Learning By DoingConsider the following scenario:
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.
- Linda is active in the feminist movement.
- Linda is a bank teller.
- Linda is a bank teller and is active in the feminist movement.
Scientists on a PlaneAfter attending a conference, I was on flight home when a woman with a poster tube sat down next to me (I had the middle seat...lucky me). Curious, I asked if she was a scientist or an architect? Her reply took me by surprise. "I'm a scientist. Don't I look like one?" I feebly tried to explain that my guess wasn't based on looks. Instead, I was trying to use information about the base rates of scientists in the US. I assumed there are more scientists than architects .
I think her question highlights a very common way of thinking. It is extremely easy to ignore the frequency of a particular class or the probability of the occurrence of a specific event. In other words, it is easy to forget about base rates (e.g., the number of female social scientists employed in the US) when there is a concrete example immediately in front of you.
Let's return to the question that opened this post. I haven't taken the time to investigate how many feminists there are in the United States, nor have I tried to mine the US Census data to figure out the number of employed bank tellers. But I know for certain that there are fewer feminist bank tellers than there are bank tellers or feminists. How do I know that? Because it is logically impossible to have more feminist bank tellers than either subgroup. The following Venn diagram proves this logical necessity (see Fig. 1).
|Figure 1. A Venn diagram depicting feminist bank tellers.|
"We are connected/Siamese twins/At the wrist" –Smashing Pumpkins, Geek U.S.A.Forgetting that overlapping populations are less frequent than their parent populations goes by the term Conjunction Fallacy. This mental slip was first introduced by Amos Tversky and Daniel Kahneman in the early 80s. They noticed that they themselves would initially make this error; however, after they had sufficient time to think about it, they realized their mistake. This dynamic duo decided that their own logical fallacy was probably symptomatic of the larger population (I mean...they were are some pretty smart dudes. If they admit falling into the conjunction fallacy, then none of us are safe). When they tested their hypothesis in a wider group, they found strong evidence that others also make the same mistake .
The S.T.E.M. ConnectionThe educational implications of the conjunction fallacy probably belong in the category of "critical thinking." We need to teach our students not to fall into the trap of thinking that data from a subpopulation is going to be more frequent than the pool from which the data are drawn. For example, the description of Linda fits closely with the stereotype that we might have about feminists. But it turns out that we should ignore that data when it conflicts with the parameters of a larger population.
If you are teaching a class on probability, the "Linda Problem" is a good way to introduce the topic of conjunctive probabilities. After the students complete the task, you can then go into a lesson of explaining the mathematics behind probabilities that depend on one another. In other words, the lesson can focus on the importance of the word "and." As a heuristic, you can explain that each time you add the word and, the probabilities are multiplied together; and when decimals are multiplied together, the entire probability decreases. As a wrap-up to the lesson, you can circle back to the Linda Problem and have a discussion about how the mathematics applies to this scenario. Once we say Linda is a bank teller and a feminist, then the proportion of people who fit that scenario drops precipitously.
Estimating the likelihood of an event or the frequency of a class (or subclass) is extremely important in assessing risk. It is used in financial estimation, medical reasoning, and in psychological experiments when defining a target population. Falling prey to the conjunction fallacy is easy, but if we remember to slow dow our thinking, we might be able to detect (and escape!) our error.
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Going Beyond the Information Given To verify my assumption, I tried to look through the US Census occupation data. It was more difficult than I initially thought because the Bureau of Labor Statistics lumps together architects and engineers. However, if we restrict our analyses strictly to architects (non-naval) and physical scientists (all other), then my assumption was correct (246,000 architects and 261,000 physical scientists).
 Tversky, A., & Kahneman, D. (1983). Extensional versus intuitive reasoning: The conjunction fallacy in probability judgment. Psychological Review, 90(4), 293.