A Role of Negation in Overcoming a Probabilistic Misconception: Focusing on the Law of Small Numbers

Abstract

　　Many difficulties in learning probability have been reported by several researchers from psychological perspectives.　One example is the Law of Small Numbers, which is a well-known misconception in probability that has been described and considered in the literature for several decades.　People who hold this misconception mistakenly assert that the Law of Large Numbers is applied to small samples as well.　The reasons why the Law of Small Numbers happens are usually explained by means of heuristics and biases. However, the question how are these heuristics and biases overcome? do not have yet any clear answer.　The purpose of this paper is to clarify a process of overcoming the Law of Small Numbers from an epistemological perspective of negation.　For this purpose, the Iwasaki’s framework on negation in concept formation is applied for describing overcoming the Law of Small Numbers.　A process of overcoming the Law of Small Numbers consists of three phases from the framework.　A first step is negation of proportion and construction of substance of proportion.　A second step is analytic negation in which substance of proportion is negated by viewpoint of necessity, and the concept of probability is constructed in this phase.　A third step is synthetic negation in which concept of probability is negated by viewpoint of contingency, and overcoming the Law of Small Numbers occurs in this phase (Fig.).

Fig. Figure of Substance: Probability