The correct answer is you should always switch. By switching you will increase your odds of finding the money from 1/3 to 2/3.
There are 3 possibilities for where the money is hidden.
|door 1||door 2||door 3|
If you choose one door and stick with that door, your chances are 1/3.
Using the switching strategy, let's say you pick door 1. If it's case A, then you lose. If it's case B, Monty shows you door 3, and you switch to door 2, so you win. If it's case C, Monty shows you door 2, and you switch to door 3, you win. It doesn't matter what door you pick in the beginning, there are always still three possibilities. One will cause you to lose, and two will cause you to win. Your chances of winning by switching are 2/3.
The trick to the solution resides in the fact that Monty knows what is behind all the doors and therefore always eliminates a door for you, thereby increasing your odds.
For example, imagine there are 1,000 doors, only one of which has a prize behind it. You pick a door and then Monty opens 998 doors with goats behind them. Do you switch? It should seem more obvious in this case, because Monty had to take care in which door not to open, and in the process basically showed you where the prize was (999 out of 1,000 times).