One of the most famous television game shows from the heyday of the genre from the 1950s to the 1980s was Let’s Make a Deal. Its host, Monty Hall, achieved a second kind of fame when a dilemma in probability theory, loosely based on the show, was named after him. A contestant is faced with three doors. Behind one of them is a sleek new car. Behind the other two are goats. The contestant picks a door, say Door 1. To build suspense, Monty opens one of the other two doors, say Door 3, revealing a goat. To build the suspense still further, he gives the contestant an opportunity either to stick with their original choice or to switch to the unopened door. You are the contestant. What should you do?
Almost everyone stays. They figure that since the car was placed behind one of the three doors at random, and Door 3 has been eliminated, there is now a fifty‑fifty chance each that the car will be behind Door 1 or Door 2. Though there’s no harm in switching, they think, there’s no benefit either. So they stick with their first choice out of inertia, pride, or anticipation that their regret after an unlucky switch would be more intense than their delight after a lucky one.
The Monty Hall dilemma became famous in 1990 when it was presented in the “Ask Marilyn” column in Parade, a magazine inserted in the Sunday edition of hundreds of American newspapers. The columnist was Marilyn vos Savant, known at the time as “the world’s smartest woman” because of her entry in the Guinness Book of World Records for the highest score on an intelligence test. Vos Savant wrote that you should switch: the odds of the car being behind Door 2 are two in three, compared with one in three for Door 1. The column drew ten thousand letters, a thousand of them from PhDs, mainly in mathematics and statistics, most of whom said she was wrong. Here are some examples:
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, PhD, 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, PhD, Georgia State University
Maybe women look at math problems differently than men.
—Don Edwards, Sunriver, Oregon
Among the objectors was Paul Erdös (1913–1996), the renowned mathematician who was so prolific that many academics boast of their “Erdös number,” the length of the shortest chain of coauthorships linking them to the great theoretician.
But the mansplaining mathematicians were wrong and the world’s smartest woman was right. You should switch. It’s not that hard to see why. There are three possibilities for where the car could have been placed. Let’s consider each door and count up the number of times out of the three that you would win with each strategy. You picked Door 1, but of course that’s just a label; as long as Monty follows the rule “Open an unselected door with a goat; if both have goats, pick one at random,” the odds come out the same whichever door you picked.
Suppose your strategy is “Stay” (left column in the figure). If the car is behind Door 1 (top left), you win. (It doesn’t matter which of the other doors Monty opened, because you’re not switching to either.) If the car is behind Door 2 (middle left), you lose. If the car is behind Door 3 (bottom left), you lose. So the odds of winning with the “Stay” strategy are one in three.
Now suppose your strategy is “Switch” (right column). If the car is behind Door 1, you lose. If the car is behind Door 2, Monty would have opened Door 3, so you would switch to Door 2 and win. If the car is behind Door 3, he would have opened Door 2, so you would switch to Door 3 and win. The odds of winning with the “Switch” strategy are two in three, double the odds of staying.
It’s not rocket surgery. Even if you don’t work through the logical possibilities, you could play a few rounds yourself with cutouts and toys and tot up the outcomes, as Hall himself did to convince a skeptical journalist. (Nowadays, you can play it online.) Or you could pursue the intuition “Monty knows the answer and gave me a clue; it would be foolish not to act on it.” Why did the mathematicians, professors, and other bigshots get it so wrong?
Many people insist that each of the unknown alternatives (in this case, the unopened doors) must have an equal probability. That is true of symmetrical gambling toys like the faces of a coin or sides of a die, and it is a reasonable starting point when you know absolutely nothing about the alternatives. But it is not a law of nature.
Certainly there were failures of critical thinking coming from sexism, ad hominem biases, and professional jealousy. Vos Savant is an attractive and stylish woman with no initials after her name who wrote for a recipe‑ and gossip‑filled rag and bantered on late‑night talk shows. She defied the stereotype of a mathematician, and her celebrity and bragging rights from Guinness made her a big fat target for a takedown.
But part of the problem is the problem itself. Many people can’t swallow the correct explanation even when it’s pointed out to them. This included Erdös, who, violating the soul of a mathematician, was convinced only when he saw the game repeatedly simulated. Many persist even when they see it simulated and even when they repeatedly play for money. What’s the mismatch between our intuitions and the laws of chance?
A clue comes from the overconfident justifications that the know‑it‑alls offered for their blunders, sometimes thoughtlessly carried over from other probability puzzles. Many people insist that each of the unknown alternatives (in this case, the unopened doors) must have an equal probability. That is true of symmetrical gambling toys like the faces of a coin or sides of a die, and it is a reasonable starting point when you know absolutely nothing about the alternatives. But it is not a law of nature.
Many people can’t swallow the correct explanation even when it’s pointed out to them. This included Erdös, who, violating the soul of a mathematician, was convinced only when he saw the game repeatedly simulated.
Many visualize the causal chain. The car and goats were placed prior to the reveal, and opening a door can’t move them around after the fact. Pointing out the independence of causal mechanisms is a common way to debunk other illusions such as the gambler’s fallacy, in which people misguidedly think that after a run of reds the next spin of the roulette wheel will turn up black, when in fact the wheel has no memory, so every spin is independent. As one of vos Savant’s correspondents mansplained, “Picture a race with three horses, each having an equal chance of winning. If horse #3 drops dead 50 feet into the race, the chances for each of the remaining two horses are no longer one in three but rather are now one in two.” Clearly, he concluded, it would not make sense to switch one’s bet from horse #1 to horse #2. But this is not how the problem works. Imagine that after you place your bet on #1, God announces, “It’s not going to be horse #3.” He could have warned against horse #2 but didn’t. Switching your bet doesn’t sound so crazy. In Let’s Make a Deal, Monty Hall is God.
The godlike host reminds us how exotic the Monty Hall problem is. It requires an omniscient being who defies the usual goal of a conversation—to share what the hearer needs to know (in this case, which door hides the car)—and instead pursues the goal of enhancing suspense among third parties. And unlike the world, whose clues are indifferent to our sleuthing, Monty Almighty knows the truth and knows our choice and picks his revelation accordingly.
People’s insensitivity to this lucrative but esoteric information pinpoints the cognitive weakness at the heart of the puzzle: we confuse probability with propensity. A propensity is the disposition of an object to act in certain ways. Intuitions about propensities are a major part of our mental models of the world. People sense that bent branches tend to spring back, that kudu may tire easily, that porcupines usually leave tracks with two padprints. A propensity cannot be perceived directly (either the branch sprang back or it didn’t), but it can be inferred by scrutinizing the physical makeup of an object and working through the laws of cause and effect. A drier branch may snap, a kudu has more stamina in the rainy season, a porcupine has two proximal pads which leave padprints when the ground is soft but not necessarily when it is hard.
People’s insensitivity to this lucrative but esoteric information pinpoints the cognitive weakness at the heart of the puzzle: we confuse probability with propensity.
But probability is different; it is a conceptual tool invented in the seventeenth century. The word has several meanings, but the one that matters in making risky decisions is the strength of one’s belief in an unknown state of affairs. Any scrap of evidence that alters our confidence in an outcome will change its probability and the rational way to act upon it. The dependence of probability on ethereal knowledge rather than just physical makeup helps explain why people fail at the dilemma. They intuit the propensities for the car to have ended up behind the different doors, and they know that opening a door could not have changed those propensities. But probabilities are not about the world; they’re about our ignorance of the world. New information reduces our ignorance and changes the probability. If that sounds mystical or paradoxical, think about the probability that a coin I just flipped landed heads. For you, it’s .5. For me, it’s 1 (I peeked). Same event, different knowledge, different probability. In the Monty Hall dilemma, new information is provided by the all‑seeing host.
One implication is that when the reduction of ignorance granted by the host is more transparently connected to the physical circumstances, the solution to the problem becomes intuitive. Vos Savant invited her readers to imagine a variation of the game show with, say, a thousand doors. You pick one. Monty reveals a goat behind 998 of the others. Would you switch to the door he left closed? This time it seems clear that Monty’s choice conveys actionable information. One can visualize him scanning the doors for the car as he decides which one not to open, and the closed door is a sign of his having spotted the car and hence a spoor of the car itself.
Adapted from Rationality by Steven Pinker, published by Viking, an imprint of Penguin Publishing Group, a division of Penguin Random House, LLC. Copyright © 2021 by Steven Pinker.