My \(n\) favorite probability puzzles

  1. \(n\) points are chosen independently uniformly at random from \([0,1]\). What’s the expected value of the maximum?1
  2. Let \(X,Y\) be random variables independently drawn from the same distribution on \([0,1]\). What’s the largest \(\mathbb{E}[|X-Y|]\) can be?2
  3. Your opponent draws \(10\) points on a piece of paper. You have \(10\) coins, and you win if you can place the coins on the piece of paper without overlap so that every point is covered. Can you win?

1 There is a solution using integration, but I recommend trying to find a solution that does not use integration. See page 29 of this paper for a solution. ↩︎

2 That is, propose an explicit constant \(c\), show that for any distribution, \(\mathbb{E}[|X-Y|]\leq c\), and prove that there are distributions that get arbitrarily close to \(c\) (or in particular, that there is a distribution with \(\mathbb{E}[|X-Y|]=c\)). ↩︎