Conjunction Fallacy (Wimbledon Example)

Predicting Wimbledon

Nisbett, Krantz, Jepson, and Kunda (1983) found that people commonly apply statistical principles in domains such as sports and gambling. The Wimbledon study asks whether that familiarity also helps people respect the conjunction rule when predicting a sports outcome.

A group of 93 subjects, recruited through an advertisement in the University of Oregon newspaper, were presented with the following problem in October 1980:

Suppose Bjorn Borg reaches the Wimbledon finals in 1981. Please rank order the following outcomes from most to least likely.

A. Borg will win the match (1.7).

B. Borg will lose the first set (2.7).

C. Borg will lose the first set but win the match (2.2).

D. Borg will win the first set but lose the match (3.5).

The average rank of each outcome (1 = most probable, 2 = second-most probable, etc.) is given in parentheses. The outcomes were chosen to represent different levels of strength for the player, Borg, with A indicating the highest strength; C, a rather lower level because it indicates a weakness in the first set; B, lower still because it only mentions this weakness; and D, lowest of all.

After winning his fifth Wimbledon title in 1980, Borg seemed extremely strong. Consequently, we hypothesized that Outcome C would be judged more probable than Outcome B, contrary to the conjunction rule, because C represents a better performance for Borg than does B. The mean rankings indicate that this hypothesis was confirmed: 72% of the respondents assigned a higher rank to C than to B, violating the conjunction rule in a direct test.

Is it possible that the subjects interpreted the target events in a nonextensional manner that could justify or explain the observed ranking? It is well-known that connectives (e.g., and, or, if) are often used in ordinary language in ways that depart from their logical definitions. Perhaps the respondents interpreted the conjunction (A and B) as a disjunction (A or B), an implication (A implies B), or a conditional statement (A if B). Alternatively, the event B could be interpreted in the presence of the conjunction as B and not A. To investigate these possibilities, we presented to another group of 56 naive subjects at Stanford University the hypothetical results of the relevant tennis match, coded as sequences of wins and losses. For example, the sequence LWWLW denotes a five-set match in which Borg lost (L) the first and the fourth sets but won (W) the other sets and the match. For each sequence, the subjects were asked to examine the four target events of the original Borg problem and indicate, by marking + or −, whether the given sequence was consistent or inconsistent with each of the events.

With very few exceptions, all of the subjects marked the sequences according to the standard (extensional) interpretation of the target events. A sequence was judged consistent with the conjunction “Borg will lose the first set but win the match” when both constituents were satisfied (e.g., LWWLW), but not when either one or both constituents failed. Evidently, these subjects did not interpret the conjunction as an implication, a conditional statement, or a disjunction. Furthermore, both LWWLW and LWLWL were judged consistent with the inclusive event, “Borg will lose the first set,” contrary to the hypothesis that the inclusive event B is understood in the context of the other events as, “Borg will lose the first set and the match.” The classification of sequences therefore indicated little or no ambiguity regarding the extension of the target events. In particular, all sequences that were classified as instances of B&A were also classified as instances of B, but some sequences that were classified as instances of B were judged inconsistent with B&A, in accord with the standard interpretation in which the conjunction rule should be satisfied.

Another possible interpretation of the conjunction error maintains that instead of assessing the probability P(B|E) of Hypothesis B (e.g., that Linda is a bank teller) in light of evidence E (Linda’s personality), subjects assess the inverse probability P(E|B) of the evidence given to the hypothesis in question. Because P(E|A&B) may well exceed P(E|B), the subjects’ responses could be justified under this interpretation. Whatever plausibility this account may have in the case of Linda, it is surely inapplicable to the present study, in which it makes no sense to assess the conditional probability that Borg will reach the finals given the outcome of the final match.