Two Goats, a Car, and the Beauty of Mathematics
There were 10,000 letters, nearly a thousand of which were written by men and women with PhD’s. But despite the rich variety of criticisms and underhanded insults, the majority agreed: Marilyn vos Savant was wrong. Not just incorrect but, as E. Ray Bobo, PhD put it, “utterly” so.
Born in Missouri in 1946, Savant rose to fame for her extraordinarily high IQ. By the time she was 40, Guinness Book of World Records credited Savant with the “Highest IQ” for her scores of 228 and 186 on the Stanford-Binet and Mega Test, respectively. (The category was discontinued a few years later in 1990 after Guinness concluded that IQ was too unreliable to designate just one record holder.) However, to this day, there are only two other people who have recorded a higher IQ than Savant.
For Savant’s readers that sent her letters in the fall of 1990, though, her intelligence added no authority to her argument. If anything, her brain power made her logical misstep even more egregious.
“You blew it, and you blew it big!” Scott Smith, PhD wrote. “There is enough mathematical illiteracy in this country, and we don’t need the world’s highest IQ propagating more. Shame!” W. Robert Smith, PhD added, “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.” Don Edwards, thankfully not a PhD, weighed in. “Maybe women look at math problems differently than men.”
The heated letters began flooding in after a reader posed the so-called “Monty Hall” problem in Savant’s Parade magazine column.
The problem goes as follows: Imagine that you are on a game show. You’re given the choice of three doors, and what lies behind them. There are goats (ostensibly unappealing) behind two of the doors but the third hides a nice, shiny, new car. You follow your gut and (randomly) choose a door. The host of the show then opens one of the two doors you did not pick, revealing… a goat. You now have the option to switch doors and take home whatever you discover. The question — whether you should switch — is what turned Savant’s readers against her.
Our instinct — by “us” I mean the charming PhD’s and myself — is that because there are two doors concealing a goat and a car, there must be a fifty-fifty shot that one of the two remaining doors is hiding the car. In fact, when Professor Bill Peterson, at Middlebury College, first handed me a page of the juiciest letters Savant received, I understood the authors’ sentiment. Even though I was stunned by the language and vehemence behind the letters, I agreed that no matter how well she scored on a test Savant simply couldn’t change the facts of basic probability.
It took me half a fall semester sitting in Warner Hall on Middlebury’s campus to realize that Marilyn vos Savant was actually correct. But what you need to know to be as smart as Savant, at least in this case, I can teach you in the remainder of this article.
The key concepts that will enable us to solve the Monty Hall problem are complementary probability and decision trees. In most disagreements it makes sense to join the most educated camp, so I will prove to you in two different ways that Savant’s camp — albeit initially less populated and less educated — is the one to get behind.
Complements (and Compliments)
If your new socks are either ugly or cool and your friend insists they are not ugly (and you trust your friend’s opinion), then your socks are cool! We will deal with complements and compliments, focussing on the former even though the ‘utterly incorrect’ PhD’s could benefit from employing the latter.
One of the most important ideas in probability is that an event will occur with probability 1. For example, let’s assume that a random Middlebury senior is a math major with probability p (there are about 45 math majors in the 2020 class of 700). Then there is a complementary probability, 1-p, that any senior I randomly stop on campus is not a math major. The unavoidable reality is that of the two possibilities — that a student is a math major and that a student is not a math major — one must occur.
I will use complementary probability to prove to you that it is advantageous to switch doors in the Monty Hall problem, just as Savant claimed. In the very beginning before you have chosen your preliminary door or the host has revealed a goat, the car and goats are randomly distributed behind the doors. That is, there is an equal chance that any door is hiding the car. Then the probability that the car is behind any particular door is ⅓: the car must be behind some door and ⅓ + ⅓ + ⅓ = 1.
It’s intuitive that there is a ⅓ chance that the car is behind the door you choose. But what is not intuitive is that the probability remains the same when the host eliminates a door: there is no shuffle of cars and goats! So when you’re deciding to switch or not, there is still only a ⅓ chance that your original door hides the car. By taking the complement of that probability, there must be a ⅔ chance that the car is behind the other remaining door.
Of course now it makes sense to switch doors!
Well, you may not be fully satisfied. I know I wasn’t. Sure, I saw that it was best to switch, but I didn’t know why. So we’ll build an intuitive answer as to why it’s advantageous to change doors by exploring another solution.
Decision Trees
Decision trees are simply a fancy mathematical way to graphically enumerate all possibilities. We’ll investigate each outcome to prove to ourselves why we should switch.
When we first pick a door, there are only two possibilities: either we have chosen the car or the goat. From our existing understanding of complementary probability, we know there’s a ⅓ chance we picked the door with the car and a ⅔ chance we picked a door with one of the goats. In either case, the host will next open one of the two remaining doors and reveal a goat.
If we are on the first branch where we correctly identified the car, the host may pick either door to show us a goat. Then switching doors will unfortunately not give us our prized car. Our strategy fails one third of the time in the case that we initially pick the car.
On the other hand, if we are on the branch with the goat, then the host can only open one door without revealing the car. He or she has effectively done the hard work for us! Now all we have to do is switch doors and, in the two thirds of cases where we initially picked a door hiding a goat, we get our car.
When I first heard the Monty Hall problem, I was so confident that there was no advantage to exchanging doors that I agreed with the ideas — although not the tone — of the PhD’s. I simply couldn’t understand how Savant could falsely tell us there was an advantage to switching! And that is exactly the beauty of the problem: it shows us that no matter who you are or how much time you have spent in school, it’s all too easy to unabashedly assume your instinct is correct.
The beauty of math is that it challenges us to arrive at the one right answer. Through the process we get to grow and, as we did with the Monty Hall problem, expand our thinking to understand multiple paths to the “objective truth.”
Unfortunately, it’s not always easy. Everett Harman, PhD, ironically told Savant. “You made a mistake, but look at the positive side. If all those PhD’s were wrong, the country would be in some very serious trouble.” Maybe, though, we’re not in such serious trouble now that the two of us — you and I — have grown our problem-solving toolkits.
