Stick Problem

Problem statement:

Two points are chosen, independently and at random, on a stick of unit length. What is the expected distance between them?

You may want to try to solve the problem, before you look at the solutions below.

Solution using calculus: show

Solution using geometry: show

Solution using symmetry: show

If you've understood how to use symmetry to solve the stick problem, you should be able to solve this one, without using a computer, a diagram, or written calculation:

I shuffle a pack of cards (52 cards including four aces), and draw cards from it, without replacement, until I find an ace. What is the expected number drawn?

This is one of several pages on using symmetry in mathematics.