If game show host opens the door at random: in 33 cases he will reveal the car and all will be over; in 33 cases the car will be behind #1; and in 33 cases it will be behind the other unopened door.
The example in wikipedia has the gameshow host always picking the goat and they come up with the 66.67%. What I have to convince myself is that the added information did not make them independent runs.
Now that I am finally waking up(dark chocolate kicking in).. they may not be independent runs because the entrance criteria to the second choice gives a 33.33% chance of being on the car and a 66.66% chance of being on the goat.
That does add another question which would take some gameshow observing. I notice that sometimes on these reveals, they throw in a completely different choice. If the probability on swap results of a 66% chance, would they give you the option to swap if you have picked the door with the car and a completely different set of choices if you haven’t?
