The Linda Problem

Conjunctive fallacies

A conjunction effect or Linda problem is a bias or mistake in reasoning where adding extra details (an "and" statement or logical conjunction; mathematical shorthand to a sentence makes it appear more likely. Logically, this is not possible, because adding more claims can make a true statement false, but cannot make false statements true: If A is true, then A+B might be false (if B is false). However, if A is false, then A+B will always be false, regardless of what B is. Therefore, A+B cannot be more likely than A.

Definition and basic example

The most often-cited example of this fallacy originated with Amos Tversky and Daniel Kahneman.

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.

The majority of those asked chose option 2. However, this is logically impossible: if Linda is a bank teller active in the feminist movement, then she is a bank teller. Therefore, it is impossible for 2 to be true while 1 is false, so the probabilities are at most equal. (...)

Tversky and Kahneman argue that most people get this problem wrong because they use a heuristic (an easily calculated) procedure called representativeness to make this kind of judgment: Option 2 seems more "representative" of Linda from the description of her, even though it is clearly mathematically less likely.

In other demonstrations, they argued that a specific scenario seemed more likely because of representativeness, but each added detail would actually make the scenario less and less likely. In this way it could be similar to the misleading vividness fallacy. More recently, Kahneman has argued that the conjunction fallacy is a type of extension neglect.

Joint versus separate evaluation

In some experimental demonstrations, the conjoint option is evaluated separately from its basic option. In other words, one group of participants is asked to rank-order the likelihood that Linda is a bank teller, a high school teacher, and several other options, and another group is asked to rank-order whether Linda is a bank teller and active in the feminist movement versus the same set of options (without "Linda is a bank teller" as an option). In this type of demonstration, different groups of subjects still rank-order Linda as a bank teller and active in the feminist movement more highly than Linda as a bank teller.

Separate evaluation experiments preceded the earliest joint evaluation experiments, and Kahneman and Tversky were surprised when the effect was observed even under joint evaluation.

Other examples

While the Linda problem is the best-known example, researchers have developed dozens of problems that reliably elicit the conjunction fallacy.

Tversky & Kahneman (1981)

The original report by Tversky & Kahneman (later republished as a book chapter) described four problems that elicited the conjunction fallacy, including the Linda problem. There was also a similar problem about a man named Bill (a good fit for the stereotype of an accountant — "intelligent, but unimaginative, compulsive, and generally lifeless" — but not a good fit for the stereotype of a jazz player), and two problems where participants were asked to make predictions for events that could occur in 1981.

Policy experts were asked to rate the probability that the Soviet Union would invade Poland, and the United States would break off diplomatic relations, all in the following year. They rated it on average as having a 4% probability of occurring. Another group of experts was asked to rate the probability simply that the United States would break off relations with the Soviet Union in the following year. They gave it an average probability of only 1%.

In an experiment conducted in 1980, respondents were asked the following:

Suppose Björn Borg reaches the Wimbledon finals in 1981. Please rank order the following outcomes from most to least likely. 

• Borg will win the match 
• Borg will lose the first set
• Borg will lose the first set but win the match 
• Borg will win the first set but lose the match

On average, participants rated "Borg will lose the first set but win the match" more likely than "Borg will lose the first set". However, winning the match is only one of several potential eventual outcomes after having lost the first set. The first and the second outcome are thus more likely (as they only contain one condition) than the third and fourth outcome (which depend on two conditions).

Tversky & Kahneman (1983)

Tversky and Kahneman followed up their original findings with a 1983 paper that looked at dozens of new problems, most of these with multiple variations. The following are a couple of examples.

Consider a regular six-sided die with four green faces and two red faces. The die will be rolled 20 times and the sequence of greens (G) and reds (R) will be recorded. You are asked to select one sequence, from a set of three, and you will win $25 if the sequence you choose appears on successive rolls of the die.

1. RGRRR
2. GRGRRR
3. GRRRRR

65% of participants chose the second sequence, though option 1 is contained within it and is shorter than the other options. In a version where the $25 bet was only hypothetical the results did not significantly differ. Tversky and Kahneman argued that sequence 2 appears "representative" of a chance sequence (compare to the clustering illusion).

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[ed. Very common. Mexican immigrant/Illegal Mexican immigrant.]