At the core of epidemiology is the use of quantitative methods to study health, and how it may be improved, in populations. It is important to note that epidemiology concerns not only the study of diseases but also of all health-related events. Rational health-promoting public policies require a sound understanding of causation. The epidemiological analysis of a disease or activity from a population perspective is vital in order to be able to organize and monitor effective preventive, curative and rehabilitative services. All health professionals and health-service managers need an awareness of the principles of epidemiology. They need to go beyond questions relating to individuals to challenging fundamentals such as ‘Why did this person get this disease at this time?’, ‘Is the occurrence of the disease increasing and, if so, why?’ and ‘What are the causes or risk factors for this disease?’

The chapter begins by looking at Beck’s theory of a Risk Society and how this has denied that risk is a term that can be applied to the ancient world. There is a survey of relevant work that has been carried out on uncertainty in antiquity, before the chapter examines how the term risk emerged and the various ways of defining the term. The chapter argues for a broader, cultural approach that sees risk as more than a modern numerical concept and the development of ideas concerning risk as a historical process.

Chapter 4 examines the requirement in Article XX GATT 1994 and Article XIV GATS (the General Exceptions) as well as Article 2.2 of the TBT Agreement that a Member’s measure must contribute to a certain policy objective. The chapter argues that this requirement is essentially a causal requirement, and raises some jurisprudence that has argued the same. The chapter also traces through parts of the jurisprudence that would seem to support a non-attribution analysis of some kind being used in this context. The chapter concludes by proposing that the Tripartite Non-attribution/Causal Link Analysis might be applied to make a determination of causation for the purposes of this requirement under the General Exceptions and Article 2.2 of the TBT Agreement. The chapter concludes by setting out in detail how the Tripartite Non-attribution/Causal Link Analysis might be applied to make causal determinations for the purposes of the General Exceptions and Article 2.2 of the TBT Agreement.

The Blanc casinos were marketed through the cultivation of an impression of honesty and mechanical universality. The study of probability, which arose historically in relationship to the calculation of gambling odds, provided a way to measure the honesty of a casino. Probability, as it was expressed in the context of nineteenth-century resort casinos, was the object of renewed interest among professional mathematicians and amateurs seeking to understand the logic of the games they played. There are three avenues through which this amplified interest in probability was expressed in the nineteenth century: the analysis of “runs” (a long sequence of identical results), the systems that gamblers developed for beating the odds, and the casino as an experimental space for mathematicians in the nineteenth century. Together, these developments suggest that the nineteenth-century casino provided a novel opportunity for inquiry into areas such as the nature of time, the limits of causation, and the science of probability.

This chapter examines suitable statistics questions for investigation by children of different ages, using a cycle of problem, plan, data, analysis and conclusion (PPDAC) (Wild & Pfannkuch 1999). The importance of variation in data and different types of variables and the difference between a population and a sample are investigated. Readers will explore different ways of displaying data to ‘tell a story’. The importance of drawing inferences from data and the uncertainty associated with these inferences are discussed. Readers will engage in activities that use technology to support the development of statistical understanding.

Uncertainty is a part of everyday life. We live with a range of situations that inherently have an element of uncertainty in them – for example, crossing the road, going on holiday, or making a major purchase – but we often ignore the embedded chance in these activities. Risk is acknowledged in many activities, and much effort is expended in identifying these risks and minimising any potential negative outcomes. In schools, for example, a risk assessment is required prior to any excursion with children. Probability is the strand of mathematics that addresses uncertainty. This chapter explores ideas relating to probability in the mathematics classroom.

For this book, we assume you’ve had an introductory statistics or experimental design class already! This chapter is a mini refresher of some critical concepts we’ll be using and lets you check you understand them correctly. The topics include understanding predictor and response variables, the common probability distributions that biologists encounter in their data, the common techniques, particularly ordinary least squares (OLS) and maximum likelihood (ML), for fitting models to data and estimating effects, including their uncertainty. You should be familiar with confidence intervals and understand what hypothesis tests and P-values do and don’t mean. You should recognize that we use data to decide, but these decisions can be wrong, so you need to understand the risk of missing important effects and the risk of falsely claiming an effect. Decisions about what constitutes an “important” effect are central.

The introduction sets out the intent of the book, an overview of the major works in the field, and a view of the arguments appearing in each chapter. Gambling is central to the cultural, social, and intellectual history of the nineteenth century. Studying casino gambling provides a way to see how nineteenth-century Europeans understood their changing world, even as it also reflected those changes itself. In this way gambling was used in an explanatory capacity, one that let contemporaries probe the inner workings of the machine and the creation of knowledge. If we want to understand the intricate dance of society, culture, politics, and ideas, then gambling is a useful tool to pry open these different stories, allowing us to see better large historical transformations.

In this chapter we identify issues pertinent to ‘out of field’ teachers of mathematics. Strategies are offered to assist ‘out of field’ teachers of mathematics while considering the invaluable knowledge a primary-trained teacher can bring to the secondary mathematics classroom. Examples are presented of teachers’ critical pedagogical content knowledge that has a high impact on students’ growth and development across the main strands of the curriculum.

People must often make inferences about, and decisions concerning, a highly complex and unpredictable world, on the basis of sparse evidence. An “ideal” normative approach to such challenges is often modeled in terms of Bayesian probabilistic inference. But for real-world problems of perception, motor control, categorization, language comprehension, or common-sense reasoning, exact probabilistic calculations are computationally intractable. Instead, we suggest that the brain solves these hard probability problems approximately, by considering one, or a few, samples from the relevant distributions. By virtue of being an approximation, the sampling approach inevitably leads to systematic biases. Thus, if we assume that the brain carries over the same sampling approach to easy probability problems, where the “ideal” solution can readily be calculated, then a brain designed for probabilistic inference should be expected to display characteristic errors. We argue that many of the “heuristics and biases” found in human judgment and decision-making research can be reinterpreted as side effects of the sampling approach to probabilistic reasoning.

The first chapter analyzes the role of verisimilar realism in two trials: the summary trial officiated by Dioneo between the servants Tindaro and Licisca (Introduction, Day 6) and the inquest of the woolworker Simona, who is falsely accused of poisoning her lover Pasquino (4.7). It argues that Boccaccio uses these trials to foreground the risks involved when rhetorical categories such as “likely” and “probable” are used to evaluate evidence. These procedural tales are driven by the tension inherent between what everyone knows on the one hand and the singular event on the other, or between normative knowledge and the “novella.” In the trial between the two servants, the judge Dioneo acknowledges that women are not always virgins on their wedding night. He updates the storytellers’ shared picture of the world, making it more realistic. The trial of the woolworker Simona suggests how an art of the probable might incorporate singular phenomena. Yet certain details always remain outside the frame.

Many of the problems that human minds need to solve – including learning concepts, causal relationships, and languages – require making informed inferences from limited data. Bayesian models of cognition consider how an ideal agent should solve these problems, drawing on ideas from probability theory, statistics, machine learning, and artificial intelligence.  The resulting models can then be used to understand human behavior, identifying in formal terms the knowledge that human minds draw on when solving these problems and identifying potential mechanisms by which their solutions might be implemented. This chapter provides an introduction to Bayesian models of cognition, starting with the basic principles of probability theory and then considering more advanced topics such as graphical models, causal learning, hierarchical Bayesian models, and Markov chain Monte Carlo. The chapter ends with a brief review of recent theoretical developments.

A review of basic probability theory – probability density, expectation, mean, variance/covariance, median, median absolute deviation, quantiles, skewness/kurtosis and correlation – is first given. Exploratory data analysis methods (histograms, quantile-quantile plots and boxplots) are then introduced. Finally, topics including Mahalanobis distance, Bayes theorem, classification, clustering and information theory are covered.