This is a collection of intriguing probability problems, similar to the author’s Digital Dice and Dueling Idiots and Other Probability Puzzlers. Each problem is introduced in a conversational, informal way, often with historical or personal anecdotes, and most of the problems are accompanied by a detailed solution. For some of the problems the author also provides computer code that simulates a solution. To be able to follow the discussion and solve the problems, readers should already know some basic combinatorics and probability, along with calculus. No mathematics more advanced than this is required.
The heart of the book: Twenty-five “curious questions” in probability, one of which lends its name to the book’s title. But the 25 questions constitute only 70% of the book. There is also a very good preface, a section of classic probability problems, and a section of challenge problems.
The preface provides an excellent feel for the book’s contents. Among other things, it describes the criteria the author used in selecting the 25 questions. To be included, a problem must possess a counter-intuitive, surprising conclusion. It must also be at least somewhat non-standard fare. So, for example, while the birthday problem receives an appreciative nod in the preface by virtue of its extremely surprising answer, it will not be treated in the main text due to its being discussed in almost every undergraduate probability course.
After laying down these criteria for inclusion, the preface then presents a few probability problems to whet the appetite and give a feel for what is to come. As an example: Assume that 100 people are to board a plane. Each passenger has a boarding pass with an assigned seat. The first passenger, a free spirit, boards the plane and takes a seat at random. Each of the remaining passengers, boarding one after the other, takes their assigned seat if it is unoccupied. Otherwise, the passenger takes an unoccupied seat at random. What is the probability that the last person to board gets their assigned seat? The answer, amazingly, is 1/2. And while this problem appears difficult to solve, the solution (which Nahin provides) is as amazingly simple as the answer.
As another example of the type of problem to expect, consider the controversial proposal to remove undocumented residents by allowing police to stop anyone at any time and demand identification. All other considerations aside (and there are many), the author asks us to focus only on measuring how inconvenient such a law would be to legal residents: On average, how many legal residents would be stopped in order to remove 50%, or 90%, or 99% of the undocumented residents? A probability analysis will give us “some numbers to discuss, and not just emotionally charged words for advocates on each side to throw at each other like rocks.” At this point, most readers will be hooked. So of course the author promises to answer the question later in the book. (It will appear as curious question #12.)
After the preface comes a selection of “classic puzzles from the past.” Here we find problems that Pascal, Galileo, Newton, and others wrestled with, such as the gambler’s ruin problem. As with the 25 curious questions, the author provides a detailed discussion of each problem along with a solution. For a few of the problems, he also provides computer code that simulates a solution.
Next, immediately preceding the 25 curious questions, is a list of twelve “challenge problems.” Readers are not expected to be able to solve all of these right away. Rather, they are encouraged to keep them in the back of their minds as they work through the 25 curious questions, keeping their eyes open for ideas or techniques they might be able to use. Solutions, along with computer code, are found at the end of the book.
Almost everyone will find something to like among the 25 curious questions. Most of the questions could be labeled as recreational mathematics, but several deal with serious topics such as medicine, voting, and public policy. One question deals with a clever method used during World War II to cut down on the number of blood tests that needed to be performed in order to detect syphilis among the troops. Another question addresses the ballot theorem, which tells us that a candidate who is thoroughly trounced in an election nevertheless has a surprisingly high probability of being at least tied at some (nontrivial) point in the ballot counting.
While the author injects his humor into the discussion throughout the book, his joke telling is at its most effusive when he is describing a fantastic movie script in Question #10, “Hollywood Thrills.” As is often the case, the humor paves the way for more serious considerations. Probability enters the script when our hero has to choose between two medicines that have been tested for their efficacy against a terrible disease that is causing an epidemic (and widespread panic, Hollywood style). In a clinical trial of infected individuals, all of the participants who received the first medicine survived, while three of the participants who received the second medicine died. Which medicine should our hero select? The surprising answer: the second one. How can this be? (Spoiler: More people were given the second medicine than were given the first.)
There is great variability in the skill level needed to solve the curious questions. For example, Question #13, “A Puzzle for When the Super Bowl is a Blowout,” is a simple question about a round-robin tournament. It can be answered via a relatively painless counting argument. At the other end of the spectrum is the very next question, “Darts and Ballistic Missiles,” which deals with the probability of two normally distributed variables being within a certain target. The solution requires facility with various aspects of random variables and multivariable calculus.
The simulation algorithms provided throughout the book are presented as Matlab code. In order to keep it simple and easily transferable to other computer languages, the Matlab code is not optimized. (The code is for the most part a bunch of control loops; seasoned Matlab users will find much to vectorize.) The author walks the reader through each piece of code, explaining how it simulates a solution to the problem under discussion. As a helpful guide, here and there throughout the book the author makes a few comments about the appropriate use of simulation, including pointers as to which kinds of problems are best suited to it.
This is a fun book, recommended for recreational math enthusiasts and undergraduates studying probability. Indeed, this collection of problems would serve as an excellent supplement for an introductory probability course. It could also serve as a good source for student presentations.
David A. Huckaby is an associate professor of mathematics at Angelo State University.