A fascinating new study, Tversky and Kahneman’s Cognitive Illusions: Who Can Solve Them, and Why?, probes into the cognitive “heuristics and biases” researched by Daniel Kahneman and Amos Tversky since the late 1960s.
If you have never encountered the “Linda brain teaser” before, please give it a try:
Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.
Which statement is more probable?
(a) Linda is a bank teller.
(b) Linda is a bank teller and is active in the feminist movement.
Quick! What’s your answer?
Solution and Explanation
If you said b) is more probable, you are in good company … and wrong.
Answer b) is what most people answer the first time they face this particular brain teaser, reflecting the pervasive cognitive bias called a “conjunction fallacy.” Statistically speaking, it is more probable that Linda is a bank teller–feminist or not–that she is both a bank teller AND also active in the feminist movement, which is obviously a subset of the whole category of bank tellers.
As the researchers elaborate in the new study (key portions bolded):
The probability of the simultaneous occurrence of two events—for example, p(Bank teller And Feminist)—can be mathematically obtained by multiplying the two involved single probabilities, that is, p(B) And p(F), or—in the case of the stochastical dependency of B and F—p(B) And p(F|B). However, the product of two numbers between 0 and 1 always becomes smaller than each of both factors, which is why (a) is the correct option. The description of Linda turns out to be irrelevant here, since it is always more unlikely that two events will happen simultaneously than that only one of both constituents will (thus the content of the events is irrelevant here, too). All that counts are the terms “probability” and “and,” which the conjunction rule interprets, respectively, as mathematical probability and the logical operator “and” (Hertwig, 1995; Gigerenzer and Regier, 1996; Hertwig et al., 2008).
Yet Tversky and Kahneman (1983) found that about 80–90% of participants judged the second option (B and F) to be more probable than the first option (B). In terms of the heuristics and biases program, the Linda problem is another instance of the representativeness heuristic, since the second option seems to be more representative of Linda than the first. The so-called “conjunction fallacy” in the form of the Linda task or similar problems has also been examined extensively since then (e.g., Fiedler, 1988; Reeves and Lockhart, 1993; Donovan and Epstein, 1997; Hertwig et al., 2008; Wedell and Moro, 2008; Charness et al., 2010). Hertwig and Chase (1998), for instance, found that the proportion of conjunction fallacies could be substantially reduced (from 78% to 42%) by changing the response format from ranking to concrete probability estimation. Interestingly, although there is no concrete probability given, the Linda problem can also be understood more easily using the natural frequency concept introduced in the context of Bayesian reasoning problems (see above). When participants are simply instructed to imagine 200 women who fit Linda’s description, they realize that there must be more women who are bank tellers than women who are both bank tellers and feminists (for details see, e.g., Fiedler, 1988; Hertwig and Gigerenzer, 1999).