Since there is no prize behind the door the host opens, in Case 1 the host could open
either door and you would win by staying with your original choice: door A. In Case 2
the host must open door C, and so you would win by switching to door B. In Case 3 the
host must open door B, and so you would win by switching to door C. Thus, in two of the
three equally likely cases, you would win by switching from A to the other closed door.
In only one of the three equally likely cases would you win by staying with your original
choice. Therefore, you should switch.