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?
- Linda is a bank teller.
- 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?
- Mark draws a spade
- Mark draws a spade that is a 7
In this case, it is clear which of the options is more probable. Clearly a is more probable since it requires less to be true. Option a would be true even if option b is true. But option a could also be true even if option b were false (i.e., Mark could have drawn any other card from the spades suit). The chances of drawing a spade of any suit is 1⁄4 (or .25) whereas the chances of drawing a 7 of spades is computed using the probability of the conjunction: P(drawing a spade) = .25 P(drawing a 7) = 4/52 (since there are four 7s in the deck of 52) = .077 Thus, the probability of being both a spade and a 7 = (.25)(.077) = .019 Since .25 > .019, option a is more probable (not that you had to do all the calculations to see this). Thus there are cases where we can easily avoid committing the conjunction fallacy. So what is the difference between this case and the Linda case? The Nobel Prize-winning psychologist, Daniel Kahneman (and his long-time collaborator, Amos Tversky), has for many years suggested a psychological explanation for this difference. The explanation is complex, but I can give you the gist of it quite simply. Kahneman suggests that our minds are wired to find patterns and many of these patterns we find are based on what he calls “representativeness.” In the Linda case, the idea of Linda being active in the feminist movement fits better with the description of Linda as a philosophy major, as being active in social justice movements, and, perhaps, as being single. We build up a picture of Linda and then we try to match the descriptions to her. “Bank teller” doesn’t really match anything in the description of Linda. That is, the description of Linda is not representative of a bank teller. However, for many people, it is representative of a feminist. Thus, our minds more or less automatically see the match between representativeness of the description of Linda and option b, which mentions she is a feminist. Kahneman thinks that in cases like these, our minds substitute a question of representativeness for the question of probability, thus answering the probability question incorrectly.7 We are distracted from the probability question by seeking representativeness, which our minds more automatically look for and think about than probability. For Kahneman, the psychological explanation is needed to explain why even trained mathematicians and those who deal regularly with probability still commit the conjunction fallacy. The psychological explanation that our brains are wired to look for representativeness, and that we unwittingly substitute the question of representativeness for the question of probability, explains why even experts make these kinds of mistakes.
6 The following famous example comes from Tversky, A. and Kahneman, D. (1983). Extension versus intuitive reasoning: The conjunction fallacy in probability judgment. Psychological Review, 90(4), 293–315.
7 Kahneman gives this explanation numerous places, including, most exhaustively (and for a general audience) in his 2011 book, Thinking Fast and Slow. New York, NY: Farrar, Straus and Giroux.