That result was all that Danny and Amos needed to make their big point: The rules of thumb people used to evaluate probability led to misjudgments. “Linda is a bank teller and is active in the feminist movement” could never be more probable than “Linda is a bank teller.” “Linda is a bank teller and active in the feminist movement” was just a special case of “Linda is a bank teller.” “Linda is a bank teller” included “Linda is a bank teller and activist in the feminist movement” along with “Linda is a bank teller and likes to walk naked through Serbian forests” and all other bank-telling Lindas. One description was entirely contained by the other.
People were blind to logic when it was embedded in a story. Describe a very sick old man and ask people: Which is more probable, that he will die within a week or die within a year? More often than not, they will say, “He’ll die within a week.” Their mind latches onto a story of imminent death and the story masks the logic of the situation. Amos created a lovely example. He asked people: Which is more likely to happen in the next year, that a thousand Americans will die in a flood, or that an earthquake in California will trigger a massive flood that will drown a thousand Americans? People went with the earthquake.
The force that led human judgment astray in this case was what Danny and Amos had called “representativeness,” or the similarity between whatever people were judging and some model they had in their mind of that thing. The minds of the students in the first Linda experiment, latching onto the description of Linda and matching its details to their mental model of “feminist,” judged the special case to be more likely than the general one.
Amos wasn’t satisfied with stopping there. He wanted to hand the entire list of Lindas to groups of people and have them rank the odds of each line item. He wanted to see if a person who decided that “Linda is a bank teller activist in the feminist movement” also thought it was more probable than “Linda is a bank teller.” He wanted to show people making that glaring mistake. “Amos really loved to do that,” said Danny. “To win the argument, you want people to actually make mistakes.”
Danny was of two minds about this new project, and about Amos. From the moment they had left Israel, they’d been like a pair of swimmers caught in different currents, losing the energy to swim against them. Amos felt the pull of logic, Danny the tug of psychology. Danny wasn’t nearly as interested as Amos in demonstrations of human irrationality. His interest in decision theory ended with the psychological insight he brought to it. “There is an underlying debate,” said Danny later. “Are we doing psychology or decision theory?” Danny wanted to return to psychology. Plus Danny didn’t believe that people would actually make this particular mistake. Seeing the descriptions side by side, they’d realize that it was illogical to say that anyone was more likely to be a bank teller active in the feminist movement than simply a bank teller.
With something of a heavy heart, Danny put what would come to be known as the Linda problem to a class of a dozen students at the University of British Columbia. “Twelve out of twelve fell for it,” he said. “I remember I gasped. Then I called Amos from my secretary’s phone.” They ran many further experiments, with different vignettes, on hundreds of subjects. “We just wanted to look at the boundaries of the phenomenon,” said Danny. To explore those boundaries, they finally shoved their subjects’ noses right up against logic. They gave subjects the same description of Linda and asked, simply: “Which of the two alternatives is more probable?”
Linda is a bank teller.
Linda is a bank teller and is active in the feminist movement.
Eighty-five percent still insisted that Linda was more likely to be a bank teller in the feminist movement than she was to be a bank teller. The Linda problem resembled a Venn diagram of two circles, but with one of the circles wholly contained by the other. But people didn’t see the circles. Danny was actually stunned. “At every step we thought, now that’s not going to work,” he said. And whatever was going on inside people’s minds was terrifyingly stubborn. Danny gathered an auditorium full of UBC students and explained their mistake to them. “Do you realize you have violated a fundamental rule of logic?” he asked. “So what!” a young woman shouted from the back of the room. “You just asked for my opinion!”
They put the Linda problem in different ways, to make sure that the students who served as their lab rats weren’t misreading its first line as saying “Linda is a bank teller NOT active in the feminist movement.” They put it to graduate students with training in logic and statistics. They put it to doctors, in a complicated medical story, in which lay embedded the opportunity to make a fatal error of logic. In overwhelming numbers doctors made the same mistake as undergraduates. “Most participants appeared surprised and dismayed to have made an elementary error of reasoning,” wrote Amos and Danny. “Because the conjunction fallacy is easy to expose, people who committed it are left with the feeling that they should have known better.”
The paper Amos and Danny set out to write about what they were now calling “the conjunction fallacy” must have felt to Amos like an argument ender—that is, if the argument was about whether the human mind reasoned probabilistically, instead of the ways that Danny and Amos had suggested. They walked the reader through how and why people violated “perhaps the simplest and the most basic qualitative law of probability.” They explained that people chose the more detailed description, even though it was less probable, because it was more “representative.” They pointed out some places in the real world where this kink in the mind might have serious consequences. Any prediction, for instance, could be made to seem more believable, even as it became less likely, if it was filled with internally consistent details. And any lawyer could at once make a case seem more persuasive, even as he made the truth of it less likely, by adding “representative” details to his description of people and events.
And they showed all over again the power of the mental rules of thumb—these curious forces that they had curiously named “heuristics.” To the Linda problem Danny and Amos added another, from work they had done in the early 1970s in Jerusalem.
In four pages of a novel (about 2,000 words), how many words would you expect to find that have the form ing (seven-letter words that end with “ing”)? Indicate your best estimate by circling one of the values below:
0 1–2 3–4 5–7 8–10 11–15 16+
Then they put to those same people a second question: How many seven-letter words appeared, in that same text, of the form n ? Of course (of course!) there had to be at least as many seven-letter words with n in the sixth position as there were seven-letter words ending in ing, as the latter was just one example of the former. People didn’t realize that, however. They guessed, on average, that the 2,000-word text contained 13.4 words ending in ing and only 4.7 words with n in the sixth position. And they did this, Amos and Danny argued, because it was easier to think of words ending in ing. Those words were more available. People’s misjudgment of the problem was simply the availability heuristic in action.