Unexpected expectations. The curiosities of a mathematical crystal ball (Q2890500)

This book consisting of ten chapters explores a range of paradoxical and seemingly surprising areas in mathematics. Most of the paradoxes studied stem from the area of probability and combinatorics, although the last chapters also consider game theory.NEWLINENEWLINEThe first two chapters provide a very useful introduction to mathematical probability which includes the historic background and the contributions by Pascal and Fermat. The set theoretical approach to probability is then presented and some of the elementary properties of probabilities are studied. The third chapter considers some practical applications of probability theory, such as insurance premiums, airline overbooking and the gambler's ruin; this is followed by a short discussion on risk aversion and ambiguity aversion and how these affect the way people make decisions.NEWLINENEWLINEThe fifth chapter looks into the classic envelope choice and envelope switching problem (or goat problem as it is sometimes known), and a number of variations of this paradox are presented. The Parrondo paradox ``winning by losing'' follows next, inspired by the ideas of perpetual motion automata. The last chapters consider paradoxes from the area of game theory such as the prisoner's dilemma or the chicken problem, but also venture into more philosophical investigations such as Newcomb's paradox or the sleeping beauty awakenings problem. The last chapter considers the puzzles associated with Benford's law and contains a very well presented proof and illustration of its applicability.NEWLINENEWLINEThe book concludes with an index and a list of very useful references.