The greatest obstacle to discovery is not ignorance— it is the illusion of knowledge.
—Daniel Boorstin, Librarian of Congress, 1984
Let’s begin with another little intuition checkup. Quick and easy now, starting with some gut checks on your intuitive physics:
1. The diagram shows a curved tube, lying flat on a table. A BB is shot into the opening and out the other end. With your finger on the page, draw the BB’s path through the tube and after it shoots out the tube.
2. While flying at a constant speed, a plane drops a bowling ball. Draw the path the ball will follow (ignoring wind resistance) and show where the plane will be as the ball hits the ground. If a BB were dropped at the same time as the bowling ball, which would hit the ground first?
3. Water is about to be poured from a glass into a bowl resting on a table below. Draw a line in the glass, representing the water’s surface (starting at the designated point)
How’s your arithmetic?
4. The Brownsons drive to Kalamazoo at an average speed of 60 mph but on their return are stuck in slow traffic and average only 30 mph. What was their average speed for the round-trip? 5. A farmer bought a horse for $60 and sold it for $70. Then the
farmer bought the horse back for $80 and sold it again, for $90. How much money did the farmer make on this horse trading?
Finally, let’s check your intuitive understanding of probabilities: 6. The people in your city have a 1 percent risk of having bone
cancer. Everyone is therefore invited to take a test that is 90 percent reliable (it spots the cancer in 90 percent of those who have it, and 10 percent of the time gives a false positive re- port). You take the test and are given the bad news: a positive report. What is the probability you have bone cancer? 7. I shuffle a deck of 80 black and 20 red cards. As I turn the
cards up (after replacing and reshuffling the last card), you re- ceive $1 each time you guess correctly whether black or red is about to appear. To pocket the most money, what percent of the time should you say ‘‘black’’ and ‘‘red’’?
8. I toss two coins, promising that if at least one of them comes up heads I will tell you. I look at both coins and volunteer that at least one is indeed a head. What’s the probability that the other is also a head?
9. Suppose you are on Monty Hall’s old Let’s Make a Deal televi- sion show and are given the choice of three doors. Behind one is a car; behind the others, goats. You pick door number 1. The host, who knows what’s behind the doors, opens number 3, which has a goat. He then says to you, ‘‘Do you want to switch to door number 2?’’ Should you switch, or does it not matter? Ready to check your intuitive physics, arithmetic, and probabili- ties? The answer key:
1. The BB exits in a straight line. About half of Johns Hopkins students, when queried in Michael McCloskey’s studies of intu-
itive theories of motion, presumed the BB would continue a curved path.
2. The ball will drop in a forward curve, with the plane directly above it as it hits the ground. Forty percent of McCloskey’s Hopkins students intuited arcs resembling the actual path (A). And contrary to Aristotle’s idea that heavy objects fall faster, a BB and a bowling ball would reach earth simultaneously. Though wrong, the Aristotelian idea intuitively felt right enough to have lasted for centuries.
3. On the ‘‘water-level task’’ devised by Jean Piaget, up to 40 per- cent of the population incorrectly intuits that the water would deviate from horizontal (indicated by the dotted line).
4. The Brownsons averaged 40 mph. If they had a sixty-mile drive each way, it would have taken them one hour going and two hours returning—thus three hours to drive 120 miles. 5. The farmer made $20. Most people, including most German
banking executives (a German colleague tells me), answer $10. But let’s do the accounting:
Buying price Selling price (amount paid) (amount received) Deal 1 $60 $70
Deal 2 $80 $90
Total $140 $160
If this isn’t convincing, reread the question with this second sentence: ‘‘Then the farmer bought some bricks for $80 and sold them for $90.’’ (Should it matter whether the second deal was bricks or a horse?) If still in doubt, get out some Monopoly money and go through the transactions.
6. With a 90 percent reliable test, the probability (given your alarming positive result) is 92 percent that you don’t have can- cer. If 1,000 people show up for the test and 1 percent—ten people—actually have the bone cancer, the test will spot it in about nine of them. So far so good. But what about the other 990? A 10 percent misdiagnosis rate would yield 99 false posi- tives (92 percent of the 108 people who were given a true or false positive outcome). Studies show that most physicians fail to comprehend these elementary mathematics.
7. You should say black 100 percent of the time, which would earn you about $80. Saying black 80 percent of the time would yield about $68 ($64 on correct guesses when black turned up and $4 for the 20 percent of the time you correctly guessed red). 8. Can we agree that there are four equally likely outcomes to the
two coin tosses: TT, HH, TH, and HT? Because I’ve revealed that the first didn’t happen, I’ve ruled out TT. Of the three re- maining possibilities, only one has a second heads. So the odds are one out of three (not 50/50) that the second coin is a head.
9. Finally, the mother of all beguiling mental puzzles, the Monty Hall Dilemma (which in a different format was introduced by Martin Gardner in a 1959 Scientific American column). When a reader posed the dilemma to Parade columnist Marilyn vos Savant, she answered, ‘‘Yes, you should switch.’’ That set off a storm of more than 10,000 letters, nine in ten disagreeing, and a series of articles in statistical journals, newspapers, and mag- azines. Nevertheless, when the dust settled it was clear from both logical analysis and empirical simulations that vos Savant was right. Think of it this way: The chances are 1 in 3 that you initially picked the right door, and 2 in 3 that it’s one of the other two. When the host eliminates one of those two (the host always opens the unchosen door that isn’t the prize door), there still are 2 chances in 3 that the correct door is not the one you picked. (Since your original guess would be wrong two out of three times, the other door—the door you switch to, if you switch—must be the right one two out of three times.) When more than 70,000 folks played the game (as you can at Na- tional Public Radio’s Car Talk website—cartalk.cars.com/ About/Monty) 33.1 percent of ‘‘stickers’’ and 66.7 percent of ‘‘switchers’’ were indeed winners.
Let’s not go overboard. Some unschooled intuitions hit the mark. As I noted in Chapter 1, even babies have an innate counter and a head for elementary physics. If accustomed to a Daffy Duck puppet jumping three times on stage, they show surprise if it jumps only twice. (They stare longer—as if doing a double-take.)
Yet these little brain teasers illustrate that intuition, even when informed by experience and observation, sometimes misses the mark. As K. C. Cole writes, ‘‘Math—that most logical of sciences— shows us that the truth can be highly counterintuitive and that sense is hardly common.’’
Okay, maybe math and physics never were our best subjects. Surely we do better when it comes to judging people, politics, and practicalities. As La Rochefoucauld observed, people may complain about their memory, but never their judgment. Indeed, thanks to our intuitive efficiency and accuracy, we generally navigate life quite well. If we had to analyze every judgment, we’d never get through the day. As Robert Ornstein writes, ‘‘There has never been, nor will there ever be, enough time to be truly rational.’’ But on judgments that really matter, and where quick and rough intuitive approximations may stray from reality, critical thinking can help.