Probability | Part IA, 2004

Let A1,A2,,ArA_{1}, A_{2}, \ldots, A_{r} be events such that AiAj=A_{i} \cap A_{j}=\emptyset for iji \neq j. Show that the number NN of events that occur satisfies

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

Planet Zog is a sphere with centre OO. A number NN of spaceships land at random on its surface, their positions being independent, each uniformly distributed over the surface. A spaceship at AA is in direct radio contact with another point BB on the surface if AOB<π2\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 NN 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 NN random great circles partition the surface of a sphere into N(N1)+2N(N-1)+2 disjoint regions with probability one.]

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