naddok and Durnominator are correct.
What you fail to take into consideration is the times the host opens the door with the prize, randomly.
We all agree that if both the contestant and the host pick doors randomly, the contestant picks the winning door straight away 1/3 of the times. So do the host.
What that means is that 2/3 of the times, its one of the doors the contestant didnt pick. Unfortunately, half of those times, the host will have picked the door with the prize. But, we dont care about that right now but focus solely on the times the host picks an empty door. Which ofcourse means the times the contestant has picked the prize, and the times its behind the last door. That means only 2/3 of the times, and should be split evenly between the two scenarions. Meaning, the contestant has 50% chance to get the prize should he switch doors in those cases.
Intent do make a difference.
Lets put it this way.
The contestant get 30 tries to get it right. Out of those tries, the prize would be located like this:
10 times behind the door the contestant picks.
10 times behind the door the host picks.
10 times behind the last door.
We ignore the times the host picks the door with the prize, since that is a prize lost. We get:
10 times behind the door the contestant picks.
10 times behind the last door.
That looks like a 50% deal to me.