Here's a favorite interesting probability problem.

Suppose you are on Monty Hall's "Let's Make a Deal!".  You are given the choice of three doors, behind one door is a car, the others, goats.  You pick a door, say 1, Monty opens another door, say 3, which has a goat.  Monty says to you "Do you want to pick door 2?"  Is it to your advantage to switch your choice of doors?