Original rules (host knows where car is and always opens a door with a goat):
- 1/3 of the time your original choice is the car, and you should stick
- 2/3 of the time your original choice is a goat, and you should switch
Alternative rules (host doesn't know where car is, and may open either the door with the car or a door with a goat)
- 1/3 of the time your original choice is the car, the host opens a door with a goat, and you should stick
- 1/3 of the time your original choice is a goat, the host opens a door with a goat, and you should switch
- 1/3 of the time your original choice is a goat, the host opens the door with the car, and you're going to lose whether you stick or switch
So even under the new rules, you still only win 1/3 of the time by consistently sticking. You're just no longer guaranteed that you can win in any given game.
Well yes, if you throw out half of the instances where your original choice was wrong, then the chance your original choice was correct will inevitably go up.
Original rules (host knows where car is and always opens a door with a goat):
- 1/3 of the time your original choice is the car, and you should stick
- 2/3 of the time your original choice is a goat, and you should switch
Alternative rules (host doesn't know where car is, and may open either the door with the car or a door with a goat)
- 1/3 of the time your original choice is the car, the host opens a door with a goat, and you should stick
- 1/3 of the time your original choice is a goat, the host opens a door with a goat, and you should switch
- 1/3 of the time your original choice is a goat, the host opens the door with the car, and you're going to lose whether you stick or switch
So even under the new rules, you still only win 1/3 of the time by consistently sticking. You're just no longer guaranteed that you can win in any given game.