Chapter 0: EG.2 Solution

Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James R. Munkres

Planet $X$ 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 the surface. Spaceships A and B can communicate directly by radio if $\measuredangle AOB<90^{\circ}$ . Show that the probability that they can keep in touch (with, for example, A communicating with B via C if necessary) is $(\pi+2)/(4\pi)$ .

Let $\alpha$ be the angle between the first two points. It is distributed so that probability density is proportional to $\sin\alpha$ . For each $\alpha\le90^{\circ}$ , the probability that all three points are connected is $\tfrac{\pi+\alpha}{2\pi}$ , and for $\alpha\ge90^{\circ}$ , the probability is $\tfrac{\pi-\alpha}{2\pi}$ . So, overall we have $\begin{eqnarray*} \tfrac{1}{2\pi}\int_{0}^{\pi/2}(\pi+2\alpha)\tfrac{\sin\alpha}{2}d\alpha & = & \tfrac{1}{4}+\tfrac{1}{2\pi}. \end{eqnarray*}$

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Chapter 0: EG.2 Solution

Chapter 0: EG.2 Solution