Chapter 12 extends students’ understanding of Statistics and introduces foundational concepts in Probability for Years 3 to 6. You will explore how to support students in collecting, organising, and interpreting data, identifying patterns, and predicting outcomes using simple probability language. The chapter also highlights strategies for integrating digital tools, adapting tasks to meet diverse learning needs, and making cross-curricular connections to enhance relevance and engagement.

Not everything we know about numbers can be interpreted as true of what they represent, so this chapter explores how far we can go in applying the numbers to real problems, and how we can be sure they are meaningful.

The book’s Conclusion develops the argument made by earlier chapters. It considers the variety of ways in which probable arguments, structured through imaginative engagements with legal forms of testimony, interacted with convictions and beliefs that were borne out of supernatural, spiritual influence. One key outcome of this discussion is the recognition that literature provided an important venue for comparing different kinds of belief and assent. Literary texts staged highly plausible legal cases, rooted in persuasively credible evidence; they could also qualify the force of these forensic arguments, especially when accounting for the ways in which religious belief was understood to work. This literary evaluation of modes of assent sheds new light on what it means to write a history of belief. The book ends by outlining a methodology that attends to texts and contexts where persuasive and probable modes of argument were afforded only provisional force.

Can de Broglie’s hypothesis be generalised to any physical object interacting with its environment? Schrödinger answered this question by introducing a complex-valued wave-function that fully characterises the state of an object as informational content. If its spatial extent is limited, it accounts for the information localisation of a classical body. We also show that physical quantities (energy, linear momentum or position) are represented by operators, and how their measurements are made in quantum physics. In particular, stationary states are eigenstates of the Hamiltonian with determined energy values. The evolution of the wave-function is governed by the Schrödinger equation, which is the fundamental equation of quantum physics. Examples are taken for a stationary (i.e., time-independent) one-dimensional interaction, when the considered physical object is in a free (or scattering) state – not classically constrained to remain in a spatially bounded domain. We consider potential energy steps that model localised interactions on which a physical object is scattered with determined probabilities of reflection and transmission.

This paper aims to assess the cogency of Hume’s famous argument against testimony for miracles. Hume starts by arguing in favour of a “general Maxim” which involves balancing the strength of the testimony “considered apart and in itself” against the inductive unlikelihood of the reported event. But although this reasoning shows real insight – anticipating what is now known as the “base rate fallacy” – it turns out that such a separation cannot work, and an adequate maxim must inevitably take into account the specific nature of the reported event when evaluating the epistemic strength of the testimony. There is also a deeper problem with Hume’s argument, which arises from his treating a miracle as an extreme example of an inductively unlikely event. For the believer can agree that miracles are inductively unlikely – or even physically impossible – whenever the world is proceeding normally. Where she will differ from Hume is in claiming that divine activity can interfere with the natural order, and can sometimes be identified through its purposive nature. Naturalist philosophers – like Hume – are likely to reject this, but their best argument for doing so comes not from theoretical probabilistic maxims, but from the hopelessly unconvincing track record of miracle reports, combined with the lack of evidence for divine purpose in the world (as revealed so artfully by Hume’s Dialogues).

This chapter starts with a discussion on models informing probability versus the case where probability is inherent in the model. The chapter also goes into detail to argue why a particular interpretation of quantum mechanics, Bohmian economics, can be useful in finance. We provide for an example of how such mechanics can be applied to daily returns on commodity prices. We also briefly look into the potential connection between Bohmian mechanics and a macroscopic fluid system.

Decisions are often made in an environment of risk, where the decision maker considers probabilities and possible outcomes. But in order to make a decision under risk, the decision maker must first determine the probabilities of the possible outcomes.

Through experience and observation, people estimate the probability of future events. The probability of an event can be determined as the number of instances of the particular event divided by the total number of possible events. With the tossing of a coin, there are two possible outcomes – a head or a tail – and the probability of either one is ½ or .5. Games of chance, including roulette and lotteries, provide clear examples of how probabilities can be calculated.

When probabilities and outcomes are known, expected value can be calculated and decision makers can use expected value when they analyze possible courses of action and make a decision.

This chapter provides a discussion on multivariate random variables, which are collections of univariate random variables. The chapter discusses how the presence of multiple random variables gives rise to concepts of covariance and correlation, which capture relationships that can arise between variables. The chapter also discussed the multivariate Gaussian model, which is widely used in applications.

This chapter lays the foundation of probability theory, which has a central role in statistical mechanics. It starts the exposition with Kolmogorov’s axioms of probability theory and develops the vocabulary through example cases. Some time is spent on sigma algebras and the role they play in probability theory, and more specifically to properly define random variables on the reals. In particular, the popular notion that ‘the probability for a real variable to take on a single value’ is critically analysed and contextualised. Indeed, there are situations in statistical mechanics where some mechanical variables on the reals do get a non-zero probability to take on a single value. Moments and cumulants are introduced, as well as the method of generating functions, which prepare the ground as efficacious tools for statistical mechanics. Finally, Jaynes’s least-biased distribution principle is introduced in order to obtain a priori probabilities given some constraints imposed on the system.