Ramesh Kapadia

A Probabilistic Perspective

There are unusual features in the conceptual development of probability in comparison to other mathematical theories such as geometry or arithmetic. A mathematical approach only began to emerge rather late, about three centuries ago, long after man’s first experiences of chance occurrences. A large number of paradoxes accompanied the emergence of concepts indicating the disparity between intuitions and formal approaches within the sometimes difficult conceptual development. A particular problem had been to abandon the endeavour to formalize one specific interpretation and concentrate on studying the structure of probability. Eventually, a sound mathematical foundation was only published in 1933 but this has not clarified the nature of probability. There are still a number of quite distinctive philosophical approaches which arouse controversy to this day. In this part of the book all these aspects are discussed in order to present a mathematical or probabilistic perspective. The scene is set by presenting the philosophical background in conjunction with historical development; the mathematical framework offers a current viewpoint while the paradoxes illuminate the probabilistic ideas.

The Educational Perspective

This opening chapter presents the aims and rationale of the book within an appropriate theoretical framework. Initially, we provide the reader with an orientation of what the book intends to achieve. The next section highlights some important issues in mathematical education, establishing a framework against which ideas in the book have been developed. Partly, the research has been inspired by the first book in this series on mathematical education: Freudenthal’s Didactical Phenomenology of Mathematical Structures. Though he considers many topics in mathematics he excludes (perhaps surprisingly) probability. Finally, summaries of each of the chapters are related to these didactic approaches.

A Historical and Philosophical Perspective on Probability

This chapter presents a twenty first century historical and philosophical perspective on probability, related to the teaching of probability. It is important to remember the historical development as it provides pointers to be taken into account in developing a modern curriculum in teaching probability at all levels. We include some elements relating to continuous as well as discrete distributions. Starting with initial ideas of chance two millennia ago, we move on to the correspondence of Pascal and Fermat, and insurance against risk. Two centuries of debate and discussion led to the key fundamental ideas; the twentieth century saw the climax of the axiomatic approach from Kolmogorov.

Modelling in Probability and Statistics

This chapter explains why modelling in probability is a worthwhile goal to follow in teaching statistics. The approach will depend on the stage one aims at: secondary schools or introductory courses at university level in various applied disciplines which cover substantial content in probability and statistics as this field of mathematics is the key to understanding empirical research. It also depends on the depth to which one wants to explore the mathematical details. Such details may be handled more informally, supported by simulation of properties and animated visualizations to convey the concepts involved. In such a way, teaching can focus on the underlying ideas rather than technicalities and focus on applications.

From Puzzles and Paradoxes to Concepts in Probability

This chapter focuses on how puzzles and paradoxes in probability developed into mathematical concepts. After an introduction to background ideas, we present each paradox, discuss why it is paradoxical, and give a normative solution as well as links to further ideas and teaching; a similar approach is taken to puzzles. After discussing the role of paradoxes, the paradoxes are grouped in topics: equal likelihood, expectation, relative frequencies, and personal probabilities. These cover the usual approaches of the a priori theory (APT), the frequentist theory (FQT), and the subjectivist theory (SJT). From our discussion it should become clear that a restriction to only one philosophical position towards probability—either objectivist or subjectivist—restricts understanding and fails to develop good applications. A section on the central mathematical ideas of probability is included to give an overview for educators to plan a coherent and consistent probability curriculum and conclusions are drawn.

Reasoning with Risk: Teaching Probability and Risk as Twin Concepts

Risk is a key aspect of life and probability is the mathematical tool to address risk. Our aim is to investigate how risk is embedded in probability and how probability can be used to solve problems of risk. Furthermore, we illustrate how risky situations are paradigmatic for the concept of probability, not least by the eminent role risk played for the emergence of probabilistic concepts. The analysis of risk forms the central section, which shows how risk is connected to probability. Decisions in risky situations draw from the various approaches to probability and from rational and behavioural views to decisions. The section on emerging concepts re-visits the development of probability and probabilistic thinking. Faced with the twin character of probability and risk, we argue that these concepts should be developed together in teaching. The conceptualisation of probability in terms of A Priori Theory (APT) Frequentist Theory (FQT) and Subjectivist Theory (SJT) forms the background to classify the pertinent constituents of the arguments. For understanding small probabilities/risks, we promote the ideas of a micromort (1 in a million chance of leading to death) and a microlife (a half-hour period, which is 1 in a million part of the average life span of a person aged 30 years).