In a very thoughtful reply to a Rebelyid posting about how intellectuals and academics miss seemingly obvious points that ordinary people with common sense seem to readily get, Roy Fickling sent this Marylin Savant math problem.

I have to confess that it took  some thought even after seeing her correct answer to understand why she was right.

I guess the point is that some things are not as obviously right as we may think they are.

from Roy’s e-mail. See if you get it.

The question is this: Monty Hall gives you three doors to chose from. Behind one of the three is a bright shiny new car. You chose door #1. Monty opens door #3, which is empty. He give you the choice: do you want to change to door #2, or stay with your first choice?

The question posed to Marilyn was, “should I change my answer?” Marilyn’s response was “of course, change to door #2. Your chances of winning will double if you switch doors.”

Marilyn was publically berated by hundreds of mathematicians and academics. They said that even a grade school math student knows that your chances go from 1/3 to 1/2 when one door is opened. The debate even made it to the front page of the New York Times. Marilyn stood fast and wouldn’t budge. It wasn’t until the results of thousands of experiments and hundreds of computer simulations were revealed that the” math geniuses” finally capitulated. The problem was that the foundations of their training were flawed. Although Marilyn had no formal training in mathematics, she did poses the highest IQ ever scored.

Just for grins, the explanation is as follows:

There are two scenarios on the first guess. There is the right answer scenario (1 in 3 chance) and the wrong answer scenario (2 in 3 chance). The odds of a successful outcome are the same at the first guess, i.e., 1/3 that you get a car, or said differently, 2/3 chance you will get encyclopedias.
Now, Monty Hall opens one of the doors that he for a fact knows contains no car.

If you chose correctly the first time (the right answer scenario) and switch your answer, there is a 100% probability that you will lose. If you chose wrong the first time (the wrong answer scenario) and switch your answer, there is a 100% chance that you will win. Since the chances that you chose wrong in the first place are still 2/3 and the chances that you chose right in the first place are still 1/3, you are twice as likely to be driving away in a shiny new car if you switch your answer.

To see it more clearly, use larger numbers. Say there were 100 doors. Your chances of choosing correctly are 1/100. Then, Monty hall opens all but 2 doors, including the one you chose. The chances are still 1/100 that you initially chose correctly and 99/100 that you chose incorrectly. So, the chances are 99/100 that the car is behind the remaining door that you didn’t initially chose. Switch your answer.