A shy, tidy man
Here is a description. Steve, we are told, is very shy and withdrawn, always helpful but not much interested in other people. He is “a meek and tidy soul,” with a need for order and a passion for detail. Is Steve more likely to be a farmer, a salesman, an airline pilot, a librarian, or a physician?
Nearly everyone says librarian. That is the example Amos Tversky and Daniel Kahneman used in their 1974 paper in Science, and it shows the first of the three “heuristics” that paper named. A heuristic is a rule of thumb, a quick way to answer a hard question by answering an easier one. The hard question was “How likely is it that Steve is a librarian?” The easier question we answer instead is “How much does Steve resemble my picture of a librarian?” This is the representativeness heuristic. The other two, which we meet below, are availability (judging by what comes easily to mind) and anchoring (starting from a number and adjusting). They are shortcuts that usually work, and in the right situations fail in predictable ways.
What the description leaves out
The trouble is that resemblance is not the same as probability. Farmers greatly outnumber librarians. Kahneman wrote that there are more than 20 male farmers for each male librarian in the United States, and the gap is still wide. In the Bureau of Labor Statistics’ 2024 survey there were about 858,000 farmers, ranchers, and other agricultural managers (27 percent of them women) and about 180,000 librarians (89 percent of them women), which works out to roughly 30 male farmers for each male librarian. That ratio is called the base rate. Even if a shy, tidy man is several times more likely to be a librarian than a farmer, there are so many more farmers that the odds can still favor farming. The activity below lets you set the numbers.
Farmers, librarians, and Steve
Each dot is a person. Choose how many farmers there are for each librarian and how well the description fits each group, and read off the chance that a man who fits is a librarian.
The default of 20 to 1 is Kahneman’s figure; the real ratio today is larger. The shares who “fit” are made up, so slide them to see how much the answer depends on each.
Base-rate neglect is not just a classroom trick. In 1982 the physician and health-policy researcher David Eddy asked physicians a question like this: a woman has a positive mammogram; breast cancer has a 1 percent prevalence, the test finds 80 percent of cancers, and 9.6 percent of healthy women also test positive; what is the chance she has cancer? Ninety-five of 100 physicians answered between 70 and 80 percent. The right answer is about 8 percent.
The psychologist Gerd Gigerenzer found a simple repair: say it in counts of people, which he calls natural frequencies. Take 1,000 people. Ten have the disease, and nine of them test positive. Of the 990 who are healthy, about 89 also test positive. So 98 people test positive, and only 9 of them, about 9 percent, are ill. Put this way, the answer is almost obvious. In Gigerenzer and Ulrich Hoffrage’s 1995 experiments, correct answers rose from 16 percent to 46 percent when the information was given as natural frequencies; in a later study, doctors’ correct answers rose from about 10 percent to 46 percent. In a 2007 study of 160 gynecologists, 21 percent chose roughly the right answer before a short training session on natural frequencies and 87 percent afterward.
Linda
The most famous example is a woman named Linda. Linda is 31, single, outspoken, and very bright. She studied philosophy, and as a student she cared deeply about discrimination and social justice and took part in anti-nuclear demonstrations. Which is more probable?
Which is more probable?
When Tversky and Kahneman asked 142 university students this question in 1983, 85 percent said “bank teller and feminist.” In a version with eight items to rank, 85 percent of doctoral students in decision science at Stanford’s business school made the same mistake. They called it the conjunction fallacy: judging a combination of two things as more probable than one of them alone. Linda’s description is so representative of a feminist that the mind answers the resemblance question and ignores the logic.
Another view
Gigerenzer and Ralph Hertwig argued in 1999 that the word “probable” is ambiguous in everyday speech: a reader may take “Linda is a bank teller” to mean “a bank teller and not a feminist.” When the question is rephrased in counts, such as “Out of 100 people like Linda, how many are bank tellers?”, the error drops sharply: across seven problems studied by Klaus Fiedler in 1988, from 73 percent to 23 percent. Kahneman and Tversky answered that frequency judgments are biased too, and a 2001 study carried out jointly by Kahneman, Hertwig, and Barbara Mellers found that frequency wording alone did not remove the error; it mostly vanished only when the question was also stripped of distracting items. The fair summary is that the effect is robust, but its size depends heavily on the wording, and “people cannot reason about probability” overstates what the experiments show.
What comes to mind
The second heuristic is availability: we judge how common something is by how easily examples come to mind. In 1973 Tversky and Kahneman asked 152 people whether the letters K, L, N, R, and V appear more often as the first letter of an English word or as the third. By their count each of the five is more common in third place, but most people, 105 of the 152, judged first place more common for most of the letters, because it is far easier to think of words that begin with R (road, red, run) than words with R in the middle (car, bird, word).
Another view
This classic example is weaker than its fame. A modern count of Google Books by Peter Norvig confirms the pattern for R (third place is 2.6 times as common as first), N (2.2), V (2.2), and L (1.5), but for K the margin is only about 1.1 to 1, which is why textbooks that use K as the example are on shaky ground. Gigerenzer and Henry Brighton pointed out that the five letters were an unrepresentative sample, and other researchers found that people’s judgments of letter and word frequency are often quite accurate. What does hold up is availability applied to risks.
In 1978 Sarah Lichtenstein and colleagues asked students to estimate how many people die each year from various causes. Dramatic, well-reported causes were overestimated and quiet ones underestimated: tornadoes were judged more deadly than asthma, which kills about twenty times more people. The same pattern appears in how we choose to travel. Per mile traveled, Americans in cars and light trucks died at about 100 times the rate of passengers on US commercial airlines between 2000 and 2009 (7.28 against 0.07 deaths per billion passenger-miles, Savage 2013). A plane crash is vivid, and a car crash is routine. After 11 September 2001 many Americans drove instead of flying; Gigerenzer estimated that about 1,500 extra people died on the roads in the following year, though other researchers dispute the size of that figure and have found no national increase.
Memory of disaster also fades faster than the risk. After the 1989 Loma Prieta earthquake in California, the share of homeowners with earthquake insurance edged up from about 22 to about 25 percent; it reached about 31 percent after the 1994 Northridge earthquake and then fell to 12 to 14 percent by the 2000s. A study of flood insurance in the United States by Justin Gallagher (2014) found that purchases jump in the year after a flood and decay over the following years, and a study of policy lengths found that the median new flood policy was dropped within two to four years.
Why do people often fear flying more than driving, though driving kills far more people per mile?
Looking for what we expect
We also tend to look for evidence that agrees with what we already think. In 1960 the psychologist Peter Wason gave volunteers a puzzle. The experimenter has a rule that the triple 2, 4, 6 obeys. The volunteer proposes other triples and is told whether each obeys the rule, then announces the rule. Try it.
Find the rule
Test as many triples as you like, then announce the rule.
Wason’s 1960 task. Of 29 volunteers, only 6 announced the right rule without first announcing a wrong one.
The real rule is simply “each number is bigger than the one before.” Most people guess “up by two” and then test triples that obey their guess (8, 10, 12; 20, 22, 24), which always get a yes, and so they feel more and more sure of a wrong rule. The way to learn is to test something your guess says should fail. This is confirmation bias, a name for the habit of seeking, noticing, and remembering what supports our view. It appears in reporters who interview only experts who agree with them, in students who search only for sources that support the thesis, and in the superstition that the phone always rings when you think of someone: we count the hits and forget the misses.
A fair qualification comes from Joshua Klayman and Young-won Ha (1987): testing cases that fit your idea is often a sensible strategy, and it fails mainly when the idea is too narrow, as in this puzzle. The remedy is not to avoid hypotheses but to look for what would prove yours wrong.
A real-life example of the better habit comes from Taiwan. In March 2011 Yang Ming-yu, a technical specialist at the government’s food and drug agency, was testing a probiotic powder for illegal slimming drugs when her instrument showed an abnormal peak. The peak turned out to be DEHP, a plasticizer, and following it led to a supplier of “clouding agent” that had been illegally mixing the chemical into products used in drinks and foods. The case became public on 23 May 2011. Instead of dismissing a result that did not fit, her team followed it up.
First impressions and loaded words
The order of information matters. In 1946 Solomon Asch gave people a list of traits describing a person: intelligent, industrious, impulsive, critical, stubborn, envious. Another group got the same words in the reverse order, ending with “intelligent.” The first group formed a more favorable impression than the second. The first words color how we read the rest. A job interview that opens badly may never recover; one that opens well gets the benefit of the doubt. Related to this is the halo effect: a person who is good on one dimension, such as looks or charm, is assumed to be good on others, as Richard Nisbett and Timothy Wilson showed in 1977 when students who watched a warm instructor rated even his accent and appearance as pleasant, and rated the same man’s accent and appearance as irritating when he acted cold.
Words matter too. In 1974 Elizabeth Loftus and John Palmer showed people films of car accidents and asked how fast the cars were going when they “smashed into” each other. Other groups heard “collided,” “bumped,” “hit,” or “contacted.” The average estimates ran from 40.8 miles per hour for “smashed” down to 31.8 for “contacted.” A week later, 32 percent of the “smashed” group said they remembered broken glass (there was none in the film), against 14 percent of the “hit” group. How a question is asked shapes what we say we remember. Critics have argued that such wording changes our answers rather than rewriting the memory itself, but the broader finding that misleading information can distort memory has been replicated many times.
Another view
Not every textbook claim holds up. Asch also reported that “warm” versus “cold” was a “central” trait that rearranged a whole impression. A 2014 pre-registered replication with 1,023 people found that swapping warm and cold did change impressions, but warmth was not more important than competence, which was the part of the claim that made it famous. The order effect and the halo effect are sturdier. What is still debated is why they happen.
The first number
The third heuristic is the most reliable. In the best-known experiment of Tversky and Kahneman’s 1974 paper, people watched a wheel of fortune stop at 10 or at 65, and then were asked what percentage of the countries in the United Nations are in Africa. Those who saw 10 gave a median guess of 25 percent; those who saw 65 guessed 45 percent. A number that everyone could see was meaningless pulled their estimates toward it. In another demonstration, high school students had five seconds to estimate 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1; their median guess was 2,250. Another group saw 1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 and guessed 512. The true answer is 40,320: both groups were far too low, but the one that saw the big numbers first guessed higher. This is anchoring. Try it.
Does the first number pull you?
Three estimates. Each starts with a number drawn at random.
One person’s answers cannot prove anything. The effect shows up in averages over many people.
Anchoring is among the best-replicated findings in the field. In the “Many Labs” project of 2014, 36 samples of people anchored on the same kind of questions: the effects were among the largest in the whole project (d = 2.4 for the number of babies born in the US each day, 2.2 for the height of Mount Everest, 1.8 for the population of Chicago, 1.2 for the distance from San Francisco to New York). Even experts are pulled. Birte Englich and colleagues found in 2006 that experienced judges gave longer sentence recommendations after throwing a pair of dice that, unknown to them, had been rigged to show a high total. In negotiations, the first offer is a strong predictor of the final price (Galinsky and Mussweiler, 2001), and a study of more than 14,000 home sales found that higher asking prices went with higher selling prices.
Why does it work? One account is “insufficient adjustment”: we start from the anchor and move toward the answer, but we stop as soon as the number seems plausible, before we have gone far enough (Nicholas Epley and Thomas Gilovich, 2006). Another is that the anchor makes some information come to mind more readily. Both are probably right in different situations.
Closer to home
If you grew up bargaining in a market, you know the move from the inside. The seller opens high, the buyer answers low, and the price settles somewhere between; whoever names the first number sets the anchor. In a supermarket or a department store, the same effect hides inside the “original price” on a sale tag. The practical advice follows from the research. When you do not know a good price, look it up before you walk in, so you have an anchor of your own. When you do know the market, make the first offer, because first offers predict final prices. And notice a trap: a fixed price can feel like a good price simply because it is lower than the number that came first.
Another view
Some of the best-known demonstrations of anchoring are shakier than the classic ones. A 2003 study that anchored people’s willingness to pay on the last two digits of their Social Security numbers has replicated only weakly (Fudenberg, Levine, and Maniadis, 2012; Maniadis, Tufano, and List, 2014). The researcher who ran it, Dan Ariely, has since been at the center of a data-integrity dispute over other papers: one of his papers, from 2012, was retracted in 2021 after its data were shown to be fabricated. This course therefore leans on the experiments of Tversky and Kahneman and on the Many Labs replication, which independent laboratories have repeated, and flags where a result rests on one source.
Are shortcuts a mistake?
Everything so far might suggest that our rules of thumb are defects. Gerd Gigerenzer argues that this is the wrong lesson. A shortcut is not good or bad in itself; it works if it fits the world it is used in, which he calls ecological rationality. His group studies “fast and frugal” heuristics, and one of the best examples is the recognition heuristic: if you recognize one of two options and not the other, pick the one you recognize. In a 2002 study by Daniel Goldstein and Gigerenzer, Germans who had barely heard of San Antonio all said that San Diego is the larger American city, which is right, while only about two-thirds of the Americans, who knew both, got it right. (The sample in the first test was a dozen or so people, so the numbers are a demonstration, not a measurement.) Knowing less helped.
A second example comes from money. In 2009 Victor DeMiguel, Lorenzo Garlappi, and Raman Uppal tested 14 sophisticated portfolio models against the naive “1/N” rule, which splits money equally among all the options. None of the models beat it consistently. For the models to win, they would have needed about 3,000 months of data for a portfolio of 25 assets. A simple rule that ignores most of the data can beat a complicated one that has to estimate a lot of numbers.
Another view
Kahneman and Gigerenzer disagreed in public for decades, and their disagreement is real: the first measures judgments against the rules of probability and finds systematic errors; the second says those rules are often the wrong yardstick and that the same shortcuts can be accurate in the right environment. Kahneman and Tversky answered in 1996 that frequency formats do not abolish biases, and that they never claimed shortcuts are generally bad. But the two sides agree on more than their rivalry suggests. In a joint paper in 2009 with the psychologist Gary Klein, whose research is on firefighters and nurses, Kahneman wrote that intuition can be trusted when it comes from a regular environment where the person has had plenty of practice and quick feedback: a chess master, a firefighter. It cannot be trusted in a noisy environment with slow feedback, such as predicting the stock market or a political future. That is Herbert Simon’s old idea again: the question is the fit between the mind and the world it works in.
What to keep
- We answer hard questions by answering easier ones: resemblance for probability, ease of recall for frequency, the first number for the right number.
- Ignoring the base rate is the commonest error. Say it in counts of people (“out of 1,000”), and the answer usually becomes clear.
- We look for evidence that agrees with us. Test your idea by trying to break it, and notice the first impression and the loaded word.
- Shortcuts are not mistakes in themselves; they work well in familiar, regular worlds and fail in strange or noisy ones. The question to ask is what kind of world you are in.
Think it over
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Sources and further reading
- Amos Tversky and Daniel Kahneman, “Judgment under uncertainty: heuristics and biases,” Science 185 (1974); “Extensional versus intuitive reasoning: the conjunction fallacy in probability judgment,” Psychological Review 90 (1983); “Availability: a heuristic for judging frequency and probability,” Cognitive Psychology 5 (1973).
- Gerd Gigerenzer and Ulrich Hoffrage, “How to improve Bayesian reasoning without instruction: frequency formats,” Psychological Review 102 (1995); Gerd Gigerenzer and others, “Helping doctors and patients make sense of health statistics,” Psychological Science in the Public Interest 8 (2007).
- Ralph Hertwig and Gerd Gigerenzer, “The ‘conjunction fallacy’ revisited,” Journal of Behavioral Decision Making 12 (1999); Barbara Mellers, Ralph Hertwig, and Daniel Kahneman, “Do frequency representations eliminate conjunction effects?” Psychological Science 12 (2001).
- Sarah Lichtenstein and others, “Judged frequency of lethal events,” Journal of Experimental Psychology: Human Learning and Memory 4 (1978); Ian Savage, “Comparing the fatality risks in United States transportation across modes and over time,” Research in Transportation Economics 43 (2013).
- Peter Wason, “On the failure to eliminate hypotheses in a conceptual task,” Quarterly Journal of Experimental Psychology 12 (1960); Joshua Klayman and Young-won Ha, Psychological Review 94 (1987).
- Solomon Asch, “Forming impressions of personality,” Journal of Abnormal and Social Psychology 41 (1946); Elizabeth Loftus and John Palmer, Journal of Verbal Learning and Verbal Behavior 13 (1974).
- Richard Klein and others, “Investigating variation in replicability: a ‘Many Labs’ replication project,” Social Psychology 45 (2014); Nicholas Epley and Thomas Gilovich, “The anchoring-and-adjustment heuristic,” Psychological Science 17 (2006).
- Daniel Goldstein and Gerd Gigerenzer, “Models of ecological rationality: the recognition heuristic,” Psychological Review 109 (2002); Daniel Kahneman and Gary Klein, “Conditions for intuitive expertise: a failure to disagree,” American Psychologist 64 (2009).