Starred exercises are more tricky. The first number in an exercise gives a rough indication of which chapter it depends on. ‘G’ stands for ‘a bit of gumption is all that's necessary’. A number of exercises may also be found in the main text. Some are repeated here. We begin with an
Antidote to measure-theoretic material – just for fun, though the point that probability is more than mere measure theory needs hammering home.
EG.1. Two points are chosen at random on a line AB, each point being chosen according to the uniform distribution on AB, and the choices being made independently of each other. The line AB may now be regarded as divided into three parts. What is the probability that they may be made into a triangle?
EG.2. 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 ∡AOB < 90°. Show that the probability that they can keep in touch (with, for example, A communicating with B via C if necessary) is (π + 2)/(4π).
EG.3. Let G be the free group with two generators a and b. Start at time 0 with the unit element 1, the empty word.
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