Effects of Temporal and Causal Schemas on Probability Problem Solving

Transcription

1 Effets of Temporal and Causal Shemas on Probability Problem Solving S. Sonia Gugga Columbia University New York, NY James E. Corter Teahers College, Columbia University New York, NY Abstrat Causal beliefs have been shown to affet performane in a wide variety of reasoning and problem solving. One type of judgment bias that an result from impliit ausal models is ausal asymmetry -- the tendeny to judge preditive inferenes as more plausible than omparable diagnosti inferenes. In the present study we investigate if the diretionality of impliit ausal models an also affet appliation of formal methods, speifially the solution of onditional probability word problems. The study examined temporal and ausal shemas, in whih the onvention is that events are onsidered in forward diretion. Pairs of onditional probability (CP) problems were written depiting events E1 and E2, suh that E1 either ours before E2 or auses E2. Problems were defined with respet to the order of events expressed in CPs, so that P(E2 E1) represents the CP in shema-onsistent, intat order by onsidering the ourrene of E1 before E2; while P(E1 E2) represents CP in shemainonsistent, inverted order. Partiipants had greater diffiulty enoding CP for events in shema-inonsistent order than CP of events in the onventional deterministi order. Keywords: Conditional probability; soial shemata; ausal shemata. Introdution Previous researh on statistial reasoning has illustrated that individuals often disregard statistially presribed proesses, employing heuristis and biases whih may produe invalid inferenes (see Barbey & Sloman, 2007; Cohen, 1981; Gigerenzer & Hoffrage, 2007; Kahneman & Tversky, 1979; Krynski & Tenenbaum, 2007; Nisbett, Krantz, Jepson, & Kunda, 1983; Stanovih, Toplak, & West, 2008; and Tversky & Kahneman, 1974, 1980 for extensive treatments of the debate). Improvement in inferential auray has been affeted by training, expression of likelihood in frequeny formats, partitive formulation of the information spae, or by expressing real-world appliation to ontexts in whih individuals are more likely to reason probabilistially with respet to ausal or soial shemas (Fox & Levav, 2004; Gigerenzer & Hoffrage, 1995, 2007; Girotto & Gonzalez, 2007; Krynski & Tenenbaum, 2007; Mahi, 2000). A mathematial word problem's over story provides realworld ontext and semanti ontent, sometimes onsidered surfae features, beause they are usually assumed not to diretly affet a problem's tehnial diffiulty or formal solution proesses. When a over story relates to soial interations, however, pragmati reasoning shemas formed in response to real-world experienes may be invoked in the mind of the problem solver (e.g., Cheng & Holyoak, 1985; Fong, Krantz, & Nisbett, 1986; Bassok, Chase, & Martin, 1998). The laim made here is that problem solving in probability is partiularly affeted by domain-speifi knowledge in probabilisti reasoning. Previous researh has found disrepanies between formal quantitative probabilisti assessments ( System 2 proessing) and on-the-fly, qualitative probabilisti judgments (tapping System 1 ). This prior researh has led to a theory of dual-systems representations for probability judgments (Evans & Frankish, 2008; Fox & Levav, 2004; Sloman, 1996; Sloman & Rips, 1998; Smith & Collins, 2009; Stanovih, Toplak, & West, 2008; Tversky & Kahneman, 1983; Windshitl & Wells, 1998). While the two proesses have been well-differentiated, it remains to be determined how they may interat in formal problem solving ontexts. Individuals have been shown to have a strong bias toward System 1 proesses for evaluating soial behavior or harateristis qualitatively, while System 2 proesses are engaged in symboli ontexts. It may be that the speifi interation of the two proesses depends on how problem elements are represented in the mind of the problem solver. The study reported below examines reasoning effets stemming from the diretion of temporality and ausation of events in an applied problem s ontent, examining whether inverting the temporal diretion of a shema affets the diffiulty of a onditional probability problem. It was hypothesized that it would be easier for individuals to reason forwards regarding temporal or ausal events given the deterministi nature of ausal and, by extension, temporal shemas (Cheng & Nisbett, 1993; Tversky & Kahneman, 1980). In other words, it should be easier for subjets to alulate the onditional probability of an event given the probability of events preeding it versus alulating the probability of an event given the probability of events ourring later. Similarly, it was expeted that problems asking for the onditional probability of an effet given the probability of its ause(s) would be easier than problems asking for the probability of a ause given the probability of its effet(s). It was supposed that inverting the diretion of determination should introdue an additional level of diffiulty to the problem. It was also expeted that the pereived ausal strength between events would mediate the effet. The results, desribed below, yield insights into how probability problems are ategorized and solved, what effets 2321

2 these pragmati shemas have on problem-solving suess, and how they may inform theories of probabilisti reasoning in general. Causality and Conditional Probability With respet to temporal and ausal shemas, it is the onvention to reason forwards, onsidering earlier events before those that happen later, and auses before effets, a preferene whih may affet the pereived strength of a ausal relationship (Fernbah, Darlow, & Sloman, 2011; Tversky & Kahneman, 1980). This onvention may, in part, explain why, when given a hoie, individuals prefer to wager on the outomes of events whih have yet to happen rather than those of past events whih may have already ourred (Brun & Teigen, 1990; Fishhoff, 1975, 1976; Rothbart & Snyder, 1970; Wright, 1982). These temporal asymmetries in reasoning faility or preferene might extend to ertain onditional probability problems. For example, if event A is a ause of B, or if A merely temporally preedes B, then the onditional probability of B given A may be easier to reason about than the reverse onditional. These reasoning effets may not extend to simple omputational problems, sine P(A B) and P(B A) are equally easy to ompute. One probabilisti reasoning error that we might expet to be related to the hypothesized asymmetries in ausal and temporal reasoning is the so-alled fallay of the transposed onditional or the inverse fallay, in whih P(A B) is onfused with P(B A) (Bar-Hillel & Falk, 1982; Díaz & de la Fuente, 2007; Krynski & Tenenbaum, 2007; Makie, 1981; Neath, 2010; Tversky & Kahneman, 1980; Villejoubert & Mandel, 2002). Among these disussions there has been speulation of onditions in whih this fallay is more or less likely to our, but little evidene demonstrating the phenomenon systematially within judgment under unertainty. If reasoning about forward-diretion ausal and temporal CPs is easier and more natural than reasoning about inverted CPs, then it might be that this error would arise more often by misinterpreting an inverse-order CP as a forward-diretion CP than vie-versa. Empirial Study If the ease of reasoning about onditional probabilities shows temporal and ausal asymmetries (but formula-based alulations do not), then we would expet to see effets of this diretionality on the diffiulty of probabilisti reasoning and problem-solving for word problems in whih the onditional probabilities are desribed or given in the problem text, not derived via formula. This study tests if suh effets are observed. Method Partiipants. Partiipants were students enrolled in one of four setions of an introdutory ourse in probability and statistial inferene during the Fall 2011 term at a graduate shool of eduation. Two instrutors eah taught two of the setions. Number of subjets per ondition for eah item is detailed in Table 1. A total of 123 students ompleted the quiz. Data from one student in the ausal-intat ondition was lost, so total N for that item is 122. Materials. Two test items were presented to eah student, embedded in an in-lass quiz. The test items are listed in the Appendix. There were two versions of the temporal-shema problem (one invoking the temporally-onsistent onditional probability, the other invoking the inverted-order probability), and two versions of the ausal shema (onsistent and inverted orders). Proedure. Both instrutors gave the quizzes during the fourth week of the ourse, after overing the topi of onditional probability. Eah student saw one temporal shema problem (either order intat or order inverted) and the other version of the ausal shema problem. Quizzes were administered during the last 30 minutes of lass. Students were permitted to use notes and a alulator. Results The dependent variables of interest were 1) orretness of numeri solution, and 2) whether the student enoded onditional probability (CP) orretly. Perentage orret for the two measures is summarized in Table 1, by ondition. Table 1: DVs: Perent auray for overall problem solution and enoding of CP (with freq), by ondition. Temporal Items Intat Inverted Total Solved the problem 58.2 (32) 52.5 (35) 54.5 (67) Coded CP orretly 74.5 (41) 57.4 (39) 65.0 (80) N Causal Items Intat Inverted Total Solved the problem 25.4 (17) 20.0 (11) 23.0 (28) Coded CP orretly 65.7 (44) 36.4 (20) 52.5 (64) N Separate generalized linear models (GZLM) were fit for the temporal and ausal problems on eah outome aording to a binary distribution with logit (i.e., log-odds) link funtion. Eah model had two preditors: Order (=intat, inverted) and Instrutor. For temporal problems, the proportion of partiipants orretly solving the problem was higher for order-intat than for order-inverted items, as hypothesized, but this differene was not signifiant: X 2 LR (1, N = 123) = 0.52, p =.472. For the seond DV, orret enoding of the given onditional probability information, the hypothesis was onfirmed: order (=intat, inverted) signifiantly predited orret enoding, X 2 LR (1, N = 123) = 6.02, p =.014. The odds of orretly enoding CP for the order-intat version relative to orderinverted was estimated as The effet of instrutor in temporal problems was signifiant for both DVs: X 2 LR (1, N 2322

3 = 123) = 3.93, p =.048 for orretness of solution, X 2 LR (1, N = 123) = 4.47, p =.034 for orretness of CP enoding. For the ausal-shema problems, order (=intat, inverted) signifiantly predited orret oding of CP, X 2 LR (1, N = 122) =11.71, p <.001, onfirming the hypothesis. The odds of expressing CP orretly for order-intat items was estimated as 3.70, relative to order-inverted items. However, order did not signifiantly predit overall problem solution suess, X 2 LR (1, N = 122) = 1.21, p =.272. The effet of instrutor was signifiant for both DVs: X 2 LR (1, N = 122) = 3.90, p =.048 for orretness of solution, X 2 LR (1, N = 122) = 3.98, p =.046 for orretness of CP enoding. One reason why onsisteny with temporal or ausal shema is related to orret enoding of CP, but not with overall solution suess, is that enoding CP is just one of many steps that must be arried out to solve a problem orretly. Table 2 presents the onditional probability of problem solution suess given orret enoding of CP. Table 2: Conditional probability of orret answer given orret CP-enoding, by ondition. Item Type Intat Inverted Total Temporal Causal Error analyses. A set of post-ho analyses examined the frequeny of several known errors in probability problem solving. Written protools were analyzed with respet to deviations from the orret solution for eah problem type (see Appendix). Table 3 summarizes the observed frequenies of the two most ommon probabilisti-reasoning errors exhibited in these data, the transposed onditional (substituting P(A B) for P(B A)) and ompound substitution, in whih the onditional probability P(A B) is onfused with the joint-event probability P(A^B). Other errors inluded misalulations, otherwise inorretly formalizing the problem, and some idiosynrati errors. Another ommonly doumented type of error, simple substitution (substituting P(A) or P(B) for P(A B)), did not our in these data. The hypothesized asymmetries in reasoning about intatand inverted-order temporal/ausal shemas are refleted in an asymmetry in the rate of transposition errors: misinterpreting P(A B) is muh more ommon than misinterpreting P(B A). For temporal items, no partiipants in the intat-order ondition made transposed onditional errors. GZLM ould not be used to analyze these data beause zero ells yield omplete separation in the data, thus unreliable estimates. A hi-square test of independene between ondition (intat, inverted) and this error type was therefore evaluated using Fisher's exat test, and was signifiant, X 2 (1, N = 43) = 11.12, p <.001. The number of students making ompound substitution errors in temporal items was too low for statistial analysis. For ausal items, more errors were of the transposed onditional type for the order-inverted ondition, and the differene was marginally signifiant, X 2 LR (1, N = 58) = 4.81, p <.057. Students in the order-inverted ondition were signifiantly more likely to ommit ompound substitution errors, X 2 LR (1, N = 58) = 6.66, p <.010. Table 3: Observed perentages (with frequenies) of seleted enoding errors, by ondition. Temporal Items Intat Inverted Total Transposed Conditional 0 (0) 51.7 (15) 34.9 (15) Compound Substitution 7.1 (1) 6.9 (2) 7.0 (3) Other Errors 92.9 (13) 41.4 (12) 58.8 (25) N Causal Items Intat Inverted Total Transposed Conditional 8.7 (2) 31.4 (11) 22.4 (13) Compound Substitution 60.9 (14) 25.7 (9) 39.7 (23) Other Errors 30.4 (7) 42.9 (15) 37.9 (22) N Disussion The rationale behind this study was to demonstrate the effet of temporal or ausal order on oding onditional probability and overall solution suess. When a problem text presents onditional probabilities in verbal form, they are more likely to be enoded orretly as onditional probability P(A B) when the natural temporal or ausal order of events is A B rather than B A. Temporal order signifiantly affeted enoding onditional probability (CP). The odds ratio for enoding CP orretly in temporal items was 2.66, indiating that a student reeiving an order-intat item was 2.66 times more likely to enode onditional probability orretly than a student reeiving an order-inverted item, ontrolling for instrutor. In an error analysis of students who did not enode CP orretly for temporal items, only students reeiving order-inverted items transposed the order of events. Order effets were also demonstrated in the ausal-shema problems. Partiipants were more than three times more likely to enode CP orretly for order-intat items than for order-inverted items, ontrolling for instrutor. Error analyses using GZLM indiated that a marginally greater proportion of students made the transposed onditional error in the order-inverted ondition, ontrolling for instrutor. For temporal-shema problems, the onditional probability for solving a problem orretly, given that onditional probability was orretly enoded, did not differ for both order onditions (Table 2). For the ausal-shema items, fewer students in the intat ondition orretly solved the problem orretly (given having enoding CP orretly) than did students in the inverse ondition. This may be explained by fat that the ausal items required alulating the probability of the omplement of the given event in addition to enoding CP, introduing an additional soure of noise into the DV of solution orretness. In fat, this omplementary event was linguistially signaled by the phrase DID NOT 2323

4 in the order-inverted problem (but not in the order-intat version), perhaps introduing a soure of bias. The results from the error analyses indiate that there may be shema-speifi effets related to temporal and ausal order. Reall that that the likelihood of ommitting a ompound-substitution error in ausal items was signifiantly higher in the order-intat ondition (Table 3). Sine types of errors oded were mutually exlusive, it makes sense that in absene of making a transposed onditional error, students erring in enoding CP may be more likely to make another systemati error. Compound substitution errors are fairly ommon when interpreting onditional probability, thus these data do not neessarily indiate that mistaking joint for onditional probability is speifially affeted by temporal or ausal order. In other words, these error analyses may demonstrate that transposed onditional errors are sensitive to order effet, while ompound substitution errors are not speifi to order effet. Findings from this study seem related to a phenomenon of judgment bias demonstrated by Tversky and Kahneman (1980). They showed that, in ases in whih P(A) = P(B), and P(A B) = P(B A), individuals were more likely to judge P(B A) as greater than P(A B) if they believed that A was a ause of B. The results of our study demonstrate that, possibly as a result of the strength of ausal diretion, students are more prone to reognize and orretly enode a onditional probability that preserves the natural forward diretion of ause to effet. General Disussion Results from the present study support the idea that there are differential shemati effets on probability problem solving. Systematially altering the ausal or temporal shema depited in a probability problem's over story is shown to affet the orret enoding of onditional probability as well as the type of errors, whih in turn affets orretly solving a problem. Importantly, effets of pragmati reasoning shemas are demonstrated for formal probability alulations and enoding usually assumed to be haraterized by System 2 reasoning proesses, whih have typially been shown to be resistant to heuristis and biases. The study showed that for both temporally- and ausallyrelated events, partiipants were more likely to enode given onditional probabilities inorretly when the events are expressed in inverse order. In addition, those errors were more likely to transpose the events from inverse order to intat order than from intat to inverse order. For problems depiting ausally-related events, partiipants were also more likely to inorretly enode CP when solving for CP expressing events in inverse order. This finding may be onsidered in light of the phenomenon of the fallay of the time axis, as illustrated by Falk (1986). In a within-subjets study, partiipants were asked to onsider the events of drawing two marbles from an urn ontaining two blak and two white marbles. Asked first to evaluate the probability of drawing a seond white marble after having drawn a first white marble without replaement, P(W 2 W 1 ), most partiipants provide the orret answer with relative ease. Next, asking the same partiipants to onsider P(W 1 W 2 ), a signifiant proportion of partiipants reply that the question is meaningless. Of those who attempt to solve the problem, many indiate that the probability is 1.00 or inorretly solve the item without onsidering the probability of the onditioning event. Although the results from the urrent study parallel Falk's (1986), the results disussed here generalize her results by demonstrating that the fallay also ours in enoding CP in formal probability problem solving, typially onsidered to be governed by System 2 proesses. Furthermore, to date there has been little evidene or explanation put forth onerning proesses influening the ourrene of the fallay of the transposed onditional. Villejoubert and Mandel (2002) doumented that frequeny formats redued the number of transposed onditional errors, however, their offered explanation was only that "people simply onfuse p(h D) with p(d H) beause the latter sounds a lot like the former." Krynski and Tenenbaum (2007) speulated that transposed onditionals are more likely when P(A B) is estimated as roughly equivalent to P(B A), as proposed by Tversky and Kahneman (1980). Our findings suggest that the transposed onditional error also an our when P(A B) and P(B A) are not equal. Rather, our results suggest that that problem solvers tend to exhibit the error only (or mainly) when the temporal or ausal order of events in the problem ontradit the natural ausal or temporal ordering of the depited realworld events. But note that these order effets in CP may not neessarily affet overall diffiulty of a problem over-andabove enoding CP. These results demonstrate that investigating the speifis of how onditional probability is interpreted in ontext may inform a bias produed by the order of events depited in probability problems and their relation to pragmati reasoning shemas indued by real-world experienes. If, in the over story of a problem (and in the underlying real-world pragmati shema), event A ours before event B or auses event B, then the onditional probability P(B A) reflets a shema-onsistent, intat order of events, translated as the probability that B ours given that A has ourred, refleting a deterministi, forward-looking time perspetive. In ontrast, translating P(A B) as the probability that A ours given that B has ourred seems nonsensial given the problem sript. Representing and reasoning about P(A B) demands onsideration of events in inverted hronologial order and may be validly translated as only the probability that A has ourred given that B has ourred, requiring retrospetive time perspetive. These two representations of onditional probability reflet two different types of reasoning about unertainty (Haking, 1975; Fox & Ülkümen, 2011). While P(B A) may be addressed in the preditive, aleatory sense, P(A B) must be onsidered with epistemi evaluation postditively. The phenomenon we doument may also have impliations for eduation. Shemati effets on probability problem 2324

5 solving may inform instrution on statistial inferene via onsideration of how the two types of probability judgments are theoretially distint. Evidene from the present study supports an interpretation whih addresses this differene in reasoning as parallel to the proess of statistial inferene and has impliations for statistial eduation and sientifi reasoning generally. Speifially, in evaluating evidene from a sample, the proess of hypothesis testing requires onsidering the onditional probability of finding a result in light of the null hypothesis H 0, a preexisting fat in the world, whih in fat either is, P(H 0 ) = 1, or is not, P(H 0 ) = 0. So while results from the present study indiate that it is easier to oneptualize P(some observed phenomenon H 0 ), hypothesis testing evaluates P(H 0 some observed phenomenon). In addition, in evaluating the validity of statistial inferenes, students are taught to onsider the likelihood of Type I and Type II errors, terms whih have beome shorthand for partiular onditional probabilities (Neath, 2010). Training in statistial inferene demands a level of mastery of understanding CP in inverted order, whih is inonsistent with ausal shemas and may warrant more lassroom disussion with respet to the nuanes and impliations for hypothesis testing. Referenes Bar-Hillel, M. & Falk, R. (1982). Some teasers onerning onditional probabilities. Cognition, 11, Barbey, A. & Sloman, S. (2007). Base-rate respet: From eologial rationality to dual proesses. Behavioral and Brain Sienes, 30, 241. Bassok, M., Chase, V. M., & Martin, S. A. (1998). Adding apples and oranges: Alignment of semanti and formal knowledge. Cognitive Psyhology, 35, Brun, W. & Teigen, K. H. (1990). Predition and postdition preferenes in guessing. Journal of Behavioral Deision Making, 3, Cheng, P. W., & Holyoak, K. J. (1985). Pragmati reasoning shemas. Cognitive Psyhology, 17, Cheng, P. W. & Nisbett, R. E. (1993). Pragmati onstraints on ausal dedution. In R. E. Nisbett (Ed.), Rules for reasoning (pp ). Hillsdale, NJ: Erlbaum. Cohen, L. J. (1981). Can human irrationality be experimentally demonstrated? Behavioral and Brain Sienes, 4, Díaz, C. & Fuente, I. de la (2007). Assessing students diffiulties with onditional probability and Bayesian reasoning. International Eletroni Journal of Mathematis Eduation, 2, Evans, J. S. B. T. & Frankish, K. (Eds.). (2008). In two minds: Dual proesses and beyond. New York: Oxford University Press. Falk, R. (1986). Of probabilisti knights and knaves. The College Mathematis Journal, 17, Fernbah, P. M., Darlow, A., & Sloman, S. A. (2011). Asymmetries in preditive and diagnosti reasoning. Journal of experimental psyhology: General, 140, Fishhoff, B. (1975). Hindsight is not equal to foresight: The effet of outome knowledge on judgment under unertainty. Journal of Experimental Psyhology: Human Pereption and Performane, 1, Fishhoff, B. (1976). The effet of temporal setting on likelihood estimates. Organizational Behavior and Human Performane, 15, Fong, G.T., Krantz, D. H., & Nisbett, R.E. (1986). The effets of statistial training on thinking about everyday problems. Cognitive Psyhology, 18, Fox, C. R. & Levav, J. (2004). Partition-edit-ount: Naive extensional reasoning in judgment of onditional probability. Journal of Experimental Psyhology: General, 133, Fox, C. R. & Ülkümen, G. (2011). Distinguishing two dimensions of unertainty. In W. Brun, G. Keren, G. Kirkebřen, & H. Montgomery (Eds.), Perspetives on thinking, judging and deision-making (pp ). Oslo: Universitetsforlaget. Gigerenzer, G. & Hoffrage, U. (1995). How to improve Bayesian reasoning without instrution: Frequeny formats. Psyhologial Review, 102, Gigerenzer, G. & Hoffrage, U. (2007). The role of representation in Bayesian reasoning: Correting ommon misoneptions. Behavioral and Brain Sienes, 30, 264. Girotto, V. & Gonzalez, M. (2007). How to eliit sound probabilisti reasoning: Beyond word problems. Behavioral and Brain Sienes, 30, 268. Haking, I. (1975). The emergene of probability: A philosophial study of early ideas about probability, indution and statistial inferene. NY: Cambridge University Press. Kahneman, D. & Tversky, A. (1979). On the interpretation of intuitive probability: A reply to Jonathan Cohen. Cognition, 7, Krynski, T. R. & Tenenbaum, J. B. (2007). The role of ausality in judgment under unertainty. Journal of Experimental Psyhology: General, 136, Mahi, L. (2000). Partitive formulation of information in probabilisti problems: Beyond heuristis and frequeny format explanations. Organizational behavior and human deision proesses, 82, Makie, J. L. (1981). Propensity, evidene, and diagnosis. Behavioral and Brain Sienes, 4, Neath, A. A. (2010). Statistial Inferene, Statistis Eduation, and the Fallay of the Transposed Conditional. Paper presented at the Joint Statistial Meetings, Vanouver, British Columbia. Nisbett, R. E., Krantz, D. H., Jepson, C., & Kunda, Z. (1983). The use of statistial heuristis in everyday indutive reasoning. Psyhologial Review, 90, Rothbart, M. & Snyder, M. (1970). Confidene in the predition and postdition of an unertain outome. Canadian Journal of Behavioural Siene/Revue anadienne des sienes du omportement, 2, Sloman, S. A. (1996). The empirial ase for two systems of reasoning. Psyhologial Bulletin, 119,

6 Sloman, S. A. & Rips, L. J. (1998). Similarity and symbols in human thinking. Cambridge, MA: MIT Press. Smith, E. R. & Collins, E. C. (2009). Dual-proess models: A soial psyhologial perspetive. In J. S. B. T. Evans & K. Frankish (Eds.), In two minds: Dual proesses and beyond (pp ). New York: Oxford University Press. Stanovih, K. E., Toplak, M. E., & West, R. F. (2008). The development of rational thought: A taxonomy of heuristis and biases. In R. V. Kail (Ed.), Advanes in hild development and behavior: Vol 36 (pp ). San Diego: Elsevier. Tversky, A. & Kahneman, D. (1974). Judgment under unertainty: Heuristis and biases. Siene, 185, Tversky, A. & Kahneman, D. (1980). Causal shemas in judgments under unertainty. In M. Fishbein (Ed.), Progress in soial psyhology. (pp ). Hillsdale, N J: Erlbaum. Tversky, A. & Kahneman, D. (1983). Extensional versus intuitive reasoning: The onjuntion fallay in probability judgment. Psyhologial Review, 90, Villejoubert, G. & Mandel, D. (2002). The inverse fallay: An aount of deviations from Bayes s theorem and the additivity priniple. Memory & Cognition, 30, Windshitl, P. D. & Wells, G. L. (1998). The alternativeoutomes effet. Journal of Personality and Soial Psyhology, 75, Wright, G. (1982). Changes in the realism and distribution of probability assessments as a funtion of question type. Ata Psyhologia, 52, Appendix: Test Items & Solutions Temporal-shema items Order Intat. You are waiting to meet your friend, who is oming from work. He phones saying he will get on the next bus. From experiene you know that if he athes an express bus his hanes of being on time are 90%, but if he athes a loal bus his hanes of being on time are 65%. You also know that 60% of the buses that stop by his work are loals, thus he has a 60% hane of athing a loal bus today. What is the probability that he arrives on time? Order Inverted. You are waiting to meet your friend, who is oming from work. He phones saying he will get on the next bus. From experiene you know that when he arrives on time, 90% of the time he has aught an express bus, but when he arrives late, 65% of the time he has aught an express bus. You also know from experiene that he is late 60% of the time, thus you figure that he has a 60% hane of being late today. What is the probability that he athes an express bus? Solutions: E: Cath express bus; OT: Is on time Order Intat P OT E. 9 POT E. P E. P OT P OT E P E P OT E P E Order Inverted PE OT. 9 PE OT. 65 POT. 6 P E P E OT P OT P E OT P OT Causal-shema items Order Intat. At a journalism shool, a professional ethis exam is given to all students at the end of their first year. Extensive researh has established that the probability that a student studies speifially for this exam is 70%. The overall proportion of students who pass the exam is 92%. Exatly 66% of the students will study for the exam and pass it. If we know that a student has studied speifially for the exam, what is the probability that the student FAILS? Order Inverted. At a journalism shool, a professional ethis exam is given to all students at the end of their first year. Extensive researh has established that the probability that a student studies speifially for this exam is 70%. The overall proportion of students who pass the exam is 92%. Exatly 66% of the students will study for the exam and pass it. If we know that a student has passed the exam, what is the probability that the student DID NOT study speifially for it? Solutions: P: Pass exam; S: Studies Order Intat P P. 92 P S. 7 PS P. 66 PP S PP S PS Order Inverted P P. 92 P S. 7 PS P. 66 PS P PS P PP