Perhaps it'll be more intuitive to you if you scale the number of doors up. If there are 100 doors, only one containing a car, you pick one, the host reveals 98/99 remaining doors as goats, it's obvious the correct choice is to switch. The correct answer is a mathematically provable probability. 1% chance you picked the right door, 99% chance the door was in the remaining pool, therefore 99% chance the last remaining door is the correct door.
You can write a simulation that does this (as MIT and several others have) and Monte Carlo it. You will find that the logic is 100% correct. The prize MUST be either the door you picked initially or the one you can switch to. There is a 99% chance it's not the one you picked.
