I love how this problem seems to fool so many people of all intelligence. An easy way to see this is to first convince yourself, using the strategy of always switching doors, that:

- If you selected the winning door to begin with you are guaranteed to lose.
- If you selected a losing door to begin with you are guaranteed to win.

Then it is easy to see that the you will win with this strategy 2/3 of the time.

Your analysis here uses the strategy of exploring all possible outcomes. The flaw in your analysis is that you do not explore all possible outcomes (for probability's sake). Basically there are a total of 6 possible outcomes (you pick 1 of 3, host "picks" 1 of remaining 2).

In your case A, the probability of either of the 2 outcomes is actually 1/6 yet for cases B and C the probability is 1/3 (each since the host does not really have a choice).

From there you should see how the correct outcome comes to light.