Monty Hall: The Problem

1. As the game begins, Monty (The Host) shows you three closed doors.
   

2. As The Host explains, behind one of the doors is a prize. The other two doors have nothing behind them.

   If you choose the door with the prize, you win the prize. If you select one of the doors with nothing behind them, you lose.

3. The Host then asks you to select one the doors.

4. You then select one of the doors. However, the door is not opened at present.

5. The Host (who knows which door has the prize behind it) then opens one of the two doors which you did not select --revealing that that door is not the door which has the prize behind it.

6. The Host now explains that you have an option. Before he opens the other doors to reveal whether or not you have won, you can chose to either:
   • Keep the door that you previously selected.
     or
   • Trade that door for the one remaining closed door.

This, then, is the puzzle. Specifically, are your chances of winning better if you:

Option A: Keep the door that you originally chose?

Option B: Trade for the other door?

Option C: It doesn't matter if you trade or keep the original door.

Now, most people would probably answer that the "common sense" answer, Option C, is the correct answer; that is it doesn't make any difference if you trade or not, your chances of winning are the same.

However, as you might suspect by this time, Option C is NOT the correct answer. The question, then, is: which of the other two Options is the correct answer? That is, should you trade or not? And why?

The Solution is available below. However, if you wish to think about this for a while, a (brief) Hint is available.