Here's a problem that is quite simple in appearance, but requires close attention to fundamental ideas in mathematical reasoning:

The contest host Monty Hall shows the contestant Joe three closed doors. He tells Joe that behind one of the doors is a valuable prize. The prize goes to Joe if Joe chooses the door with the prize. Joe doesn't know which door holds the prize. He chooses one of the doors, maybe by some random process or perhaps by whim. Call his choice door A. Monty then opens up one of the remaining two doors -- the one not holding the prize (Monty knows which one -- call this door B).

*Replace the previous sentence by: "Monty, who knows where the prize is, then opens up one of the remaining two doors -- one not holding the prize -- call this door B".*Then Monty asks Joe to stick with the original choice (A) or make a switch to the third door (C).

Question: Should Joe switch his choice to the third door (C)?

Use simulation or mathematical reasoning (e. g. principles of probability theory) to decide whether Joe should switch or not. It would be a service to the rest of us to explain your reasoning or your simulation.

This puzzle can be extended to 4, 5, ..., n doors, and the answer to the question will be the same.