The Monty Hall problem needs to be far more explicit for the answer to be correct. The host must ALWAYS open a door that he knows has a goat, and the contestant must know that the host is bound by that rule. If the contestant believes it is possible that the host opened a door with a goat by chance, the contestant has too little information to make a decision. But if the contestant has all the information he needs, the game isn't as much fun.