Monty Hall.

This is a continuation of the previous posts under this title.

Yes, I have difficulty in coming to terms with the fact that if MH doesn't know where the car is, that makes a difference. But really, what it comes down to is information. We have seen that different information can lead to different probability assessments. And this means that if there is any change in the information you have, you may need to reassess your calculations. In the original scenario, when MH opens a door and shows it to be empty, this changes nothing - because you know that's what's going to happen. But if he doesn't know where the car is, and opens an empty door, that tells you something you didn't know before, and alters the probabilities. Look at it this way - suppose we have three contestants, X, Y and Z. We put X in front of Door A, Y in front of Door B and Z in front of Door C. Nobody knows where the car is. Each contestant then has a one in three chance of being in front of the right door. Y now opens Door B, and shows it to be empty. I think it's obvious that X and Z now have an equal chance of being in front of the right door. Y of course is Monty Hall and X and Z are both you - if he doesn't know where the car is and opens an empty door, swopping doors doesn't improve your chances. Put it another way - let's simply refer to the doors as (1) the door you choose (2) the door he opens and (3) the other door. The probabilities have to add up to one, yes? So in the original problem the door you choose has a 1/3 chance of being the "right" door, the door he opens has a zero chance (he will never open the door the car is behind) so the other door has to have a 2/3 chance. If he doesn't know where the car is, then the door you choose has a 1/3 chance, the door he opens has a 1/3 chance, and the other door also has a 1/3 chance. And that I think is quite enough, save to say that although the problem bears his name, this was not the game Monty Hall was associated with - in his game you weren't allowed to swop!