Prize insights in probability, and one goat of a recycled error: Jason Rosenhouse's The Monty Hall Problem

Jason Rosenhouse’s ‘great big Monty Hall book’ [12] is published by Oxford; but it was on one of my occasional pilgrimages to Cambridge that I stumbled upon it. As one who has taken an unnatural interest in this subject over the years, I found it unsurprising that enough material existed out there to fill a book on the Monty Hall problem, but Rosenhouse is to be commended for having taken the trouble to write it. And he has done so with suitable breadth, from the raw maths of the problem and others like it, to its psychological and philosophical implications. The Monty Hall problem is the TV game scenario where you, the contestant, are presented with three doors, with a car hidden behind one and goats hidden behind the other two. After you select a door, the host (named Monty Hall after the actual host of Let’s Make a Deal, a 1960s American TV show) opens a second door to reveal a goat. You are then invited to stay with your original choice of door, or to switch to the remaining unopened door, and claim whatever you find behind it. Assuming your objective is to win the car, is your best strategy to stay or switch, or does it not matter? Rosenhouse provides the definitive analysis of this game, along with several intriguing variations, and discusses many of the wider issues that emerge. Some reviewers, such as David Spiegalhalter in the LMS Newsletter [13] and Donald Granberg in Science [5] have queried the mathematical level of the book. Indeed the maths content is in a seldom-acknowledged middle place: uncomfortably informal for those used to an academic standard of rigour, but still undeniably heavy for those expecting a popular treatment. Yet the style of presentation is not as off-beat as might be thought by either ‘experts’ or non-mathematicians: it will be quite familiar to engineers, for example. The “contention” in Rosenhouse’s title clearly doesn’t reside on one side or other of C.P. Snow’s Two Cultures—where one is either a robotic theorem-prover or a learned hater of equations—and nor can a book like this. Those who take enough interest in brainbenders like Monty Hall to read books about it tend to be generalists who’ve kept closely in touch with their high school or undergrad mathematics. Recreational maths tragics like myself, in other words. I’ve certainly had fun digesting The Monty Hall Problem and engaging critically with its contents. The following—a much longer article than I had intended, for which I apologise in advance—is an attempt to share some of my reflections on the book’s contents, as well as to detail just one or two criticisms.