2.II.12F

Let $A_{1}, A_{2}, \ldots, A_{r}$ be events such that $A_{i} \cap A_{j}=\emptyset$ for $i \neq j$. Show that the number $N$ of events that occur satisfies

$P(N=0)=1-\sum_{i=1}^{r} P\left(A_{i}\right)$

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 $\angle A O B<\frac{\pi}{2}$. 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.]

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