Let's Make A Deal

I was reading a book this weekend about randomness and probability.  There was a probability “riddle” that I’ve seen several times before in other places.   I hesitate to write about it because it’s become such common knowledge that I doubt I’ll be exposing you to something new, but I’m going to anyway because it is an interesting thought exercise.

The problem was first popularized in the Ask Marilyn column of Parade Magazine.   I do not read Parade, so I didn’t hear about it until it was used in my MBA Statistics class.   My teacher used it as his opening to what would become, by far, the worst class of my program.   The teacher over complicated the proof and consequently screwed it up thus confusing the entire class.   We pretty much stayed confused by everything he said for the rest of the term.

The problem was then used in the movie, 21.  In the movie Kevin Spacey’s character presents the scenario to his MIT class and one student stands out as a genius by knowing the right answer.   He then goes on to learn how to count cards and makes millions in Las Vegas before getting caught and blacklisted by the casinos.  

Then scenario is taken straight from the game show, Let’s Make a Deal.  You have three curtains and behind one is a new car and behind the other two are goats.  You are asked to choose one.   After making your choice, the host opens one of the remaining two curtains and reveals that it is a goat.   You are then offered the opportunity to switch from your currently selected curtain and change to the other unopened one.   Should you change?  

Many people will say (if they have not heard the problem before) that it does not matter whether you change.   There are two unopened doors and there is a 50/50 chance that the car is behind the one you already chose.   This is wrong.   You are better off to change curtains. 

The reason I wrote this today was for this paragraph.   After having heard the explanation 3 different ways and given this problem a lot of thought, I think I have a very easy way to explain it.   When you were originally presented your choice, you had a 33% chance of being right and a 66% chance of being wrong.   The subsequent events do not change those original odds.  Therefore, you should change doors because you are effectively switching from the ability to choose one door to the ability to choose 2 doors.  Simple.    The question that I've always had though is whether the game show host would have offered you the choice if you had not already selected the curtain with the car.   This then changes the question to one of manipulation rather than probability.