# 3.6: The Conjunction Fallacy

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

In this and the remaining sections of this chapter, we will consider some formal fallacies of probability. These fallacies are easy to spot once you see them, but they can be difficult to detect because of the way our minds mislead us—analogous to the way our minds can be misled when watching a magic trick. In addition to introducing the fallacies, I will suggest some psychological explanations for why these fallacies are so common, despite how easy they are to see once we’ve spotted them.

The conjunction fallacy is best introduced with an example.6

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.

Given this information about Linda, which of the following is more probable?

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

If you are like most people who answer this question, you will answer “b.” But that cannot be correct because it violates the basic rules of probability. In particular, notice that option b contains option a (i.e., Linda is a bank teller). But option b also contains more information—that Linda is also active in the feminist movement. The problem is that a conjunction can never be more probable than either one of its conjuncts. Suppose we say it is very probable that Linda a bank teller (how boring, given the description of Linda which makes her sound interesting!). Let’s set the probability low, say .4. Then what is the probability of her being active in the feminist movement? Let’s set that high, say .9. However, the probability that she is both a bank teller and active in the feminist movement must be computed as the probability of a conjunction, like this:

.4 × .9 = .36

So given these probability assignments (which I’ve just made up but seem fairly plausible), the probability of Linda being both a bank teller and active in the feminist movement is .36. But .36 is a lower probability than .4, which was the probability that she is bank teller. So option b cannot be more probable than option a. Notice that even if we say it is absolutely certain that Linda is active in the feminist movement (i.e., we set the probability of her being active in the feminist movement at 1), option b is still only equal to the probability of option a, since (.4)(1) = .4.

Sometimes it is easy to spot conjunction fallacies. Here is an example that illustrates that we can in fact easily see that a conjunction is not more probable than either of its conjuncts.

Mark is drawing cards from a shuffled deck of cards. Which is more probable?