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In another famous experiment by Kahneman and Tversky (1973), half of the participants read a story, ‘cover story’, which said that psychologists prepared 100 personality descriptions on the basis of interviewing and testing 30 engineers and 70 lawyers. The other half of the participants read almost the same cover story, except that the number of engineers and the number of lawyers were switched; 70 engineers and 30 lawyers. Then participants were presented with personality descriptions – supposedly randomly selected from the 100 personality descriptions – and were asked to judge the probability of that person being an engineer. Here is one such description:

Jack is a 45-year-old man. He is married and has four children. He is generally conservative, careful, and ambitious. He shows no interest in political and social issues and spends most of his free time on his many hobbies which include home carpentry, sailing, and mathematical puzzles.

One finding was that participants made a judgment on the basis of the stereotypes associated with the two occupations – engineer and lawyer – which is consistent with what we saw in the Linda experiment. Linda was regarded as most likely to be active in the feminist movement, which nicely fits the stereotype provided in her personal description. Similarly, in this study, Jack was most likely judged to be an engineer rather than a lawyer because his personality description nicely fits the stereotype of engineers. The crucial finding was that the base-rate information was largely neglected; the judgment was independent of the base rates provided in the cover stories.

This error constitutes a violation of Bayes’ rule, which says that the probability of an hypothesis H given an observation O (‘posterior probability’, P(H/O)) is determined by both how likely O is if H is true (‘likelihood’, P(O/H)), and how probable H is without the observation (‘prior probability’, P(H)). For example, the probability of Jack being an engineer given the personality description is determined by both how likely the personal description (being generally conservative, careful, ambitious, etc.) is if Jack is really an engineer, and how probable it is that Jack is an engineer without the personal description. What happened in the experimental results was that the participants largely ignored the prior probability (i.e., how probable it is that Jack is an engineer without the personal description) which is determined by the base-rate. Thus, Kahneman and Tversky summarize their finding as follows:

One of the basic principles of statistical prediction is that prior probability, which summarizes what we knew about the problem before receiving independent specific evidence, remains relevant even after such evidence is obtained. Bayes’ rule translates this qualitative principle into a multiplicative relation between prior odds and the likelihood ratio. Our subjects, however, fail to integrate prior probability with specific evidence. […] The failure to appreciate the relevance of prior probability in the presence of specific evidence is perhaps one of the most significant departures of intuition from the normative theory of prediction. (Kahneman & Tversky 1973, 243)