Almost by definition of risk, rare events play a crucial role. We tackle this problem by presenting some basic tools from extreme value theory (EVT). From a statistical point of view, the workhorses are the block maxima method (BMM) and the peaks over threshold method (POTM). Besides giving the mathematical formulation, we exemplify both approaches via simulated examples. Once these tools are in place, we can provide estimators of the relevant risk measures such as high-exceedance probabilities, quantiles and return periods. In a crucial part of the book, we then estimate these quantities for sea-level data at Hoek van Holland near Rotterdam. We obtain estimates, including confidence intervals, for a necessary dike height withstanding a required 1 in 10 000 years storm event. Further applications concern financial data and data from the L’Aquila earthquake. For the latter, we present dynamic models for earthquake aftershocks. After an excursion to the world of records in athletics, we present the signature application of EVT through the story of the sinking of the MV Derbyshire. We show how an application of EVT techniques has saved many lives at sea.

In this chapter we end our more strenuous hike and start to enjoy a technically more relaxed stroll through the landscape of risk. An important feature throughout the book is that, for all the data examples given, we first start with a section “About the data”. In this way, you gain experience in finding and preparing the relevant data before starting with a statistical analysis. In order to discuss the consequences of climate change, we analyze the Hadley Centre Central England Temperature (HadCET) dataset. For the implications of a rise in temperature to the loss in volume of alpine glaciers, we apply the techniques learned to the case of the Lower Arolla Glacier in Switzerland. For an application to the realm of agricultural production, we look at the consequence of climate change to wine production. For this, we analyze data from a specific wine producer in France. We end this stroll with an interesting and perhaps somewhat surprising story of the astronomer Johannes Kepler. The story relates to his second marriage and the measurement of the volume of wine barrels.

We have reached the end of our stroll. We find ourselves in the company of Alexandre Dumas who, in 1850, wrote “The Black Tulip”. In it, he combines the stories of the tulip mania in the Netherlands with the tragic story of the brothers de Witt. In our final example of “About the data” we reconstruct the historic trading data of tulip bulbs, which turns out to be a detective story in its own right. Prices for tulip bulbs crashed on February 3, 1637. We also include the story of the growing of the first black tulip in 1986. Johan de Witt was tragically lynched by a politically motivated mob on August 20, 1672. With him, we meet a politician who, through his mathematical training, was able to solve an important problem from the realm of life insurance risk, the pricing of annuities. His publication “Waerdye” is our final example on risk communication. We leave the closing lines of our book to Shakespeare’s Hamlet, who spoke the following words to Horatio “There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy.” We hope that we were able to convince you that these words very much apply to the realm of risk.

When it comes to natural disasters, earthquakes and tsunamis all too often top the list of worst calamities. Using several examples we will try to improve our understanding of how they occur. In later chapters, we discuss whether science indeed has techniques that can lead to statistical modeling. The examples discussed include the 2004 Boxing Day tsunami, killing more than 220 000 people, the 2011 Tōhoku earthquake and tsunami, which included the major nuclear disaster in Fukushima, and the volcanic explosion at the kingdom of Tonga on January 15, 2022. From each of these events we discuss specifics concerning risk, both in understanding as well as communication. We start the chapter with a brief, non-technical discussion of (Daniel) Bernoulli’s principle in incompressible fluids. This allows us to learn how tsunamis are formed and propagate across oceans causing catastrophic inundations to lower-lying coastal areas, often very far away. Especially for the Tōhoku and Fukushima case, we discuss the crucial difference between an "if" approach to risk management versus a "what if" one. The Tonga explosion highlights the importance of modeling such extremal events, taking the global geometric shape of our planet into account.

This rather long chapter constitutes part of the hike in our walk/hike/stroll set-up. We introduce the reader to the basics of stochastics (representing both probability and statistics) necessary for the more technical discussions on risk later. The path followed starts from probability space (a theoretical concept we quickly leave aside); we then move to the notion of a random variable and,, its distribution function, including the most important discrete as well as continuous examples. Historical examples as well as pedagogical ones are always included in order to support the understanding of the new concepts introduced. These examples often show that there is more to randomness than meets the eye. For the applications discussed later, we will measure statistical uncertainty through the concept of confidence intervals. These can be based either on some asymptotic theory involving the famous bell curve, the normal distribution, or on some form of resampling known under the name of bootstrapping. Further, we add some tools that are very important for measuring and communicating risk; these include the concepts of return periods and quantile functions.

The 2007 – 2008 financial crisis has been the subject of many articles, books, movies and even a theater play. We take you for a walk along Wall Street, pointing at some aspects of the story that touch our personal experiences and interests. We look critically at some of the main underlying causes. Besides political blindness, we highlight the unbridled growth of power of relatively small groups of investment bankers and how these brought financial institutions to (and in some cases beyond) the edge of bankruptcy. We expose some of the over-complicated financial instruments together with their astronomical volumes traded. We also look more critically at the role played by quants (financial engineers) in general and mathematicians in particular. What is the truth behind “The formula that killed Wall Street”? An important aspect concerns early warnings not heeded to. In summary, this is a chapter on greed, power, complexity, volume and stupidity.

Finding the elixir of life has always been an important quest of humanity. The topic of longevity, hence answering the question about the maximal age attainable (now and in the future) of humans has fascinated civilizations throughout the ages. Starting from the example of the oldest person on record, Jeanne Calmant, we introduce basic statistical techniques from survival (also reliability) analysis in order to study the process of aging from a statistical point of view. Recent findings on the topic are applied to the question of whether there is a plateauing of the hazard rate at high age. We then turn to the important problem of risk communication in the case of hurricanes. In particular, we compare and contrast the two hurricanes Katrina (2005) and Ida (2016) that hit the coast of Florida around New Orleans. Especially in the case of Ida, we comment upon the communication around the risk due to the ensuing flash floods over the northeast coast of the United States. New architectural and city planning initiatives, such as sponge cities, deserve our attention.

During the night of January 31 to February 1, 1953, the southwest coast of the Netherlands experienced a ferocious storm, killing over 1800 people, causing untold suffering and a major economic loss. As a consequence, the Dutch government initiated the Delta Project, which, through a combination of engineering works, should make the country safe for years to come. As part of this project, risk measures were introduced, like the so-called Dutch standard of a 1 in 10 000 years safety measure. Their statistical estimation was worked out and embedded in major engineering projects. These resulted in the construction of numerous new dikes along the coast. Through this example, we highlight several aspects of hazard protection. First, mathematics has an important role to play. Second, interdisciplinarity is key. Third, with such major and costly projects, spanning several generations, a clear communication to politicians as well as the public is both demanding as well as necessary.

It is January 28, 1986. While the world was watching, just 73 seconds after take-off, the Challenger Space Shuttle exploded, killing all seven astronauts on board. The crew included the teacher Christa McAuliffe who would have lectured schoolchildren from space. An important factor that contributed to the disaster was the extremely low temperature at launch. “Extreme” here means “well below temperatures experienced at previous launches”. In this chapter, we give a short overview of the errors that contributed to the explosion. These errors range from purely managerial errors to technical as well as statistical errors. Our discussion includes a statistical analysis of the malfunctioning of so-called rubber O-rings as a function of temperature at launch. As a prime example of efficient risk communication we also recall the press conference at which the physics Nobel Prize winner, Richard Feynman, made his famous “piece-of-rubber-in-ice-water” presentation. This exposed the cause of the accident in all clarity.

Our book was written during the COVID pandemic. As a result, it was natural to include a chapter on this topic. In line with the overall theme of our book, we highlight aspects close to the understanding and communication of risk. Topics included in more detail are the inherent danger of exponential growth and the need for adhering to the precautionary principle when faced with a new, possibly catastrophic and hence not yet widely understood, type of risk. The precautionary principle enables decision-makers to adopt measures when scientific evidence about an environmental or human health hazard is uncertain and the stakes are high. A question we address to some extent is whether this pandemic happened totally unexpectedly; was it a so-called Black Swan? We present evidence that it most certainly was not. We give examples of early warnings from scientific publications, highly visible presentations in the public domain as well as regulatory measures in force to absorb the consequences of a possible pandemic. In discussions around risk, numbers, especially large ones, and also units of measurement play an important role; we offer some guidance here.

Time for a break! Chapter 7 takes you for a guided walk through a tiny part of mathematical wonderland. We will encounter several mathematical personalities. An important one is Andrew Wiles, who solved Fermat’s Last Theorem. The story about how he finally obtained a proof is a must-read. We learn about the Fields Medal, the equivalent of a (non-existing) Nobel Prize in mathematics. We also tell you about the four-yearly International Congresses of Mathematicians and their influence on the field. There will be a first step on the ladder towards a theory of randomness; key names here are Jacob Bernoulli and Andrei Nikolajewitsch Kolmogorov. Randomness also comes to us through the famous discussion between Niels Bohr and Albert Einstein on “God throwing dice”. Of course, we include Leonhard Euler and his most beautiful formula of mathematics.

We single out the 2006 L’Aquila earthquake in Italy as it yields a dramatic perspective on the problem of evidence-based communication. In the aftermath of this earthquake, several scientists were sentenced to jail for insufficiently clear communication related to an imminent earthquake. Though the sentences were later overturned, we can all learn from this example. It is interesting that this court case took place in the country that also tried Galileo Galilei in the seventeenth century for his defense of heliocentrism. A wonderful example in this context is provided by Galileo’s Dialogo published in 1632. In this publication, Galileo communicates his findings to a wider public through a series of dialogues between two philosophers and a layman. We present several parallels to present-day discussions on risk and science communication.

One of the most counterintuitive examples involving randomness is the birthday problem. From 23 persons onwards, the probability of finding at least two people in a group with the same birthday is above 50%. Leonhard Euler’s solution of the Koenigsberg bridge problem heralded the start of the fascinating field of graphs and networks with applications to numerous applied problems across many disciplines. In 1929 the Hungarian writer Frigyes Karinthy highlighted the world’s smallness through his wonderful story “Chains” where he introduced the by now well-known “separation by six” idiom. Starting from these examples, we discuss some risks due to network effects present on the World Wide Web and social media. We present the reader with a glimpse of the fascinating world of coincidences. For instance, the law of truly large numbers states that, with a large enough sample, any outrageous thing is likely to happen. Real-life examples highlight the meaning of this law.