There is a flaw in the chart. For the three listed Lose cases, it clumps two distinct choices into one line item and counts them as such, and thus provides an incorrect count of the possibilities.
Moreover the basic argument is flawed. The knowledge of Monty has no barring on the knowledge of the contestant. If Monty simply presented you with three doors that you were to pick from, but opened an incorrect door before you choose your first door, then what would your odds be? The fact that he opens one after you choose does not matter. Monty has total knowledge of the setup and you still have zero.
After you choose a door, Monty cannot choose it -- he's stuck choosing one of the two doors that you did not. So, the order of choosing the doors absolutely matters.
OK. You are right. I worked through the whole problem meticulously and figured it out. It's actually rather easy to explain once worked through, as follows:
The odds of initially choosing correctly are 1 in 3. In that case if you switch you will loose, but if you stay you will win. The odds of choosing wrong initially are 2 in 3. In which case if you stay you will loose, but if you switch you will win. So it is actually that simple. If you always stay you will win 1/3rd of the time and if you always switch you will win 2/3rds of the time.
It only seems like it should be 50/50 at first blush b/c that is what the odds would be if you choose to stay or switch arbitrarily. That is also why if you write down all the possibilities and simply add them up it will appear to be 50/50 too (which is what I did).
The only thing I still can't quite figure out is exactly how the exposure of one of the renaming two doors increases one's knowledge. I thought I could take the list of possibilities and eliminate those that no longer apply and then add them up, but that doesn't seem to work.
Moreover the basic argument is flawed. The knowledge of Monty has no barring on the knowledge of the contestant. If Monty simply presented you with three doors that you were to pick from, but opened an incorrect door before you choose your first door, then what would your odds be? The fact that he opens one after you choose does not matter. Monty has total knowledge of the setup and you still have zero.