Academic journal article Journal of Educational and Developmental Psychology

The Cognitive Processes of Probability Estimation in Random Sampling from Discrete Uniform Spaces

Academic journal article Journal of Educational and Developmental Psychology

The Cognitive Processes of Probability Estimation in Random Sampling from Discrete Uniform Spaces

Article excerpt

Abstract

The research examines how high school students solve simple probability problems, and the probability estimations of students, boys and girls, in a discrete and uniform probability space. In discrete uniform probability spaces problem solving using estimations of probabilities and strategies of solving problems in a variety of situations were described. Solving probability problems among elementary intermediate and high school students were studied, however, reviewing the literature in probability; I am not familiar with any research on estimating probabilities or characterizations of rules people use for estimating and solving probability problems relating specifically to numerically symmetric and asymmetric objects. The research describes the rules and heuristics that student use to solve probability problems and estimate probabilities. The research has shown that students at different levels of proficiency in solving probability problems take a different approach when solving and estimating probabilities. When students are asked to estimate the probability of selecting two beads with different colors from a bag containing beads of different colors some calculate probabilities and other estimate them using a variety of rules. The use of rules for estimating probabilities is directly related to numerical relations and symmetries of objects. I describe a variety of rules students use for estimating probabilities, e.g., the "pair rule: if the number of pairs of objects in a group is greater than the number of the other pairs the probability of drawing a pair of first kind is greater than drawing pairs of the second kind". I present a mechanism for the estimations and decisions taken by students when estimating probabilities.

Keywords: Asymmetry, Cognition, Mechanism, Perception, Probability, Sampling, Symmetries, Unified, Uniform

(ProQuest: ... denotes formulae omitted.)

1. Introduction

When students solve problems which have a probabilistic characterization, many of them "put aside" formal rules and use instead heuristics to solve these problems.

Preview research gives strength to these views. In their work (Piaget & Inhelder, 1975; Schnarch, 1998), which investigated also students that studied probability, found that when these students solved probability problems which included numerically symmetric and asymmetric sets of objects many of them have used rules for solving problems. An interesting strengthening in the use of heuristics in solving probability problems is shown by an experiment conducted by Tversky and Kahneman (1969). They found that many senior academic people from areas such as mathematical psychology who were asked to solve problems presented to them in a questionnaire based their judgment on small samples, showed insensitivity to the size of the sample, and used a heuristics approach to solve these problems, which made it impossible for them to solve it correctly (Kahneman & Frederick, 2002; Tversky & Kahneman, 1971).

Gestalt psychologists, who have studied the visual system, have shown that visual perception is not a direct reading of the world. For example a dense set of points ...is conceived as a continuous line compared to a separated set of points...There is a tension between observing material and our perception (Goldmeier, 1972; Wertheimer, 1950). In this case the point represents the material and the line is the figure. Minimizing distances between the points improves the chances that we see a line or a figure and not just a collection of points (material). A similar behavior of the brain is described by the mar-poggio model for stereograms; when looking at a complex collection of points the brain produces a picture (Pinker, 1999; Poggio, 1984).

In a preview research (Schreiber, 2003) a direct relation between the representation of the problem and the use of intuitive rules is shown with respect to the set of objects being drawn. …

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