Many studies have demonstrated that intuitive evaluations regarding the probability of uncertain events are often misconceived. Researchers agree that knowledge of the sources of the intuitive background of probabilistic reasoning will help in the correction of misconceived probabilistic judgments.

Various aspects of the psychological mechanisms for probabilistic reasoning have thus far been studied.

Tversky and Kahneman claimed that intuitive errors arise from the use of some heuristic principles which often lead to misconceived probabilistic judgments. The principle of Representativeness, by which people evaluate the probability of an event according to the extent to which its description reflects the manner in which they perceive the population of its basic possible outcomes, or alternatively, the process producing it, is an example of such a principle. In a lottery game, for example, a sequence such as 1, 16, 13, 20, 6 will appear to have better odds than such a sequence as 1, 2, 3, 4, 5, because it better represents the randomness of the lottery results (Tversky & Kahneman, 1974).

Many studies with adult subjects have shown that there is a tendency to erroneously evaluate two outcomes of a stochastic experiment as equiprobable. For example, in tossing two dice together, the probability of throwing the pair 6-6 is perceived as equal to that of throwing the pair 5-6. This tendency has been found to resist age, knowledge of probability theory and experience in games of chance. This misconceived judgment was thought by the researchers to result from operating a cognitive model by which random events tend to be assumed to be equiprobable by nature (Lecoutre, 1992, Lecoutre & Durand, 1988).

As mentioned above, various aspects of the psychological mechanisms of probabilistic intuitions have thus far been studied. Researchers, however, have ignored the possibility that probabilistic intuitions are linked to what Piaget (1976) called "the operational (logical, analytical) capacities of the individual". These capacities depend, according to Piaget's theory, on age, general intellectual development, acquired knowledge and experience. Also, no comprehensive study of the evolution of misconceived probabilistic intuitions with age, over a broad range of ages, has yet been performed.

Accordingly, we decided upon two basic objectives:

1. To determine the relationship between schemata and intuitions.

2. To investigate the evolution with age of intuitively based misconceptions, and to determine the extent to which such evolution is related to schemata.

Concomitant to these main objectives, we investigated the influence of the level of mathematical studies, and the gender of the subject, upon the extent to which misconceptions are applied. As mentioned above, our study focused on the domain of probabilistic intuitions.

Intuition is defined in this study in accordance with the approach of Fischbein (1987) as immediate cognition which appears subjectively as self-evident, is accompanied by a feeling of confidence, and is global, coercive and extrapolative. The axiom which states that the shortest distance between two points is the straight line joining them is, for example, intuitively acceptable by this definition.

In this study, the term schema will be used in the sense of structural schema: a program, basic way, approach or general principle of reasoning, interpretation or solution. All the operational schemata described by Piaget, such as the proportion schema, the combinatorial schema and the probability schema are, in fact, structural schemata (Fischbein, 1997).

These two cognitive structures are characterized by stability on the one hand, and by flexibility in accordance with the circumstances on the other hand. There is, however, an essential difference between them. Schemata are analytical programs of interpretation and response, while in contrast, intuitions are global and immediate approaches.

Several classical probabilistic intuitively based misconceptions appear in the literature, such as:

(1) Representativeness, (2) Gambler's fallacy, (3) Conjunction fallacy, (4) Availability, (5) Time axis fallacy - 'The Falk Phenomenon', (6) Insensitivity to sample size and (7) Simple and compound events - Equiprobability bias.

To investigate the issues presented in the study, a questionnaire was composed. The questionnaire contained nine probability problems, recognized in the literature as problems which lead to the use of the above-mentioned probabilistic intuitively based misconceptions.

A total of 367 subjects, of four age groups (grades 5, 7, 9 and 11), who had not studied probability, were requested to solve the various problems in writing. The problems were formulated so that the students would be requested to provide spontaneous answers. The students were asked to explain their answers.

The study of intuitions is a complex methodological problem mainly because the (unconscious) psychological mechanisms which shape the intuitions are hidden. To obtain more information regarding these mechanisms, we interviewed 32 subjects (8 students from each of the 4 age groups). These interviewees were selected from the 367 subjects who filled out questionnaires. The interviews were semi-structured and flexible, and were held a short time after the completion of the questionnaires.

The analysis of the students' replies and explanations revealed that their spontaneous responses were directed by various schemata and general principles. Only for 10% of the replies were we unable to identify a schema or general principle of the spontaneous responses.

This finding indicates that students' intuitive responses are generally not random, but rather they are given under the influence of general tacit logical schemata or principles which are identifiable by means of posterior analysis of the explanations given by the subjects.

The findings regarding the evolution of probabilistic intuitively based misconceptions with age reveal that some misconceptions progressively decrease (i.e., representativeness, gambler's fallacy and conjunction fallacy), while others become stronger (i.e., insensitivity to sample size, availability and time axis fallacy). Only one misconception, intuition related to simple and compound events, was found to remain relatively stable across ages.

The analysis of the explanations provided by subjects in the questionnaires and during interviews suggests that probabilistic intuitions are influenced by general principles and schemata. These schemata were found to become more dominant in the subjects' intellectual activity with age. However, their use is sometimes adequate and at other times inadequate for solving a particular problem. When the use of a schema was adequate for solving the problem, misconception declined with age (e.g., representativeness decreased with age under the influence of the principle of independence of outcomes in a stochastic experiment). However, when the problem was complex, the subject ignored important information required for solving the problem, and tended to use a general principle which was inadequate to the problem's specific constraints. The general principle was used owing to its superficial similarity to the problem’s apparent data in this case. Consequently, the frequency of misconceptions increased with age (e.g. the fallacy related to insensitivity to sample size became more frequent with age under the influence of the proportion schema).

In the misconceived solutions to the problem dealing with simple and compound events, we identified two dominant logical principles. One was a simple, basic principle, devoid of probabilistic model (the principle of chance, according to which random events are equiprobable by nature). The other was a complex, subtle principle, constituting a synthesis of two probabilistic concepts (the principle of equiprobability in a condition of symmetry). These principles evolved in opposite directions with age and consequently, the main misconception cowthis problem remained relatively stable with age.

The trend in the evolution of each misconception with the level of mathematical education was similar to the trend in the evolution of this misconception with age. Some of the misconceptions were more frequent among advanced level students, while others were less frequent among these students. The frequency of two of the misconceptions studied was fairly equal among the advanced and the intermediate level students. The analysis of the explanations provided by students in questionnaires and interviews revealed that advanced level students tended to use more subtle schemata and logical principles, such as the combinatorial schema or the principle of independence of outcomes in a stochastic experiment. In contrast, intermediate level students preferred using more basic principles, such as the "more of A, more of B" schema, or the chance principle.

We interpreted this finding as follows: Studying mathematics at a higher level develops logical reasoning, i.e. general logical principles and schemata evolve as mathematics is studied at a higher level. However, the use of a more complicated logical principle was not always adequate for the specific constraints of the problem. The major misconception was thus more frequent among the higher level students.

As mentioned above, age and level of mathematical education had a similar influence on the evolution of probabilistic intuitively based misconceptions. However, age was found to have a stronger effect on probabilistic intuitively based misconceptions than the level of mathematical education.

In the present study, no significant gender differences were found in the level of use of probabilistic intuitively based misconceptions.

The major contribution of this study is that it reveals the relationship between schemata and intuitions. These components of cognition, which play an important role in the process of learning and teaching, have previously been considered in the literature as separate components. The exploration of the relationship between them sheds light on the origin of probabilistic intuitively based errors in particular, and upon misconceptions in general.

The findings of this research may assist in formulating curricula and teaching techniques adapted for age and level, based on the relationship found between schemata and intuitions.