16.4: The Conjunction Fallacy

We begin this section with two puzzles.

1. 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 participated in antinuclear demonstrations. Which is more likely:
   • Linda is a bank teller.
   • Linda is a bank teller who is active in the feminist movement.
2. Which alternative seems more likely to occur within the next ten years?
   • An all-out nuclear war between the United States and Russia.
   • An all-out nuclear war between the United States and Russia in which neither country intends to use nuclear weapons, but both sides are drawn into the conflict by the actions of a conflict that spirals out of control in the Middle East.

In both cases, the second alternative is a conjunction that includes the first alternative as one of its conjuncts. So, for the second option to be right (in either case), it must be possible for a conjunction to be more probable than one of its conjuncts. But this can never happen. But these examples attest to our tendency to judge some conjunctions more probable than their conjuncts. Since this involves bad reasoning, we will call it conjunction fallacy.

There are three ways to see that such reasoning is fallacious. First, we can think about what would have to be the case if it were right. How could the probability of two things happening together be greater than the probability of either one happening by itself? After all, for both to occur together, each of the two must occur.

Second, the point is a consequence of the rule for probabilities of conjunctions. This is easiest to see in the case where the conjuncts are independent, though the same idea applies in cases where they are not. Unless one of the conjuncts has a probability of 1, we will be multiplying a number less or equal to 1 by a number less than 1, and the result will have to be smaller than either of the two original numbers.

When A and B are independent, Pr(A & B) = Pr(A) x Pr(B). If the probability of each conjunct is 1, then the probability of the conjunction itself will be 1. But in most real-life situations, the probabilities of the two conjuncts is less than 1. In that case, the conjunction will be less probable than either conjunct. For example, if Pr(A) = .9 and Pr(B) = .9, the probability of the entire conjunction is only .81. If Pr(A) = .7 and Pr(B) = .6, the probability of the conjunction is .42. Third, and best, we can draw a diagram to represent the situation.

Figure \(\PageIndex{1}\): Feminist Bank Tellers

The crosshatched area where the circles overlap represents the set of bank tellers who are also feminists. Clearly, this area cannot be larger than the entire circle on the left which represents bank tellers.

Specificity and Probability

As we add detail to a description, it often becomes more specific. And as it becomes more detailed and specific, it becomes less probable. For example, suppose that you are going to toss a quarter once. The probability of it landing heads is 1/2. But the probability it will land heads with Washington looking generally north is less, and the probability he’ll be looking due north is very small. Indeed, the probability he’ll be looking in any direction that you specify precisely before the flip is minuscule.

This relates directly to the conjunction fallacy, because adding more detail is just a matter of adding more conjuncts, and adding more conjuncts typically adds more detail. To say that the quarter will land heads with Washington looking north is to say that it will land heads and Washington will be looking north. And as always, a conjunction cannot be more probable than either of its conjuncts. Finally, note that we don’t need to know precise probability numbers to appreciate the fundamental point that the probability of the conjunction can never be greater than the probability of its least probable conjunct, whatever its probability might turn out to be.