"

 You can also solve the problem by saying, right up front, ""by symmetry, the identity (the numbers on, names of...) of the doors is irrelevant, What's important are their roles"". Several respected academic writers do it that way. You can then name the doors by role. Door with car CA; door chosen CH, door left closed CL, door opened O. CA = CH or CL. Probabilities 1/3 and 2/3. That's all   "