Few of us calculate probabilities in daily life. We judge by feel, using the shortcuts Tversky and Kahneman described in 1974, which are fast and often reasonable but fail in predictable ways.

Availability

The availability heuristic means judging how common or likely something is by how easily examples come to mind. Often this works, but ease of recall also depends on how vivid, recent or widely reported something is.

In one early study, people were asked whether English has more words beginning with the letter K or more words with K as the third letter. Most chose the first, because words starting with K are easier to call up. In fact, typical English text contains more words with K in the third position. Studies of risk found a similar pattern: people tend to overestimate dramatic, heavily reported causes of death, such as tornadoes, floods or botulism, and underestimate common, quiet ones, such as diabetes or stroke.

Representativeness

The representativeness heuristic means judging probability by how closely something resembles a typical case. In a famous 1983 study, Tversky and Kahneman described “Linda”: 31, single, outspoken, a philosophy graduate deeply concerned with discrimination and social justice. Which is more probable: that Linda is a bank teller, or that she is a bank teller and active in the feminist movement? Most participants chose the second.

That answer breaks a basic rule. Two conditions together can never be more probable than one of them alone, because every feminist bank teller is also a bank teller. The mistake is called the conjunction fallacy. The second option simply resembles the description more.

Base-rate neglect

A close relative is base-rate neglect: ignoring how common something is overall when judging a particular case. Suppose a man is described as shy, tidy and fond of detail. Is he more likely a librarian or a farmer? The description sounds like a librarian, but there are far more male farmers than male librarians, so the odds may still favor farmer. A good judgment starts with the base rate and then adjusts for the specific evidence.

Anchoring

Anchoring is the tendency for an initial number to pull later estimates toward it. In the 1974 paper, participants watched a wheel of fortune, rigged to stop at either 10 or 65, and were then asked what percentage of African countries belonged to the United Nations. The wheel's number pulled their estimates toward it even though it was obviously irrelevant: the typical answer was 25 percent after seeing 10 and 45 percent after seeing 65. Later studies found that even experienced judges, in experiments, were swayed by numbers produced by rolling dice. Anchors matter in negotiations and prices (“was $200, now $99”).

The gambler's fallacy

The gambler's fallacy is believing that past independent outcomes change the odds of the next one, so that after a run of reds, black is “due.” A fair coin or roulette wheel has no memory. After five heads in a row, the chance of heads on the next toss of a fair coin is still one half. A widely repeated story holds that in 1913 a roulette wheel at Monte Carlo landed on black 26 times in a row, while gamblers lost fortunes betting that red was overdue.

The opposite belief, that a streak will keep going, appears in the idea of a “hot hand” in sports. Level 7 explains why research on that idea has recently taken a surprising turn.

A citizen who knows these shortcuts can ask: Am I judging by vivid examples or by data? What is the base rate? Where did my starting number come from?

Worked example: the taxi problem

Tversky and Kahneman posed a problem like this one. A taxi was involved in a hit-and-run accident at night. Two taxi companies operate in the city: 85 percent of the taxis are Green and 15 percent are Blue. A witness said the taxi was Blue. Tested under similar conditions, the witness identified colors correctly 80 percent of the time and was wrong 20 percent of the time. What is the probability the taxi was really Blue?

Most people answer about 80 percent, relying on the witness's accuracy and neglecting the base rate. Natural frequencies make the problem much clearer. Imagine 100 such accidents:

• About 15 involve Blue taxis. The witness correctly says “Blue” for 80 percent of them: 12 accidents.
• About 85 involve Green taxis. The witness wrongly says “Blue” for 20 percent of them: 17 accidents.

So the witness says “Blue” in 12 + 17 = 29 of the 100 accidents, and the taxi really is Blue in only 12 of those. The probability is 12 out of 29, about 41 percent. Despite a fairly reliable witness, the taxi is more likely Green than Blue, simply because Green taxis are so much more common.

The testimony still matters: it raised the probability of Blue from 15 percent to about 41 percent. Good reasoning combines the base rate with the specific evidence.

The same structure appears in medical tests and security screening: when the thing you seek is rare, even a fairly accurate signal produces many false alarms. Counting cases out of 100 or 1,000 is one of the best ways to avoid being misled.