The Story Behind the Most Heated, Controversial, and Downright Toxic Debate in the History of Mathematics

Imagine you are a contestant on the classic game show Let’s make a Deal. The host, Monty Hall, presents you with three closed doors. Behind one is a brand-new sports car, while behind the other two is a goat.

You make your choice and prepare to discover whether you will leave the studio the proud owner of a set new ride or of a four-legged garbage disposal. But at the last moment, Monty offers to make your choice a little easier; he opens one of the doors, revealing a goat, and gives you the option to either change your guess or stick with your original choice. What do you do?

This is the Monty Hall Problem, first posed by American statistician Steve Selvin in a 1975 letter to the journal American Statistician. For 15 years after its introduction, the puzzle was discussed by a number of academic publications but failed to make much of an impact. But when in September 1990 reader Craig F. Whitaker submitted the question to Parade Magazine’s Ask Marylin column, it unexpectedly triggered one of the most heated, controversial, and downright toxic debates in the history of mathematics.

Ask Marylin, which has run in Parade since 1986, is written by Marylin sos Savant, who for many years was widely known as the “World’s Smartest Person.” Born in St. Louis, Missouri in 1946, in 1956 at the age of 10 vos Savant took the standard 1937 Stanford-Binet Test, achieving an unprecedented IQ score of 228. Later in the 1980s she scored a 46/48 on the Hoeflin Mega Test, which revised her IQ down to a more reasonable but still impressive 186. Based on these two scores, vos Savant held the Guinness World Record for the highest recorded IQ from 1986 to 1989 when the category was finally retired.

At first glance the answer to the Monty Hall Problem appears obvious: your chances of picking the car, originally 1 in 3, have now been increased to 1 in 2. However, as you’ve been given no additional information as to which door the car sits behind, it makes no difference whether you stick with your original guess or switch; your odds of winning remain the same. However, in her reply to Craig Whittaker’s statement of the problem, vos Savant gave an altogether different answer, arguing that the best strategy was, in fact, to switch your first guess:

“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?”

The reaction to this counter-intuitive solution was swift and surprisingly hostile, with vos Savant being bombarded with letters from hundreds of readers – several with PhDs in statistics and related fields – staunchly refuting her analysis. These ranged from the relatively polite but dismissive:

“Your answer to the question is in error. But if it is any consolation, many of my academic colleagues have also been stumped by this problem.”

-Barry Pasternack, Ph.D.

California Faculty Association

…to the arrogant and condescending:

“Since you seem to enjoy coming straight to the point, I’ll do the same. You blew it! Let me explain. If one door is shown to be a loser, that information changes the probability of either remaining choice, neither of which has any reason to be more likely, to 1/2. As a professional mathematician, I’m very concerned with the general public’s lack of mathematical skills. Please help by confessing your error and in the future being more careful.”

-Robert Sachs, Ph.D.

George Mason University

“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!”

-Scott Smith, Ph.D.

University of Florida

…to the downright misogynistic:

“Maybe women look at math problems differently than men.”

-Don Edwards

Sunriver, Oregon

In response to this deluge of criticism, vos Savant devoted her three next columns to patiently re-explaining the logic of her solution, but the majority of her respondents remained unconvinced, with one writing nearly a year later:

“I still think you’re wrong. There is such a thing as female logic.”

So, is it possible that the “world’s smartest person” actually got it wrong? Well, actually, no. Much of the confusion regarding the Monty Hall Problem stems from the ambiguous manner in which it was stated in Craig Whittaker’s original letter to Parade Magazine. Most of those who disagree with vos Savant’s answer assume that the host’s choice of which door to open is entirely random, and in this case the conclusion that the contestant’s chances become 50/50 would be correct. However, in Steve Selvin’s original formulation of the problem this is not the case. After all, if the host’s choice of door were random, there is a chance he would open the door with the prize behind it, ruining the game. Thus, the host must always open a door with a goat, and it is this detail which makes all the difference.

To understand why, imagine the three possible scenarios when playing the game: you can either guess the Prize, Goat 1, or Goat 2. If you guess the Prize, then your best strategy is to stay put, since switching will lose you the game. But if you choose Goat 1 or Goat 2, then the best strategy is to switch. As the optimal strategy in 2 out of 3 possible scenarios is to switch, your chances of winning are 2/3 if you switch compared to only 1/3 if you stick to your original guess. So your best bet is always to switch.

So how did a seemingly innocuous probability problem manage to ignite such fierce and passionate condemnation? The Monty Hall problem is what American Philosopher Willard Quine called a veridical paradox – a result that intuitively looks false but can nonetheless be logically proven to be true. Human intuition is particularly ill-suited to dealing with problems of probability, and when presented with a counter-intuitive solution a common reaction is simply to reject it outright. In the case of Marilyn vos Savant this knee-jerk incredulity – along with the egos of top athematicians and not a small amount of sexism – appear to have combined into the perfect storm of academic controversy.

Thankfully, however, history appears to have vindicated Marilyn vos Savant, with polls showing that by 1992 56% of readers and 71% of academics had accepted her solution, compared to only 8% and 35% two years before. And even Robert Sachs of George Mason University, once among vos Savant’s harshest critics, eventually wrote her to repent his former arrogance:

“After removing my foot from my mouth I’m now eating humble pie. I vowed as penance to answer all the people who wrote to castigate me. It’s been an intense professional embarrassment.”

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