The scope of probability
Probability is the formalization of the study of the notion of uncertainty. The effects of blind chance are apparent everywhere. Biologically, we are all a random mixture of the genes of our parents. Catastrophes, like oil spills, volcano eruptions, tsunamis, or earthquakes, and happier events such as winning lottery prizes, randomly and dramatically change peoples’ lives.
Many people have a good intuitive understanding of probability. But this understanding can go astray when you have an initial idea about the likelihood of something, but then some new fact, whose relevance is not wholly apparent, is revealed. There are indeed a few notorious ‘trick questions’, about birthdays, or families with two children, or television game shows with three choices, that seem to have been designed to persuade you that the subject defies common sense. It does not. So long as any hidden assumptions in these questions are flushed out, and taken account of, sensible answers emerge. But probability does require clear thought processes.
The development of its ideas and methods has been driven by its wide applicability. The D-Day invasion of Normandy went ahead in June 1944 only because the probability of favourable weather was deemed sufficiently high. Engineers in the Netherlands must take account of the chances of severe floods when they build the dykes that protect their country from the sea. Is a new medical treatment more likely than present methods to enable a patient to survive for five years? How much you pay to insure your life, car, house, or possessions depends on the chances of an early claim being made. Most decisions you make – what to study at school, who to select as a life partner, where to live, which career to follow – are made under conditions of uncertainty. As Pierre-Simon Laplace wrote in 1814:
…the most important questions in life are, for the most part, only problems in probability.
Whenever the phrase ‘the probability is…’ appears, some assumptions (that may inadvertently have been omitted) are being made. If those assumptions are unwarranted, little reliance should be placed on the claim. I hope that, in this book, these assumptions are clear, either implicitly or explicitly. Before we look at how probability statements can be interpreted, we will describe different ways in which they may arise.