Memory & Cognition
2002, 30 (2), 171-178
In this article, we examine posterior probability judg-
ment, which involves one’s assessing the likelihood of
an event by updating a prior probability in light of new
evidence. A normative model for calculating posterior
probabilities is Bayes’s theorem. This theorem states that
p(H |D), the posterior probability that hypothesis H is
true given datum D, can be calculated as follows:
(1)
where p(D |H) and p(D |,H) refer to the conditional
probability of observing D, given that hypothesis H is true
and given that the mutually exclusive, alternative hypoth-
esis, ,H, is true, respectively. In Bayesian terms, these
probabilities are called likelihoods, whereas the probabil-
ities p(H) and p(,H) are called prior probabilities. Pos-
terior probability judgments are fundamental to belief re-
vision and are involved in many consequential real-world
situations such as medical diagnosis or juror decision mak-
ing. Consider, for example, a physician who knows, prior
to the examination of an individual patient, the probability
that a person will have disease X. If the patient presents
a diagnostic symptom, she will have to update the prob-
ability that he has the disease, given this new observation.
Suppose the physician knows that (1) only 5% of the over-
all population suffers from disease X, (2) 85% of patients
who have the disease show the symptom, and (3) 25% of
healthy patients show the symptom. According to Bayes’s
theorem, the posterior probability that a patient who shows
the symptom has the disease can be calculated as follows:
p(disease | symptom) = (.85 3 .05) / [(.85 3 .05) 1 (.25 3
.95)] 5 .15. Thus, there is only a 15% chance that the pa-
tient examined has the disease even though he presents a
highly diagnostic symptom.
The Inverse Fallacy
That both lay and expert judges often confuse a given
conditional probability with its inverse probability has been
noted in many studies. This tendency has been alterna-
tively labeled the conversion error (Wolfe, 1995), the con-
fusion hypothesis (Macchi, 1995), the Fisherian algorithm
(Gigerenzer & Hoffrage, 1995), and the inverse fallacy
(Koehler, 1996a). In the present article, we adopt Koehler’s
term to refer to the tendency for judges to confuse any of
the following: p(H |D) with p(D |H), p(,H |D) with
p(D |,H), p(H | ,D) with p(,D |H), or p(,H |,D)
with p(,D |,H). Although there are other algorithms that
participants can use when they estimate posterior proba-
bilities (see, e.g., Gigerenzer & Hoffrage, 1995), the in-
verse fallacy is often the most frequent error observed.
As early as 1955, Meehl and Rosen reported that clin-
icians considered that the probability of the presence of a
symptom given the diagnosis of a disease was on its own
a valid criterion for diagnosing the disease in the presence
of the symptom. This result was later experimentally dem-
onstrated by Hammerton (1973), who observed that me-
p p p
p p p p
( | ) ( ) ( )( ) ( ) ( ) ( ) ,D H
D |H H
D |H H D |~H ~H
= ×
× + ×
171 Copyright 2002 Psychonomic Society, Inc.
We thank Vittorio Girotto, Evan Heit, Jay Koehler, David Over, and
Frédéric Vallée-Tourangeau for their feedback on this research. The
data for this research were collected in 1999 at the Department of Psy-
chology, University of Hertfordshire, Hatfield, United Kingdom while
the first author was completing her doctoral thesis. Portions of this re-
search were presented at the Fourth International Thinking Conference,
Durham, United Kingdom. Correspondence concerning this article
should be sent to G. Villejoubert, Leeds University Business School,
Maurice Keyworth Building, University of Leeds, Leeds, West York-
shire LS2 9JT, England (e-mail: gv@lubs.leeds.ac.uk) .
—Accepted by previous editorial team
The inverse fallacy: An account of deviations from
Bayes’s theorem and the additivity principle
GAËLLE VILLEJOUBERT
Leeds University Business School, Leeds, England
and
DAVID R. MANDEL
University of Victoria, Victoria, British Columbia, Canada
In judging posterior probabilities, people often answer with the inverse conditional probability—a ten-
dency named the inverse fallacy. Participants (N = 45) were given a series of probability problems that
entailed estimating both p(H |D) and p(,H |D). The findings revealed that deviations of participants’ es-
timates from Bayesian calculations and from the additivity principle could be predicted by the corre-
sponding deviations of the inverse probabilities from these relevant normative benchmarks. Method-
ological and theoretical implications of the distinction between inverse fallacy and base-rate neglect and
the generalization of the study of additivity to conditional probabilities are discussed.

172 VILLEJOUBERT AND MANDEL
dian judgments of p(disease |symptom) were almost equalto the presented value of the inverse probability, p(symp-tom | disease). Liu (1975) replicated those results by vary-ing the value of p(D |H) in a between-subjects design.Similarly, Eddy (1982) investigated how physicians esti-mated the probability that a woman has breast cancer,given a positive result of a mammogram. Approximately95% of clinicians surveyed gave a numerical answer closeto the inverse probability. In Kahneman and Tversky’s (1972) taxicab problem (seealso Bar-Hillel, 1980; Lyon & Slovic, 1976; Tversky &Kahneman, 1980), participants were asked to estimate theprobability that a cab had been involved in an accidentgiven that it was Blue rather than Green. When asked toestimate p(H |D), most participants answered with a valuethat matched the inverse probability, p(D |H). More re-cently, Dawes, Mirels, Gold, and Donahue (1993) demon-strated that this fallacy extended to individuals’ beliefs in-herent to their implicit personality theory.Some researchers have interpreted these findings in termsof a base-rate fallacy. The inverse fallacy is then under-stood to be the result of people’s tendency to consistentlyundervalue, if not ignore, the base-rate information pre-sented as a proxy for prior probabilities (e.g., Bar-Hillel,1980; Dawes et al., 1993; Kahneman & Tversky, 1973;Pollard & Evans, 1983). Other researchers, however, haveproposed that the base-rate effect was in fact originatingfrom the inverse fallacy and not the reverse (e.g., Hamm,1993; Koehler, 1996a; Wolfe, 1995). In support of this no-tion, Wolfe (1995, Experiment 3) found that participantswho were trained to distinguish p(D |H) from p(H |D)were less likely to exhibit base-rate neglect compared witha control group. We agree that base-rate and inverse fal-lacies are different. The inverse fallacy entails not only theneglect of the base-rate information but also that of p(D |,H). To illustrate this argument, consider the diagramsshown in Figure 1. Each diagram depicts two categories Hand ,H. Their base rates are the proportion of space oc-cupied by each category, respectively. The sample spacedelimited by the hatched areas represents the proportionof elements having the feature D. These diagrams indicatethat the inverse fallacy relies on a different representationand integration of available information than does thebase-rate fallacy. Moreover, as shown in Diagram 2, theintegration of base rates into the final judgment is unnec-essary when they are equal. Therefore, if judgment accu-racy were only undermined by the neglect of base-rate in-formation, judgments involving equal base rates shouldbe normative.If people commit the inverse fallacy in judging poste-rior probabilities, one would expect that posterior prob-ability estimates would be systematically biased as a func-tion of the deviation between the posterior probability andits inverse probability. Consider the physician who needsto estimate p(disease | symptom). If she commits the in-verse fallacy, she will answer with the value of the inverseprobability p(symptom | disease) = .85, rather than the
Bayesian value of p(disease | symptom) = .15. In this case,the deviation between the estimate and the Bayesian valueof p(disease | symptom) is .70. Although some preliminaryresearch indicates that the inverse fallacy is a distinct con-tributor to deviations from Bayesian judgment, no study hasyet examined whether the deviations between posteriorprobabilities and their inverse probabilities can be used topredict people’s deviations from Bayesian judgment. Thiswas a key objective of the present study.The Additivity PrincipleThe additivity principle states that the judged probabil-ities for complementary events should sum to unity. For in-stance, if one judges that p(disease | symptom) is .75, onealso should judge that p(no_disease | symptom) is .25.From a descriptive standpoint, there is disagreement withinthe literature as to whether we should expect that judg-ments of two complementary probabilities will be additive.Some researchers (e.g., Rottenstreich & Tversky, 1997;Tversky & Koehler, 1994) have reported that the judgedprobability of a hypothesis and that of its complement areadditive. Other researchers (e.g., Ayton, 1997) have re-ported that people’s probability judgments tend to be sub-additive (i.e., the sum of the individual estimates is greaterthan one). And evidence of superadditivityin the case of bi-nary complementarity also has been found (see, e.g., Mac-chi, Osherson, & Krantz, 1999).Although previous research has examined unconditionalprobability judgments, the additivity principle also appliesto conditional probabilities because p(H |D) and p(,H |D)always add to one when H and ,H are exhaustive andmutually exclusive. In the domain of conditional proba-bility judgments, Baratgin and Noveck (2000) suggestedthat the participants in Kahneman and Tversky’s (1973)lawyer–engineer problem violated the additivity principle.The authors demonstrated that the participants integratedbase rates more efficiently when they were induced tomake complementary estimates. In the present study, wetested a stronger, empirically verifiable claim. Namely, thatparticipants’ complementary estimates will not be additive,mainly because people commit the inverse fallacy. Fur-thermore, the inverse fallacy offers a basis for a systematicprediction of the patterns of deviation from additive judg-ment. Our second objective, then, was to examine whetherpeople’s estimates of complementary posterior probabil-ities, and any deviations from the additivity principle, canbe predicted on the basis of the inverse conditional prob-abilities. EXPERIMENT 1In this experiment, we examined whether evidence forthe inverse fallacy would emerge even when base rate ne-glect could be ruled out. We used a problem adapted fromSlowiaczek, Klayman, Sherman, and Skov (1992, Exper-iment 1A) in which participants were asked to estimate theposterior probability that an encountered alien creature

THE INVERSE FALLACY 173
Figure 1. Sample-space representations of the information provided in textbook problems.
was one of two mutually exclusive types in light of thepresence or absence of a diagnostic feature. Skov andSherman (1986) noted that the use of natural groups im-poses restrictions on the likelihood of a particular feature
in these groups. In such cases, diagnosticity and likeli-hood are often confounded: A diagnostic trait (e.g., likesparties) would be frequent in the focal group (e.g., extro-verts) and infrequent in the alternative group (introverts).

174 VILLEJOUBERT AND MANDEL
Thus, the use of unnatural categories allowed us to controlfor the likelihood of the features.Our first hypothesis was that the majority of partici-pants would commit the inverse fallacy. Second, we hy-pothesized that the deviation between the participants’estimates and Bayesian answers could be predicted bythe deviation between p(D |H) and p(H |D). Thus, par-ticipants were expected to overestimate p(H | D) whenp(D |H) . p(H |D), and they were expected to under-estimate the Bayesian posterior when p(D |H) , p(H |D).Moreover, by experimentally manipulating the size ofthe deviation between posterior and inverse probabilities,we were also able to predict the magnitude of the partic-ipants’ judgment inaccuracies. The third hypothesis testedwas that when p(D | H) and p(D |,H) sum to less thanone, the sums of participants’ judgments of p(H |D) andp(,H |D) would be superadditive. Conversely, the sumsof their judgments of p(H |D) and p(,H |D) were ex-pected to be subadditive when p(D |H) and p(D |,H)exceeded one.Our experimental design also allowed us to distinguishbetween the inverse fallacy and a simpler, matching heuris-tic. In the domain of logical reasoning, Evans (1998) de-fined a matching bias as a tendency to only consider asrelevant the information whose lexical content matchesthat of the information presented in the propositional ruleto be tested. By extension, the tendency to equate p(H |D)with p(D |H) could be defined as the tendency to esti-mate p(H |D) on basis of the match between the experi-mental question and the information presented. Such astrategy, however, would lead people to answer with thedisplayed value of p(D |H), when that value was explic-itly provided, even when they were asked to estimatep(H |,D). By contrast, the inverse fallacy account pro-poses that people will estimate p(H |,D) with the valueof p(,D |H). We therefore expected to be able to distin-guish between the inverse fallacy and the matching heuris-tic in cases in which diagnostic response information in-dicated the absence of D (i.e., via no responses). MethodForty-five University of Hertfordshire undergraduates participatedin the experiment for course credit. The participants were provided
with a 13-page questionnaire. The first page presented their task asfollows:Imagine you are visiting a planet called Vuma. There are two and onlytwo types of invisible creatures that live on this planet. There are 1 mil-lion Gloms, and 1 million Fizos.You will randomly meet 12 creatures. Imagine you are particularly in-terested in guessing their identity. Each time you meet one of the invis-ible creatures, you want to know whether it is a Glom or a Fizo. You willwalk with an interpreter who will ask each creature whether or not itpossesses a certain feature. Each time, you will be provided with thepercentages of Gloms and Fizos on Vuma possessing the target feature.The creatures cannot help but to tell the truth, so you can be sure youwill get a truthful answer, which will provide you with some informa-tion about the creature’s identity.The participants were then provided with an example and were told:“On the basis of the creature’s answer, you will be asked to estimateboth the likelihood that it is a Glom and the likelihood that it is a Fizo.Turn the page for your first encounter.” The participants “met” 12creatures one by one on the subsequent pages. For each of these 12encounters, the creature was asked about a different feature, and theparticipants were provided with a reminder of the number of Glomsand Fizos on the planet as well as the percentages of each type of crea-ture that possessed the feature. In order to be consistent with Slo-wiaczek et al. (1992), posterior probability judgments were elicitedby using a frequency question, and the participants were asked toestimate the “chances in 100” rather than the “probability ” that H(or ,H) was true. A summary of the stimuli is presented in Table 1.Figure 2 shows the questionnaire layout corresponding to the firstline of Table 1.Stimuli and Design. Each of the 12 encounters represented aunique stimulus condition in a 2 (creature’s response: no, yes) 3 2[expected direction of deviation: p(D |H) , p(H |D), p(D |H) .p(H |D)] 3 3 (expected magnitude of deviation: small, medium, large)fully crossed repeated-measures design with two dependent variablesmeasured for each stimulus. Each of the 12 stimuli required two es-timates, and each single estimate corresponded to distinct Bayesian andinverse values (see Table 2). The diagnostic probabilities presented
Table 1Percent of Gloms and Fizos Presenting Each of Twelve Featuresand Encountered Creature’s AnswerGloms Fizos Features Response98 58 plays the harmonica Yes92 42 exhales fire No90 50 wears hula hoops Yes10 50 gurgles a lot No80 40 have a flying license Yes20 60 gulp bluebottles down No42 92 smokes maple leaves Yes58 98 drinks petrol No50 10 has gills Yes50 90 eats iron ore No60 20 breeds scampi Yes40 80 climbs walls No
Figure 2. Representation of one of the stimuli used.

THE INVERSE FALLACY 175
were chosen so that the creature’s response would have no effect onthe participants ’ estimates. Thus, the set of yes stimuli were perfectlymatched to the set of no stimuli in terms of the Bayesian and inversevalues. The order of stimulus presentation was randomized, and ques-tion order was counterbalanced across participants .ResultsClassification of estimates. In order to test whether theparticipants committed the inverse fallacy, each estimatewas compared with the value of its corresponding inverseand Bayesian probability, respectively. An estimate wasclassified as Bayesian or as inverse if it was equal to its cor-responding Bayesian or inverse value, respectively, withina margin of error of 6.02. Figure 3 shows the distributionof participants according to the number of their estimatesclassified as Bayesian or as inverse. The vast majority(80%) of participants provided no more than 4 Bayesianestimates out of a total of 24. By contrast, and in supportof our first hypothesis, as many as 51% of the participantshad 20 or more estimates that were equal to the corre-sponding inverse value. This is a powerful result that dem-onstrates the prevalence of committing the inverse fallacyin judging posterior probabilities. Finally, as expected, theparticipants did not rely on a matching heuristic, sincemost estimates of p(H |,D) and p(,H |,D) relied on thevalues of p(,D |H) and p(,D |,H), respectively. Withinthe set of no stimuli, 55% of the participants provided atleast 7 (out of 12) estimates equal to the inverse value,whereas only 1 participant provided 7 estimates consis-tent with a matching heuristic.Deviations from Bayesian judgment. The partici-pants’ estimates were converted to deviation scores, d, bysubtracting the corresponding Bayesian value. Thus, d =0 for accurate Bayesian estimates, d . 0 for overestima-tions of Bayesian probabilities, and d , 0 for underesti-mations of Bayesian probabilities. Analyses includingorder of stimulus presentation and order of question pre-sentation as between-subjects variables revealed no sig-
nificant effect of this manipulation, and the data were col-lapsed across order. The deviation scores were subjected toa 2 (creature’s response) 3 2 (expected direction of devi-ation) 3 3 (expected magnitude of deviation) doubly mul-tivariate repeated-measures analysis of variance (ANOVA).Hypothesis 2 stated that the participants’ deviations fromBayesian judgment would be predicted by the deviation be-tween the inverse and the Bayesian values. Accordingly,a significant two-way (expected direction of deviation 3expected magnitude of deviation) interaction was observed[multivariate F(4,41) = 47.29, p , .0005]. Univariate Ftests revealed that this interaction was statistically sig-nificant only for the Glom measure [F(2,88) = 106.69,MSe = 0.02, p, .01, h2 = .71]. The 2 (expected directionof deviation) 3 3 (expected magnitude of deviation)interaction is shown in Table 2 (in the “Bayes’s theorem-
Table 2Summary of Stimuli, Design, and Results: Bayesian and Inverse Probabilities, Expected and Observed Deviations as a Function of Magnitude and Direction of Deviation for Glom and Fizo MeasuresDeviationsExpected Probabilities Bayes’s TheoremMagnitude Bayesian Inverse Glom Fizo Additivityof Deviation Glom Fizo Glom Fizo Exp. Obs. Exp. Obs. Exp. Obs.Expected Overestimation of Bayesian ValuesLarge .63 .37 .98 .58 .35 .14 .21 .09 1.56 1.23Medium .64 .36 .90 .50 .26 .10 .14 .06 1.40 1.16Small .67 .33 .80 .40 .13 .00 .07 .05 1.20 1.05Expected Underestimation of Bayesian ValuesLarge .96 .04 .42 .02 2.54 2.53 2.02 .05 .44 .56Medium .83 .17 .50 .10 2.33 2.33 2.07 .00 .60 .72Small .75 .25 .60 .20 2.15 2.22 2.05 .10 .80 .78Note—Bayesian probabilities refer to p(type | response) and inverse probabilities refer to p(response | type). Exp.,Expected; Obs., Observed. The expected deviations from Bayes’s theorem are given by the deviations betweenBayesian and inverse values. The expected deviations from additivity are given by the deviations between the sumof the inverse values for Gloms and Fizos, and unity.
Figure 3. Distribution of participants as a function of the num-ber of their estimates equating either the inverse probability orthe Bayesian probability.

176 VILLEJOUBERT AND MANDEL
Glom-Observed” column). Deviation scores were positiveand negative when overestimation and underestimationwas expected, respectively. Moreover, the magnitude ofthese deviations varied as predicted. The multivariateanalysis also revealed a significant main effect of re-sponse on deviation scores [multivariate F(2,43) = 5.45,p, .01]. It has already been suggested and observed thatparticipants tend to be more influenced by positive answersthan by negative answers (see Sherman & Corty, 1984; Slo-wiaczek et al., 1992), leading to a higher level of confir-mation of the hypothesis tested.Deviations from the additivity principle. Our third hy-pothesis was that the observed patterns of additivity wouldbe a function of the sum of inverse probabilities. The sumof each participant’s estimates of the complementary prob-abilities p(Glom | response) and p(Fizo | response) wascomputed. These sums were subjected to a 2 (creature’s re-sponse) 3 2 (expected direction of deviation) 3 3 (expectedmagnitude of deviation) repeated-measures ANOVA. Asexpected, response did not significantly affect the scores[F(1,44) = 0.12, MSe = 0.05, p. .05]. The significant maineffects of expected direction and expected magnitude ofdeviation from unity are better explained by the predictedtwo-way (expected direction of norm deviation 3 expectedmagnitude of norm deviation) interaction [F(2,88) = 36.09,MSe = 0.05, p , .001, h2 = .45]. This interaction is illus-trated in the last column of Table 2. As predicted, the par-ticipants’ estimates were subadditive when complementaryinverse probabilities summed to more than 1, and theirestimates were superadditive when these probabilitiessummed to less than 1. The magnitude of these deviationsalso varied as predicted by the sum of inverse probabilities.Finally, a significant two-way (expected direction of normdeviation 3 creature’s response) interaction was obtained[F(1,44) = 5.96, MSe = 0.14, p , .02]. Further analysesrevealed that this effect was due to a significant effect ofresponse when superadditivity was expected [Myes = .65,SD = .22 vs. Mno = .73, SD = .24, Bonferronni t(88) =14.44, MSpooled error = .11, p , .001, with a corrected alevel of .025]. DISCUSSIONConsistent with previous research (e.g., Bar-Hillel, 1980;Eddy, 1982; Hamm, 1993; Wolfe, 1995), the results ob-tained in the present study reveal that roughly half of thesample equated the posterior and inverse probabilities onover 80% of the judgment trials. This is a powerful find-ing because, in past studies demonstrating the inverse fal-lacy, typically only one judgment per participant was so-licited. By contrast, we have demonstrated that a sizeableproportion of judges consistently used an inverse fallacy al-gorithm over a set of judgment tasks that varied in terms ofboth the question asked and the available probability infor-mation. We were able to demonstrate that the direction andmagnitude of deviations of participants’ estimates fromboth Bayesian judgment and the additivity principle were
successfully predicted by the deviation between the inverseand posterior probabilities, thus supporting our second andthird hypotheses, respectively. Furthermore, deviationsfrom normative benchmarks could not be accounted for bythe base-rate fallacy because base rates for the two relevantcategories were always equal.Methodological ImplicationsGirotto and Gonzalez (2001) showed that some of Cos-mides and Tooby’s (1996) observations were erroneouslyclassified as accurate judgments by these authors on thebasis of parity between participants’ posterior probabilityestimates and the numerical values computed by Bayes’stheorem. In the present study, the collection of multiple es-timates for each participant militated against such unwar-ranted conclusions resulting from a confusion between theoutcome of a test and the computational process leadingto this outcome. Such methodological precautions pre-vented us from mistakenly concluding that judgments ofp(Fizo | response) demonstrated normative judgment (asthey showed little deviation from the Bayesian norm, seeTable 2). A more likely explanation, supported by the re-sults observed for the Glom measure, is that those judg-ments were based on the inverse fallacy, whose outputs hap-pened to be similar to those arising from Bayes’s theorem.The fact that the inverse fallacy is associated with non-additive posterior probability judgments also has signif-icant methodological implications. For instance, Slowia-czek et al. (1992, Experiment 1A) assumed that theirparticipants’ judgments were additive and combined es-timates of p( H |D) with estimates of p(,H | D) sub-tracted from 1. This recoding procedure may have in-duced a bias in their results. For instance, when p(D |H) =.5, p(D |,H) = .10 and D is present (a stimulus that wasalso used by Slowiaczek et al., 1992), p(H |D) = .83 andp(,H |D) = .17. A judge committing the inverse fallacy,would estimate p(H |D) and p(,H |D) to be .50 and .10,respectively. Recoding p(,H |D) as a .90 (1 2 .10) esti-mate of p( H |D) results in a near-normative answer,whereas the judge committing the inverse fallacy wouldhave been more likely to estimate p(H |D) to be .50 [thevalue of p(D |H)], thus underestimating the normativevalue by more than a 30% difference.Theoretical ImplicationsAdditivity. The present experiment extended the studyof additivity to conditional probability judgments. Specif-ically, we demonstrated that the pattern of subadditivityand superadditivity observed for the participants’ judg-ments could be predicted from the sum of the inverse prob-abilities. Rottenstreich and Tversky (1997) specified thatthe binary complementarity predicted by support theory(Tversky & Koehler, 1994) applies to cases in which thealternative hypothesis is explicitly described as such. Thisprecision implies that, in the present study, support theorywould only predict additivity for measures of p(Glom |response) and p(,Glom | response). Still, in the present

THE INVERSE FALLACY 177
context, it was made clear to the participants that the crea-ture could only be a Glom or a Fizo (and obviously, notboth). Rottenstreich and Tversky suggested that “additiv-ity is likely to hold” (p. 407) in such a case. Our results,however, strongly indicate that the latter proposition doesnot hold, at least in the context of conditional probabilityjudgment. Future research might examine the effect of im-plicitly versus explicitly negating the alternative hypoth-esis on the additivity of posterior probability judgments incases of binary complementarity.Origin and extent of the inverse fallacy. The questionof the underlying bases and the scope of the inverse fallacystill require investigation. The sample-space framework(Gavanski & Hui, 1992; Hanita, Gavanski, & Fazio, 1997;Sherman, McMullen, & Gavanski, 1992) proposes that theinverse fallacy is the result of a memory-based process.People access sets of information (sample spaces) frommemory when judging probabilities. When participants areasked to estimate p(H |D), they are required to base theirjudgments on the sample space defined by feature D. Ga-vanski and Hui argued that this sample space is unnaturalbecause knowledge is partitioned by categories rather thanby features. So, people may replace the unnatural samplespace with a more readily accessible one—namely, thesample space of category H. This process would result inthe inverse fallacy. The sample-space explanation, how-ever, cannot fully account for the present results becausethe categories used were hypothetical, and judgmentscould not be based on sample spaces stored in memory. Alternatively, Macchi (1995, 2000) proposed that theformulation of diagnostic information plays a key role inthe interpretation of the data. Consider the following for-mulations: (1) The percentage of elements presenting thefeature D is three times higher among H elements thanamong ,H elements. (2) In the group of elements pre-senting the feature D, the percentage of H elements isthree times higher than the percentage of ,H elements.(3) The feature D is present in x% of H elements, the fea-ture D is present in y% of ,H elements, and x is three timeshigher than y. Macchi (1995) proposed that a formulationsuch as (1) is interpreted as (2) by participants, as opposedto what it logically implies—namely formulation (3). It isthis misinterpretation, Macchi argued, that leads to theinverse fallacy. Consequently, the tendency to estimatep(H |D) with p(D |H) results from the lack of clarity ofthe independence of base rate p(H) and p(D |H). Macchirecommended the use of formulations such as (3) toavoid ambiguity and demonstrated that it could reducethe proportion of inverse fallacies. Yet, this explanation forthe origin of the inverse fallacy has its shortcomings. It isnot clear why it is the ambiguous independence of p(D |H)and p(H) that would lead people to mistake p(D |H) forp(H |D), given that the combination of the base rate andthe diagnostic information [ p(D |H ).p( H)] results inp(D ù H), which is not the formal equivalent of p(H |D),as is demonstrated by the diagrams presented in Fig-
ure 1. Furthermore, this explanation conflicts with thesample-space account because Macchi’s suggestion thatparticipants’ interpretations [i.e., formulation (2)] rely onthe sample space defined by feature D. Yet, according tothe sample-space explanation, this is an unnatural and un-likely basis for probability judgments. Finally, even thoughour formulation of the diagnostic information was in linewith Macchi’s (1995) recommendations for reducing theinverse fallacy, we still found that 51% of participantsmade almost all their judgments in accordance with the in-verse fallacy.An alternative account relies on the frequency hypoth-esis. Gigerenzer and Hoffrage (1995; see also, e.g., Cos-mides & Tooby, 1996) demonstrated that Bayesian answerscan be elicited with the use of a frequency format for boththe information and the question asked. Moreover, Thomp-son and Schumann (1987) showed that frequency formatsreduced the number of inverse fallacies committed. Aprobability presentation format might increase the use ofheuristics, but that still does not explain why most peopletend to commit the inverse fallacy rather than use anotherheuristic (e.g., the joint-occurrence algorithm describedin Figure 1). Finally, it is plausible that people simplyconfuse p(H |D) with p(D |H) because the latter soundsa lot like the former (J. J. Koehler, personal communica-tion, April 2001). However, at present, no empirical re-search has tested this account. Concluding RemarksThe present study demonstrated how the inverse fallacycan account for deviations from Bayes’s theorem and theadditivity principle. This fallacy might also explain otherresults on probabilistic reasoning observed within the lit-erature. Koehler (1996b) showed that people confuse pos-terior odds ratios with likelihood ratios. Such confusioncould be explained by the tendency to commit the inversefallacy. Doherty, Mynatt, Tweney, and Schiavo (1979) dem-onstrated that people tend to choose diagnostically worth-less information such as p(D1 |H) and p(D2 |H) to revisethe probability that hypothesis H is true. This “pseudo-diagnosticity” phenomenon may be explained by the factthat participants seeking p(D1 |H) and p(D2 |H) thinkthat they are given p(H |D1) and p(H |D2 ). Finally, thepresent research spotlighted the need to distinguish judg-ment output from the judgment process and demonstratedthat the examination of judgment over multiple trials wasan effective method to do so. REFERENCESAyton, P. (1997). How to be incoherent and seductive: Bookmakers ’ oddsand support theory. Organizational Behavior & Human Decision Pro-cesses, 72, 99–115.Baratgin J., & Noveck, I. A. (2000). Not only base rates are neglectedin the Engineer–Lawyer problem: An investigation of reasoners’ un-derutilization of complementarity. Memory & Cognition, 28, 79-91.Bar-Hillel, M. (1980). The base rate fallacy in probability judgments.Acta Psychologica, 44, 211–233.

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Cosmides, L., & Tooby, J. (1996). Are humans good intuitive statisti-cians after all? Rethinking some conclusions from the literature onjudgment under uncertainty. Cognition, 58, 1-73.Dawes, R. M., Mirels, H. L., Gold, E., & Donahue, E. (1993). Equat-ing inverse probabilities in implicit personality judgments. Psycho-logical Science, 4, 396-400.Doherty, M. E., Mynatt, C. R., Tweney, R. D., & Schiavo, M. D.(1979). Pseudodiagnosticit y. Acta Psychologica, 43, 11-21.Eddy, D. M. (1982). Probabilistic reasoning in clinical medicine: Prob-lems and opportunities. In D. Kahneman, P. Slovic, & A. Tversky (Eds.),Judgment under uncertainty: Heuristics and biases (pp. 249-267) .Cambridge: Cambridge University Press.Evans, J. St. B. T. (1998). Matching bias in conditional reasoning: Dowe understand it after 25 years? Thinking & Reasoning, 4, 45-82.Gavanski, I., & Hui, C. (1992). Natural sample spaces and uncertainbelief. Journal of Personality & Social Psychology, 63, 766-780.Gigerenzer, G., & Hoffrage, U. (1995). How to improve Bayesian rea-soning without instruction: Frequency formats. Psychological Review,102, 684–704.Girotto, V., & Gonzalez, M. (2001). Solving probabilistic and statis-tical problems: A matter of information structure and question form.Cognition, 78, 247-276.Hamm, R. M. (1993). Explanation for common responses to the blue/green cab probabilistic inference word problem. Psychological Reports,72, 219-242.Hammerton, M. (1973). A case of radical probability estimation. Jour-nal of Experimental Psychology, 101, 252-254.Hanita, M., Gavanski, I., & Fazio, R. H. (1997). Influencing proba-bility judgments by manipulating the accessibility of sample spaces.Personality & Social Psychology Bulletin, 23, 801-813.Kahneman, D., & Tversky, A. (1972). On prediction and judgment .ORI Research Monographs, 12.Kahneman, D., & Tversky, A. (1973). On the psychology of predic-tion. Psychological Review, 80, 237-251.Koehler, J. J. (1996a). The base rate fallacy reconsidered: Descriptive,normative and methodologica l challenges. Behavioral & Brain Sciences,19, 1-53.Koehler, J. J. (1996b). On conveying the probative value of DNA ev-idence: Frequencies, likelihood ratios and error rates. University of Col-orado Law Review, 67, 859-886.Liu, A. Y. (1975). Specific information effect in probability estimation.Perceptual & Motor Skills, 41, 475-478.Lyon, D., & Slovic, P. (1976). Dominance of accuracy information and
neglect of base rates in probability estimation. Acta Psychologica, 40,287-298.Macchi, L. (1995). Pragmatic aspects of the base rate fallacy. QuarterlyJournal of Experimental Psychology, 48A, 188-207.Macchi, L. (2000). Partitive formulation of information in probabilisticproblems: Beyond heuristics and frequency format explanations. Or-ganizational Behavior & Human Decision Processes, 82, 217-236.Macchi, L., Osherson, D., & Krantz, D. H. (1999). A note on super-additive probability judgment. Psychological Review, 106, 210-214.Meehl, P., & Rosen, A. (1955). Antecedent probability and the effi-ciency of psychometric signs of patterns, or cutting scores. Psycholog-ical Bulletin, 52, 194-215.Pollard, P., & Evans, J. St. B. T. (1983). The role of representative-ness in statistical inference. In J. St. B. T. Evans (Ed.), Thinking andreasoning (pp. 309-330). London: Routledge & Kegan Paul.Rottenstreich, Y., & Tversky, A. (1997). Unpacking, repacking, andanchoring: Advances in support theory. Psychological Review, 104,406-415.Sherman, S. J., & Corty, E. (1984). Cognitive heuristics. In R. S. Wyer& T. K. Srull (Eds.), Handbook of social cognition (Vol. 1, pp. 189-286) .Hillsdale, NJ: Erlbaum.Sherman, S. J., McMullen, M. N., & Gavanski, I. (1992). Naturalsample spaces and the inversion of conditional judgments. Journal ofExperimental Social Psychology, 28, 401-421.Skov, R. B., & Sherman, S. J. (1986). Information-gathering processes:Diagnosticity, hypothesis-confirmatory strategies, and perceived hy-pothesis confirmation. Journal of Experimental Social Psychology, 22,93-121.Slowiaczek, L. M., Klayman, J., Sherman, S. J., & Skov, R. B. (1992).Information selection and use in hypothesis testing: What is a good ques-tion, and what is a good answer? Memory & Cognition, 20, 392-405.Thompson, W. C., & Schumann, E. L. (1987). Interpretation of statis-tical evidence in criminal trials: The prosecutor’s fallacy and the de-fense attorney’s fallacy. Law & Human Behavior, 11, 167-187.Tversky, A., & Kahneman, D. (1980). Causal schemata in judgmentsunder uncertainty. In M. Fishbein (Ed.), Progress in social psychology(Vol. 1, pp. 49-72). Hillsdale, NJ: Erlbaum.Tversky, A., & Koehler, D. J. (1994). Support theory: A nonexten-sional representation of subjective probability. Psychological Review,101, 547-567.Wolfe, C. R. (1995). Information seeking on Bayesian conditional prob-ability problems: A fuzzy-trace theory account. Journal of Behav-ioral Decision Making, 8, 85-108.
(Manuscript received January 23, 2001; revision accepted for publication September 6, 2001.)

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ErratumVillejoubert, G., & Mandel, D. R. (2002). The inverse fallacy: An account of deviations fromBayes’s theorem and the additivity principle. Memory & Cognition, 30 (2), 171-178.In the first paragraph of the text on p. 171, the letters D and H were incorrectly transposed inEquation 1. The correct equation is printed below:
p p pp p p p( | ) ( | ) ( )( | ) ( ) ( |~ ) (~ )H D D H HD H H D H H= ×× + ×