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Here is my proof that it is a straight 50/50 shot. In this PDF of a spreadsheet, I have done 1/3 of the exhaustive analysis. Here's how you read it.
The "Car" column shows which door hides the car. I have started from door #1. If I did door #2 and door #3, I would have two more groups of the same kind, 18 rows of which 10 are precluded by rules.
The "Pick" column shows which door the player picks. I have equal numbers of picks for cases 1, 2, and 3.
The "Reveal" column shows which door Monty reveals when he offers the option to switch. I have equal numbers of picks there, too.
The "Switch" column is 1 for switching, 0 for not switching.
The "Win" and "Lose" columns are simple - 1 in the Win column or Lose column but not both.
But... the game that Monty plays has rules. It can never occur that if you picked the car, he would reveal the car and ask you to switch. So in the game, that row can't happen. It ALSO can never occur that he would reveal your pick first and then invite you to switch. He would ALWAYS give you the chance to switch before revealing your door, so the rows where he immediately reveals your pick are ALSO precluded by the rules.
The bottom line is just that you have equal chances for all possible scenarios, of which there are 8 when the car is behind door #1. The same situations would apply for car behind #2 or #3.
The "Car" column shows which door hides the car. I have started from door #1. If I did door #2 and door #3, I would have two more groups of the same kind, 18 rows of which 10 are precluded by rules.
The "Pick" column shows which door the player picks. I have equal numbers of picks for cases 1, 2, and 3.
The "Reveal" column shows which door Monty reveals when he offers the option to switch. I have equal numbers of picks there, too.
The "Switch" column is 1 for switching, 0 for not switching.
The "Win" and "Lose" columns are simple - 1 in the Win column or Lose column but not both.
But... the game that Monty plays has rules. It can never occur that if you picked the car, he would reveal the car and ask you to switch. So in the game, that row can't happen. It ALSO can never occur that he would reveal your pick first and then invite you to switch. He would ALWAYS give you the chance to switch before revealing your door, so the rows where he immediately reveals your pick are ALSO precluded by the rules.
The bottom line is just that you have equal chances for all possible scenarios, of which there are 8 when the car is behind door #1. The same situations would apply for car behind #2 or #3.