Probability Matching: Exploring Mathematical Simplification in Uncertain Contexts

• In Event: 46.084 - The Bright Side of Uncertainty, Variability, Forgetting, and Bias: Rethinking the Cognitive Processes Involved in Learning

Abstract

Probabilistic reasoning refers to the ability to use probabilities to make optimal inferences. Thus, probabilistic reasoning combines both numerical comprehension (use of fractions, decimals, percentages, and/or proportions) and decision-making in uncertainty. Probabilistic reasoning, therefore, can be thought of as a bridge between mathematical cognition (and math education) and scientific reasoning (and science education). It combines mathematical content (percentages, decimals, arithmetic operations) with scientific processes (evaluation of claims, generation of hypothesis and explanations). Therefore, the educational implications of when and how people operate with probabilistic information abound (Jones et al., 2005).

Despite the importance of this ability, years of psychological research has demonstrated that individuals demonstrate robust biases in evaluating outcome potentials based on probabilities, often favoring oversimplified solutions that neglect relevant information (Kahneman & Tversky, 1973). While these biases are problematic in particular situations, they also demonstrate a certain level of efficiency in cognitive processing of numerical information. Instruction aimed at addressing these biases, with the goal of training individuals to suspend rushing to overly simplified conclusions, improves the more general capacity to evaluate evidence.

One bias is the tendency to neglect independence. Independence refers to the concept that across a series of outcomes, no one example reflects the underlying probability alone (i.e. when flipping a coin, getting two heads in a row does not increase the chance of getting tails on the next role). The tendency to makes one’s predicts to “correct” or “even” out expectations with known underlying probabilities is called probability matching.

The current study was designed to further explore the probability-matching phenomenon. Across two studies with 120 participants, we used colored dice (with only two possible colors) and roulette wheels to elicit participants’ predictions. In a novel design, we alternated probabilities between 4/6 vs. 5/6 for the dominant color, using 1/2 as a pseudo control condition, and alternate the number of colors to be selected under each probability condition among 3, 6, and 8. In this manner, we could explore whether increasing the dominate color’s probability towards 1 and limiting the easy of “matching” the probability might decrease this bias to probability match. Results demonstrated marginal decreases in probability matching in our experimental conditions. The key finding was the participants maintained a bias towards probability matching, even when the condition required them to convert the probability to a different base (i.e. participants reduced 4/6 chance of blue to 2 out of 3 when only 3 “bets” were allowed).

In short, the ambiguity of decision-making when a priori probabilities are the primary evidence available creates a situation in which individuals rely heavily on the surface features of the numerical information provided, while ignoring the difference between probabilities and their conditional nature and other similar numerical information (fractions, percentages, proportions, etc.). These results underscore the robustness of errors in probabilistic reasoning and also suggest that the oversimplified application (and in some cases misapplication) of underlying mathematical concepts such as proportional reasoning may be driving many of these errors.

Author

• James Houseworth, University of Minnesota