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Effects of Temporal and Causal Schemas on Probability Problem Solving

Effects of Temporal and Causal Schemas on Probability Problem Solving S. Sonia Gugga (ssg34@columbia.edu) Columbia University New York, NY 10027 James E. Corter (jec34@columbia.edu) Teachers College, Columbia University New York, NY 10027 Abstract Causal beliefs have been shown to affect performance in a wide variety of reasoning and problem solving. One type of judgment bias that can result from implicit causal models is causal asymmetry -- the tendency to judge predictive inferences as more plausible than comparable diagnostic inferences. In the present study we investigate if the directionality of implicit causal models can also affect application of formal methods, specifically the solution of conditional probability word problems. The study examined temporal and causal schemas, in which the convention is that events are considered in forward direction. Pairs of conditional probability (CP) problems were written depicting events E1 and E2, such that E1 either occurs before E2 or causes E2. Problems were defined with respect to the order of events expressed in CPs, so that P(E2|E1) represents the CP in schema-consistent, intact order by considering the occurrence of E1 before E2; while P(E1|E2) represents CP in schema- inconsistent, inverted order. Participants had greater difficulty encoding CP for events in schema-inconsistent order than CP of events in the conventional deterministic order. Keywords: Conditional probability; social schemata; causal schemata. Introduction Previous research on statistical reasoning has illustrated that individuals often disregard statistically prescribed processes, employing heuristics and biases which may produce invalid inferences (see Barbey & Sloman, 2007; Cohen, 1981; Gigerenzer & Hoffrage, 2007; Kahneman & Tversky, 1979; Krynski & Tenenbaum, 2007; Nisbett, Krantz, Jepson, & Kunda, 1983; Stanovich, Toplak, & West, 2008; and Tversky & Kahneman, 1974, 1980 for extensive treatments of the debate). Improvement in inferential accuracy has been affected by training, expression of likelihood in frequency formats, partitive formulation of the information space, or by expressing real-world application to contexts in which individuals are more likely to reason probabilistically with respect to causal or social schemas (Fox & Levav, 2004; Gigerenzer & Hoffrage, 1995, 2007; Girotto & Gonzalez, 2007; Krynski & Tenenbaum, 2007; Macchi, 2000). A mathematical word problem's cover story provides real- world context and semantic content, sometimes considered “surface features”, because they are usually assumed not to directly affect a problem's technical difficulty or formal solution processes. When a cover story relates to social interactions, however, pragmatic reasoning schemas formed in response to real-world experiences may be invoked in the mind of the problem solver (e.g., Cheng & Holyoak, 1985; Fong, Krantz, & Nisbett, 1986; Bassok, Chase, & Martin, 1998). The claim made here is that problem solving in probability is particularly affected by domain-specific knowledge in probabilistic reasoning. Previous research has found discrepancies between formal quantitative probabilistic assessments (“System 2” processing) and on-the-fly, qualitative probabilistic judgments (tapping “System 1”). This prior research 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; Stanovich, Toplak, & West, 2008; Tversky & Kahneman, 1983; Windschitl & Wells, 1998). While the two processes have been well-differentiated, it remains to be determined how they may interact in formal problem solving contexts. Individuals have been shown to have a strong bias toward System 1 processes for evaluating social behavior or characteristics qualitatively, while System 2 processes are engaged in symbolic contexts. It may be that the specific interaction of the two processes depends on how problem elements are represented in the mind of the problem solver. The study reported below examines reasoning effects stemming from the direction of temporality and causation of events in an applied problem’s content, examining whether inverting the temporal direction of a schema affects the difficulty of a conditional probability problem. It was hypothesized that it would be easier for individuals to reason forwards regarding temporal or causal events given the deterministic nature of causal and, by extension, temporal schemas (Cheng & Nisbett, 1993; Tversky & Kahneman, 1980). In other words, it should be easier for subjects to calculate the conditional probability of an event given the probability of events preceding it versus calculating the probability of an event given the probability of events occurring later. Similarly, it was expected that problems asking for the conditional probability of an effect given the probability of its cause(s) would be easier than problems asking for the probability of a cause given the probability of its effect(s). It was supposed that inverting the direction of determination should introduce an additional level of difficulty to the problem. It was also expected that the perceived causal strength between events would mediate the effect. The results, described below, yield insights into how probability problems are categorized and solved, what effects