The Conjunction Fallacy

Taxonomy: Probabilistic Fallacy > The Conjunction Fallacy

Alias: The Conjunction Effect1

Thought Experiment:

Which of the following events is most likely to occur, or are they equally likely?:

  1. Vice President Mike Pence will become the next president.
  2. President Donald Trump will be impeached and Vice President Mike Pence will become the next president.

Answer

Exposition:

A conjunctive statement, or "conjunction", for short, is a sentence of the form: "…and―." For instance, the sentence: "Today is Saturday and the sun is shining" is a conjunction. A conjunct is a statement that is part of a conjunction. For example, "Today is Saturday" and "The sun is shining" are both conjuncts of the example sentence.

The probability of a conjunction is never greater than the probability of its conjuncts. In other words, the probability of two things being true can never be greater than the probability of one of them being true, since in order for both to be true, each must be true. However, when people are asked to compare the probabilities of a conjunction and one of its conjuncts, they sometimes judge that the conjunction is more likely than one of its conjuncts. This seems to happen when the conjunction suggests a scenario that is more easily imagined than the conjunct alone.

Exposure:

Exception:

A few readers4 have pointed out that in questions such as the Thought Experiment, above, or the Linda Problem, people may assume that an unstated conjunct is implicitly denied. For instance, in the Thought Experiment, readers may interpret the alternatives in the following way, where the implicit part is in parentheses:

  1. Vice President Mike Pence will become the next president (and President Donald Trump will not be impeached).
  2. President Donald Trump will be impeached and Mike Pence will become the next president.

Given this interpretation, some readers may correctly think that 2 is more likely than 1. If this is how anyone interprets the Thought Experiment, then that person did not commit the conjunction fallacy. However, such a person is guilty of an unwarranted assumption. The reason I stated the alternatives in the order that I did, above, is to forestall any tendency to interpret the first alternative as saying how Pence will become the next president. Kahneman and Tversky did something different in testing the Linda Problem, namely, the two relevant statements about Linda were included among a group of eight statements, with an intervening one.5 It may, for this reason, be that the Thought Experiment is more subject to this kind of misinterpretation than the Linda Problem, but I didn't want to clutter it up with several alternatives.6

Kahneman and Tversky were aware of this issue and addressed it by using a "between-subjects" design with some test subjects, that is, some subjects were given only the conjunction while others were given only the conjuncts to evaluate7. The conjunction effect still occurred in the between-subjects tests, that is, the subjects still tended to rank the conjunction as more probable than a conjunct. It is hard to see how this result could be explained in terms of the implicit assumption since the subjects could not compare the conjunction with its conjunct as can be done with the Thought Experiment.

Moreover, even if all of those who rank the conjunction as more probable than its conjunct are actually interpreting the problem as a comparison of the probability of two conjunctions, this would mean that the conjunction fallacy is less common in everyday reasoning than the experiments suggest. Nonetheless, the conjunction effect remains a formal fallacy of probability theory.

Technical Appendix:

Here is a proof of the theorem of probability theory that a conjunction is never more probable than its conjuncts. For the axioms cited, see the entry for Probabilistic Fallacy.

Theorem: P(s & t) ≤ P(s)

Proof: By Axiom 4 and the fact that P(s & t) = P(t & s), it follows that P(s & t) = P(t | s)P(s). Now, 0 ≤ P(t | s) ≤ 1, by Axiom 1 and the fact that P(s) ≤ 1, for all s. The theorem follows from a general fact about inequalities: if a = bc and 0 ≤ b ≤ 1, then a ≤ c.

Answer to the Thought Experiment:

2 is no more likely than 1, and probably less likely, because a conjunction is never more likely than either of its conjuncts―see the Exposition, above. That Trump is impeached and removed from office is one scenario in which Pence could become the next president, but there are other scenarios, such as that Pence should run for the office and win.

Notes:

  1. Amos Tversky & Daniel Kahneman, "Judgments of and by Representativeness", in Judgment Under Uncertainty: Heuristics and Biases, Kahneman, Paul Slovic, and Tversky, editors (1985), p. 90.
  2. P. 92.
  3. P. 93.
  4. Stanislav Shchekin, Jeremy Wenisch, and Andy Stout.
  5. P. 92.
  6. Kahneman and Tversky also tested some "statistically naive" subjects with the conjunction and its conjuncts alone, and the effect was still observed as strongly as with the additional statements; see pages 93-94. So, the Thought Experiment, above, may not be so bad. Of course, it's possible that for these blatant cases statistical sophistication may make a difference.
  7. Pp. 95-96. That is, they were given the set of eight targets with either the conjunction or the conjuncts removed.