It isn't the Monty Hall problem if the host doesn't know where the prize is. The problem is generally stated, up funny, the host knows where the prize is. The issue isn't poorly worded, it is that a lot of people can't understand how two doors don't represent half and half. You are correct, in that if that just randomly selects one of the two doors, occasionally he will reveal the prize, changing to odds. Which is why it is usually started that the host knows and selects she empty door.