Linda the Bank Teller Case Revisited

Source: Wikimedia Commons

Amos Tversky and Daniel Kahneman are famous for their work on a large number of cognitive fallacies that we all tend to commit. One is what they call the conjunction fallacy. The most famous illustration of this fallacy is the "Linda the Bank Teller" case. Tversky and Kahneman (1983) asked participants to solve the following problem:

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 is more probable?

1. Linda is a bank teller.
2. Linda is a bank teller and is active in the feminist movement.

More than 80 percent of participants chose option 2, regardless of whether they were novice, intermediate, or expert statisticians. However, the probability of two events occurring in conjunction is always less than or equal to the probability of either one occurring alone.

Compare the "Linda" case to the following case: What is more likely: that (1) you will have a flat tire tomorrow morning or that (2) you will have a flat tire tomorrow and that a man in a black car will stop to help you out. In this case, it should be evident that (2) is not the most likely outcome.

One explanation for why we commit the conjunction fallacy is that we incorrectly employ what Tversky and Kahneman call the representative heuristic. Even though logically we should not pick option 2, we consider option 2 more likely to be correlated with what Linda did in college. With her special background, Linda is representative of someone who is a feminist.

But maybe Tversky and Kahneman are not quite right about the Linda case. Research participants exposed to the exercise might be making a mistake because they didn’t assign a literal reading to the information provided.

In most everyday contexts, people do not attempt to communicate what they express semantically but are attempting to convey non-literal unexpressed information as well. A well-known example is "John and Mary got married and had a baby" versus "John and Mary had a baby and got married." The two sentences are logically equivalent and have the same semantic meaning but in ordinary discourse, the sentences normally also convey a temporal order, as in "John and Mary got married, and then they had a baby" versus "John and Mary had a baby, and then they got married."

In the "Linda the Bank Teller" case there are two ways in which a non-literal reading may be assigned to the case. Consider the difference between 1 and 2:

1.

(a) Linda is a bank teller.

(b) Linda is a bank teller and a feminist.

2.

(a) Linda is only a bank teller (that is, a bank teller but not a feminist).

(b) Linda is a bank teller and a feminist.

1(a) does not exclude any feminist bank tellers. So, any person who falls into the 1(b) category also falls into the 1(a) category. Since it is logically impossible for a person to fall into the 1(b) category without falling into the 1(a) category, it cannot be more likely for a person to be in the 1(b) category than the 1(a) category. All else being equal, it is more likely for a person to be a bank teller but not a feminist than it is for a person to be both a bank teller and a feminist.

Now let's consider the second formulation. If someone falls into category 2(b), then as a matter of necessity they do not fall into category 2(a). Yet if we pick an arbitrary individual, it is more likely that they are a bank teller and not a feminist than it is that they are a bank teller and a feminist.

However, Linda is not a randomly chosen individual. The reader is given background information about Linda. The background information tells us that when Linda was in college, she was a devoted feminist. If the reader assumes that the majority of people who are devoted feminists in college continue to be feminists later on, then the only rational response to the question of Linda’s occupation is is to say that there is a greater chance that Linda is a bank teller and a feminist than a bank teller and not a feminist.

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Granted, the task that the research participants were asked to complete was that of determining the probability concerning a case in which the semantic meaning given is that of the first formulation above (1). So, if the task is correctly followed, then the answer is that it is more likely that Linda is a bank teller (rather than a bank teller and a feminist).

But the skill of providing answers based on the meaning that is literally given to us is not typically a useful skill. If the host at a conference asks you to find out whether the keynote speaker has already had breakfast, and you discover that the keynote had breakfast on the previous day but not that same morning, you would commit no semantic errors if you reported back to the host that the keynote already had breakfast. Saying this to the host, however, would be a mistake. Even though the host did not mention it, she was interested in knowing whether the keynote had breakfast that same morning and not whether she had breakfast yesterday or a week ago.

The upshot is that people’s intuitive answer is grounded in a useful intellectual skill: that of being able to determine what the speaker is attempting to convey rather than what is semantically expressed in real-world environments.

The so-called cognitive flaw made by research participants in the Linda case also turns on the formulation of the problem. Suppose the task really is to determine which of the two options is more probable in (1). People may be more likely to provide the correct answer if the literal meaning is made explicit. For example, the two answer options could have been formulated as follows:

3.

(a) Linda is a bank teller (and we are not saying that she is only a bank teller and not also a feminist. We are leaving that option open).

(b) Linda is a bank teller and a feminist.

Given this way of articulating the problem, if the background information about Linda’s college days is given a lot of weight, we would expect research participants to assign equal probability to 3(a) and 3(b). If the instructions further include a remark to the effect that it is not the case that most college feminists continue to be feminists, people might assign a higher probability to 3(a), which is the desired outcome in this particular case.