This paper discusses the Hot Hand as a cognitive illusion.
This paper discusses the Hot Hand as a cognitive illusion.
Whether the sexes of children born in the same family are independent or whether they are positively or negatively correlated, is an important biological problem, that is interesting also for statisticians.
States and local schools face the implementation of the Chance and Data part of a curriculum with no Australian research base from which to make recommendations for the preparation of teachers or for the suggestion of methods and topics realistic to the developmental level of the students. It is the purpose of this paper to suggest such a base.
In this paper we explore the possibility of using computer simulations in relation to the correction of misconceptions in probability judgments. But before considering this issue, we will look at some of possible difficulties that could come along with using computer simulations.
Although I don't want here to defend my point of view, it will serve my purposes to exemplify what I regard as a contrary one. My assumption is that students have intuitions about probability and that they can't check these in at the classroom door. The success of the teacher depends on large part on how these notions are treated in relation of those the teacher would like the student to acquire. Additionally, I think it is a myth that mathematics, either as a discipline or in the mind of a mathematician, develops independently from concerns about objects and relations that are believed to have real-world referents. This was certainly not so in the case of the development of probability theory.
There are two main reasons for our interest in statistical reasoning in children. The first one is that research has shown that understanding of statistical principles, and their appropriate usage, are related to the quality of decisions, judgments and inferences people make. The second reason is that American children learn very little about statistics in school.
My interest is in the relationship between probability and statistics (data analysis) with regard to teaching and learning. Ideas for teaching (exploratory) data analysis with no preparation in probability emphasize, among other things, finding relationships in sets of variables, identifying relevant variables, interpreting data with regard to sources of variation, possible explaining factors and causes. Probability is often introduced as an antithesis to deterministic situations. Some empirical research even blames children for looking for causes where there is "really" randomness. There is other research taking positions against stochastics.
A number of theoretical positions in psychology - including variants of case-based reasoning, instance-based analogy, and connectionist models - maintain that abstract rules are not involved in human reasoning, or at best play a minor role. Other views hold that the use of abstract rules is a core aspect of human reasoning. We propose eight criteria for determining whether or not people use abstract rules in reasoning, and examine evidence relevant to each criterion for several rule systems. We argue that there is substantial evidence that several different inferential rules, including modus ponens, contractual rules, causal rules, and the law of large numbers, are used in solving everyday problems. We discuss the implications for various theoretical positions and consider hybrid mechanisms that combine aspects of instance and rule models.
This paper explores a heuristic - representativeness - according to which the subjective probability of an event, or a sample, is determined by the degree to which it: i) is similar in essential characteristics to its parent population; and ii) reflects the salient features of the process by which it is generated.
Although P(A&B|X) can never exceed P(A|X) (the conjunction rule), it is possible for P(X|A&B) to exceed P(X|A). Hence, people who rank A&B as more probably than A are not necessarily violating any normative rule if the ranking is done in terms of the probability of these events to yield an event X. Wolford, Taylor, and Beck (1990) have argue that this indeed is what happens in some problems (e.g. Tversky & Kahneman's (1983) Linda Problem). The claim made here is that the Linda problem is hard to reconcile with this interpretation; that there is little if any evidence that subjects utilize this interpretation; and that in any case, representativeness can account for all Linda problem results.