Olympiad Probability

Mark problem about sphere: "How large is the largest coordinate of a random point on the sphere? What is the distribution for a fixed coordinate?"

Cf. https://math.stackexchange.com/questions/87230/picking-random-points-in-the-volume-of-sphere-with-uniform-probability

See Scott Aaronson's nice blog post "How to upper-bound the probability of something bad" for his personal list of techniques.