Think about the Monty Haul problem this way. What are your odds of picking the prize door using each of the methods? Consider the entire process from start to finish, rather than the individual choices.

If you stick with the door you pick at the beginning, you only get it right if you pick the prize door immediately. Your chance of picking the prize door out of 3, is 1/3.

If you switch doors, you end up with the prize door whenever you don't pick the prize door at first. If you start with the prize door, you'll switch to a false door. If you start with a false door, the second false door gets eliminated, and you end up on the prize door. So your chance of ending up on the prize door is equal to your chance of initially picking a false door - i.e., 2/3.