1. An urn contains two balls, each black or white with even odds. You pick a ball from the urn. It is black. You replace the ball. You pick another ball from the urn. What are the odds that it is also black?
2. Do the odds change if prior to picking the first ball in problem 1, you are told that one of the balls is black, then again your first pick is black?
3. There are two empty urns in a room. You have 50 white balls and 50 black balls. After you place the balls in the urns, a random ball will be picked from a random urn. Distribute the balls (all of them) into the urns to maximize the chance of picking a white ball.
4. You have one white ball and one black ball to place in an urn. What picking method would you propose to create a probability of 1/3? That is, if I am picking between A and B based on the outcome of your method, I will pick A an average of 1 time out of 3. Multiple picks are allowed (and probably necessary.) This is similar to asking how to simulate a probability of 1/3 with only a fair coin.

Source: Internet newsgroup rec.puzzles, used as Ken's Puzzle of the Day 3/22&29/94