Home/BlogA diversion from global warming topics.
The simple little probability problem below has apparently been debated for many years. It came to prominence when Marilyn vos Savant answered a reader’s question about it. Her answer was believed to be wrong by some of the greatest statistical minds in the world, and eventually most of them admitted she was correct after all.
A story about that debate is here.
But I maintain that the answer depends upon an unstated assumption, and so there is no correct answer. Of course, I could be wrong. Disagreeing with a person having the highest IQ in the world is, statistically speaking, not a smart thing to do.
The Monty Hall Problem
There are three doors, and behind one of them is a new car, and behind the other two doors are goats. You want the new car. You choose door #1, knowing you have a 1 in 3 chance of winning.
Monty Hall then opens door #3 and shows you a goat there. Should you change your pick from door #1 to door #2? Most people said no, that you still don’t know whether the car is behind the first or second door, and all that has happened is your chance of winning has simply improved from 1/3 to 1/2.
But Marilyn vos Savant said “yes”, that you should switch. Experts disagreed.
From what I can tell, through, it entirely depends upon why Monty Hall showed you what was behind door #3.
If there is a goat behind door #3, then clearly the new car is behind either door #1 or door #2. If Monty Hall was going to show you door #3 no matter what was behind it, then your chances are still 50/50… you might as well stay with door #1.
BUT…if Monty Hall was only going to show you a remaining door that had a goat behind it, then you should switch to door #2. The reason is you would have new information you didn’t have before…that if he knew that the new car was behind one of the remaining doors, he was going to in effect tell you that by not opening that door.
In that case, you actually have a 2 in 3 chance of winning by switching doors.
But, as far as I can tell, which of these two assumptions is in effect was never stated, and so there is no correct answer to the problem.
(RIP, Monty Hall).
It has nothing to do with ‘framing the question’ or assumptions. Mathematicians routinely ask hypothetical questions It is a simple question of listing the possible outcomes. Example: Car in Door A, Player picks Door A, Monty opens Door B. If the player stays with Door A- win. Just list the possible outcomes. Six wins from not switching, six wins from switching- as you expect. No Genius required.