It's not correct. P(A|B)=P(A) only if A and B are independent.
Requiring independence in this case literally means "the host opens the door regardless of the player making the right or wrong choice first time". It's a core assumption in your calculations, without it the math is not correct.
Requiring independence in this case literally means "the host opens the door regardless of the player making the right or wrong choice first time". It's a core assumption in your calculations, without it the math is not correct.