Imagine you are attending the filming of a game show when suddenly you hear your name called by the PA announcer. You’re the next contestant! The host greets you and makes you laugh nervously with his charming wit before presenting you with the game. You have the choice of three doors. Behind each door is a prize that will become yours. On the other side of one of the doors is a new car. Behind the other two is a goat, as in the grass-munching farm animal. You’d prefer to drive home in the car.
The game seems straight forward at the beginning. You appear to have a one-third chance of selecting the car. You randomly select a door, just going with your gut. The host then surprises you by opening one of the other doors and revealing a goat. As game show hosts are wont to do, he complicates this decision-making process by allowing you the chance to switch your chosen door.
Should you do it?
This is the problem proposed to not only several contestants by Monty Hall, host of the 1970s game show “Let’s Make a Deal,” but also Parade Magazine columnist Marilyn vos Savant in a 1990 column.
Vos Savant, according to her official webpage, held the Guinness World Record for highest IQ for five years, for both adult and child scores. On a variety of IQ tests, she scored between 186 and 228. Though the methods of measuring intelligence are dubious across the board, and testing methods resulting in Savant’s scores have received special criticism even from herself, the fact remains that she has been widely considered very intelligent.
After her rise to fame, she began writing a column for Parade magazine in 1986 entitled “Ask Marilyn” in which fans would ask her to solve puzzles or answer difficult questions.
Fan Craig F. Whitaker of Columbia, Md., wrote the letter that sparked a very heated debate between Savant and many academic mathematicians.
So back to the original problem. Many people would respond to the problem by stating that they now have a 50 percent chance of being correct whether they stay or switch doors. Here is vos Savant’s reply:
“Yes; you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance. Here’s a good way to visualize what happened. Suppose there are a million doors, and you pick door #1. Then the host, who knows what’s behind the doors and will always avoid the one with the prize, opens them all except door #777,777. You’d switch to that door pretty fast, wouldn’t you?”
Did you get it right?
This was less than convincing for 92 percent of the people who took time to reply to Savant’s answer, including 65 percent of the respondents who were identified as university academics. People were outraged by her assertion. Vos Savant’s website even posts some of the replies. This from Scott Smith, Ph.D., from the University of Florida:
“You blew it, and you blew it big! Since you seem to have difficulty grasping the basic principle at work here, I’ll explain. After the host reveals a goat, you now have a one-in-two chance of being correct. Whether you change your selection or not, the odds are the same. There is enough mathematical illiteracy in this country, and we don’t need the world’s highest IQ propagating more. Shame!”
Vos Savant, to her credit, took to her column to respond with confidence, explaining that at first selection, you have a 1 in 3 chance of selecting the correct door. Those odds do not increase simply because the host opened a door with a goat. When you make your choice, the odds are 2 in 3 that the car is behind a door you did not choose. The host is guaranteed to select a door behind which you will find a goat. The odds are 1 in 3 of your door being a winner, 0 in 3 behind the door the host chooses, and therefore 2 in 3 that the other door is correct.
This follow-up still was not enough to convince her readers. The responses took an increasingly acerbic tone.
“May I suggest that you obtain and refer to a standard textbook on probability before you try to answer a question of this type again?”
Charles Reid, Ph.D., University of Florida
“I am sure you will receive many letters on this topic from high school and college students. Perhaps you should keep a few addresses for help with future columns.”
-W. Robert Smith, Ph.D., Georgia State University
“You are utterly incorrect about the game show question, and I hope this controversy will call some public attention to the serious national crisis in mathematical education. If you can admit your error, you will have contributed constructively towards the solution of a deplorable situation. How many irate mathematicians are needed to get you to change your mind?”
-E. Ray Bobo, Ph.D., Georgetown University
“You are the goat!”
-Glenn Calkins, Western State College
And maybe most dismissively of all:
“Maybe women look at math problems differently than men.”
-Don Edwards, Sunriver, Ore.
Vos Savant implored her readers to test her answer by actually running trials of the game. The resulting tests finally led to her conclusion beginning to sink in, with junior high and high school math teachers writing in that her answer worked out. Mathematics professors worked with colleagues, coming around to Savant’s way of thinking, as well. Over the course of many trials, people reported winning approximately 33 percent of the time when they stayed and 66 percent chance when they switched.
Over time, and due to her Vos Savant’s persistence, sentiment toward her view began to shift. Zachary Crockett on Pricenomics.com writes that by the end of 1992, vos Savant reported that 56 percent of her general readership accepted her answer, while 71 percent of academics agreed with her.
There were holdouts, for sure, like the aforementioned Don Edwards of Sunriver, Ore., who just couldn’t accept the limits of the female brain, and re-wrote to Savant to express his disagreement:
“I still think you’re wrong. There is such a thing as female logic.”
And obviously there’s such a thing as male stubbornness, too.