Since there is some critics here about my own version, here's a respond:

"Peter assumes that we MUST switch to door C, and does not provide for sticking with door A."

Sorry for the omitting the variant of sticking, however:
"If we redraw his tree to show the sticking option in all cases, suddenly the probability of winning becomes 1/2, not 1/3."

Is completely nonsense and not true!!!!! Draw it again and you'll see.

The only critics about the general statement might be that often in the story two things are left out:
a) The game host knows where the price is
b) He will always first open a door that is not the original chosen one and does not contain the price.

Kind regards,
Peter Stikker

p.s. Also a computer simulation will show the same thing.