Conned Again, Watson! Cautionary Tales of Logic, Math, and Probability

Bruce's approach is to teach concepts in statistics and probability through mystery stories written around the characters of Sherlock Holmes and Dr. Watson. At first I was a bit skeptical, wondering if something this non-traditional might be just a gimmick. I was pleasantly surprised to discover that the book not only has real intellectual merit, but that Bruce is a pretty good mystery writer to boot.

Holmes solves most of the mysteries in this book by using analysis grounded in the mathematics of statistics. Some of the solutions to these mysteries are non-intuitive, and may trip up even those who consider themselves to be experts. Gambling fallacies are a common theme, including the mistaken idea that the "law of averages" somehow decrees that, after a string of one type of random event, another type of independent random event becomes more probable. This error is rooted in the mistaken notion that if the ratio of two numbers approaches 1, then the difference between the two numbers approaches 0. For example, if you toss a fair coin N times, the ratio of the total number of heads, divided by the total number of tails, approaches 1 as N becomes very large. However, the difference between the number of heads and the number of tails can (and usually does) diverge. There is nothing in the laws of statistics that says that, after a string of 10 heads, the next throw of the coin is more likely to come up tails (if the coin is fair). Yet this common fallacy persists among many gamblers. This is closely related to the mathematics of the drunkard's walk, which is the centerpiece of another mystery unraveled by Holmes as he investigates the case of an unfortunate sailor and the insurance money pursued by his distraught sister.

In another caper, Holmes uses his knowledge of the well-known birthday paradox (given N people in a room, what is the probability that two or more of them will share a common birthday) to expose a fake genealogy at the heart of a dispute over a wealthy inheritance. The real lesson of this mystery, however, is that the human mind is a poor random-number generator that inevitably fails to appreciate the nuances of truly random events. In this story, Holmes uses the tell-tail signs of a concocted distribution of birthrates to deduce that a particular document is a forgery.

Who hasn't been exposed to supposed messages of seemingly profound importance, found encoded in the Bible? In the case of the foolish graduate student, Holmes exposes the mathematics of hidden messages and prophecies coded in religious texts (or any other type, for that matter). The main point is that, in almost any large body of text, the number of possibilities is so large as to make such coded messages a virtual certainty - if you look long and hard enough (you can even find them in things like software manuals).

There is hardly a more common human tendency than placing complicated entities in a linear hierarchy. Witness, for example, the Sunday college-football rankings and the linear ranking of IQ scores. The tendency is possibly rooted in our basic understanding of such things as elementary mathematics, where we are taught that if A is greater than B, and B is greater than C, then A is greater than C. This is true for the set of real numbers, but is hardly true in general. Bruce points out that in higher mathematics, A may be greater than B, and B may be greater than C, yet C may be greater than A. It's really not a difficult concept. Every child knows it well. Paper wraps rock, rock breaks scissors, and scissors cuts paper. The problem comes in internalizing the concepts and understanding where linear hierarchies don't apply. Con men make use of this error with simple games in which the mark gets to pick one of three dice. Unsuspectingly, he fails to appreciate that, no matter which dice he picks, the con man can pick one of the remaining two, that will beat (on average have a higher score) whichever one the mark chooses. The villain is no match for Holmes, though, who sees through the scam with clarity and dispatches his trademark logic to save a friend from his folly.

Many of the mysteries solved by Holmes have implications for public policy. One such example is a case in which Holmes calculates the probability of a particular outcome of a drug test involving one of Dr. Watson's patients. The results have wide application in public policy regarding drug testing. The central theme is that it's possible for some tests to sound very reliable, and yet a large number of the positive tests are false, or a large number of the negative tests are true. The results depend, in part, on the relative number of samples in the population that are using the drug, or have the illness that the drug is supposed to cure.

This book is easy to read, has no equations, and only a few figures. It looks like, feels like, and reads like an honest-to-gosh mystery novel, but manages to illuminate many important aspects of logic and statistics at the same time. I enjoyed reading it, and I'll bet you will, too.


Holmes as a master educator in logic and deduction   May 25, 2004
Charles Ashbacher (Marion, Iowa United States(cashbacher@yahoo.com))
19 out of 20 found this review helpful

Some time ago, Lamarr Widmer, the editor of the problem column of "Journal of Recreational Mathematics" submitted a review of this book to me, in my capacity as book reviews editor of JRM. As soon as I read the first two paragraphs of the review, I knew that I had to read the book. Sherlock Holmes is without question the greatest character to appear in fiction, the style of the stories still inspire many spin-offs. In the science fiction television series, "Star Trek: The Next Generation", the Holmes style of problem solving is used in many episodes. This book presents several stories where Holmes solves problems with a mathematical theme. Each of them is a delight to read and I did a good deal of head scratching as I tried to anticipate the solution to the puzzle.
My favorite story in the collection is "The Case of the Martian Invasion", which, set at the turn of the twentieth century, covers the possibility of heavier-than-air flying machines, "Martian" images on the Moon, crop circles and secret messages being embedded in biblical verse. The proponent of a Martian invasion believes that heavier-than-air machines are possible, putting forward the fundamental principle of using complex machines. That is of course redundancy, where multiple engines are placed on the aircraft in such a way that it can fly with any subset above a certain size. The explanation of the "secret messages" is easy, nothing more than a simple exercise in the probability of the frequency of the appearance of letters and looking hard enough.
The other stories were nearly as interesting and cover many areas of life, the probability of various events being the most common scenario. Game theory and decision theory is also used to solve the cases brought before the greatest detective of all time. Although they are set in the time of Holmes, the events described in the puzzles can still be applied to life in the twenty-first century.
I found this to be one of the best demonstrations of logical deduction based on sound mathematical principles that I have ever seen. Although he is constantly praised for his skill in logical deduction, Holmes also possesses another talent, that of a master teacher.

Published in the recreational mathematics newsletter, reprinted with permission.


puzzles in probability explained by detective   April 28, 2001
Michael R. Chernick (Malvern, PA)
14 out of 15 found this review helpful

The author does a marvelous job of presenting Sherlock Holmes stories through the thought of Dr. Watson, never much in the style of Sir Arthur Conan Doyle. However instead of simple detective mysteries each story has a probabilistic theme. After reading first couple of chapters I thought this is great for me but I am a statistician. Could a novice understand the complex explanations and story that enhances ones memory about the principles as the author suggests? I think so. The later chapters convince me. There they over the waiting time paradox, capture-recapture methods and other related problems in the chapter on the poor observer. The famous Monte Hall problem and the birthday problem are also covered and well explained through the eyes of Watson based on the work of Sherlock Holmes and his brother.


A Wonderful, enjoyable book!   December 30, 2002
C. Hardwick (Houston TX, USA)
12 out of 13 found this review helpful

Unlike some other reviewers, I am neither a statistitian nor a Sherlock Holms lover. I never cared much for murder mysteries perse, but as a tool for exploring such interesting concepts I thought it worked well. Yes he took a few liberties with history (as he pointed out in the end notes)--so what?

The stories were not designed to top those of doyle but to make some interesting probability and decision making concepts approachable, relevent, and enjoyable. This they did wonderfully. As someone who was turned off to math after years of dull, abstract school lecture, my interest arose from my work in business and computer science. Some of these concepts were not new to me, but all were from new angles. I found .the math easy to follow(depressingly difficult to predict!) and only wished I had not run out of pages. I plan not only to check out the author's other work, but some of the additional reading he kindly suggests in the notes. Thank you Mr. Bruce for and enjoyable read.