Monty Hall problem keeps popping back up recently, most of them with a catchy headline “even geniuses/ Ph.D. got it wrong”.
Well since it was originated from one of the Berkeley profs, where I study now. I might jump in to support the intellectual exercise.
Here is why: (according to Wikipedia)
But, Marilyn's solution would be correct if she spells out "The HOST will NEVER open a door with the car"!!!
(in other words, there WAS NO chance that the door the host opens first would be the car. This is the key, you did NOT lose a 1/3 chance because the HOST will never give it to you, which means this had not been a 1 chose 3 game ever.
Now, the host will open a door, BUT he would *NEVER* open the door you chose!!! (That's why you never had a 1/3 chance of winning, he will NOT doors randomly, instead, he is using his knowledge to play tricks on you, hence you now have a chance to play it back at him, but this was not explicitly spelled out to begin with).
There SHOULD be the following possibilities:
Here, knowing you are not going to win right away (he will always pick a goat door), you LIMITED
the host's choice to 2 doors, (because he will NOT open door 1 whatsoever.)
So remember, if you were correct, the HOST has 2 options to play tricks on you.
Yes, remember, here, the host HAS to open the goat that you did not select!!! He has NO other options, because he would NEVER open the car in the first round.
OK, you have 1/3 chance choosing a door with the car, and allow the host play tricks on you with 2 doors.
HOWEVER, you have 2/3 of the chance choosing a door with a goat, which FORCE the host to open the other goat door for you (remember, in this situation, the host actually has no options to play tricks other than open the OTHER door with goat).
Because in the cases (#2 & #3 you selected a goat door) you forced the host to open the other goat door, you know in this 2/3 of the chances, switching is a sure-fire of the car.
It's a little bit complicated for #1, because if you already chose the door with the car if you switch, you would have lost the car, so in this 1/3 of the cases, you should stay.
BUT since you don't know which you chose the first round, you really could not make the adjustment like we did here, BUT you know in 2/3 of the cases (where your chose was a goat), switch is a sure-fire of the car, but in the 1/3 of the cases (your selection was the car), staying would be better... Since you don't know, you might just switch, because it has higher chance 2/3 .vs. 1/3
IF he opened the door to a goat by chance, then you just missed your 1/3 chance of winning, and were given a second chance.
This is the key to her "scam", this "always" was not given in the original question (mathematically this means she is adding additional conditions and restrictions and changed the actual model).
And here is the Commonsense Solution: