Monty Hall problem
A true classic, and probably the best example of how counter-intuitive conditional probability can be.
You are on the game show Let's Make a Deal and host Monty Hall has chosen you to play a game. He tells you that behind one of the three closed doors before you there is a fabulous new car. But behind the other two, there are less than fabulous goats. You can pick any one of the doors, and take home the prize there concealed. You pick door 2. Before revealing your prize, Monty Hall, as always, says "It's sure a good thing you didn't pick that door there." He points to one of the remaining doors as it opens, revealing a baying goat. And now the suspense builds, as the host asks, "So, do you want to stay with your original choice, or change your mind?" What should you do and does it matter?