spaceships $A$ and $B$ can communicate directly by radio if  $\angle AOB  < \pi /2$, and similarly for spaceships $B$ and $C$ and spaceships $A$ and $C$. Given angle $\angle AOB =\gamma < \pi /2$, calculate the probability that $C$ can communicate directly with either $A$ or $B$. Given $\angle AOB =\gamma > \pi /2$, 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 $( \pi +2)/(4\pi )$.

Spoiler alert! Here is the answer.

The second question is rather different. Here you are being tested on the idea that a sample space can be partitioned into disjoint events.

2004/II/12
Let $A_1,A_2,\dotsc,A_r$ be events such that $A_i\cap A_j = \emptyset$; for $i \neq j$. Show that the number $N$ of events that occur satisfies

$P(N = 0) = 1 −\sum_{i=1}^rP(A_i) .$

Planet Zog is a sphere with centre $O$. A number $N$ of spaceships land at random on its surface, their positions being independent, each uniformly distributed over the surface. A spaceship at $A$ is in direct radio contact with another point $B$ on the surface if $\angle AOB < \pi/2$. Calculate the probability that every point on the surface of the planet is in direct radio contact with at least one of the $N$ spaceships.

[Hint: The intersection of the surface of a sphere with a plane through the centre of the sphere is called a great circle. You may find it helpful to use the fact that $N$ random great circles partition the surface of a sphere into $N(N − 1) + 2$ disjoint regions with probability one.]