Introduction to Probability: Second Revised Edition by Charles M. Grinstead
My rating: 5 of 5 stars

"One may summarize these results by stating that one should not get drunk in more than two dimensions."

The above is the most extraordinary sentence one can find in a university textbook on advanced mathematics! Charles M. Grinstead's and J. Laurie Snell's Introduction to Probability (Second Edition, 1997) is indeed a most remarkable textbook, by far the best text that I have ever used in my almost 40 years of teaching undergraduate mathematics and computer science. Probability was my favorite field of mathematics during my own studies in the early 1970s, but I somehow avoided teaching it, most likely because I had not found a textbook that I really liked. Until Grinstead and Snell.

Standard textbooks heavily focus on the combinatorial aspects of probability, which do not interest me too much. When I teach the upper-division probability course I love to emphasize the calculus-based approach, particularly when it involves multidimensional calculus and its applications to joint probability distributions. Grinstead and Snell's approach is virtually tailor-made for my probability course.

The second factor that makes me love the textbook is the emphasis on random numbers and pseudo-random variables generation. Having worked in the field of mathematical modeling and simulation for over 40 years I believe this is a natural approach to ground the probability course in. Grinstead and Snell's geometry-based problems that use the cumulative distribution functions to find the densities are a wonderful teaching tool: the students can also appreciate the applications of calculus: many of my students seemed to like discovering the connections.

Yet another great feature of the textbook is its emphasis on the moment generating function. Naturally, it is used to prove the Central Limit Theorem, the fundamental theorem of probability and the foundation of statistics. I follow the mathematical argument in class every time I teach the course so that math majors can appreciate a little more elaborate proof than the usual toy ones.

I also love the inclusion of a chapter of random walks (from which the epigraph is taken). When teaching partial differential equations (another of my favorite fields in math) I often discuss the Tour du Wino method of numerically solving the Laplace equation, which uses the random walks approach. The authors provide the famous proof by Pólya, which shows that a random walk must eventually return to the origin in one or two dimensions, but not necessarily for higher dimensions.

Of many other nice features of the textbook I should mention the authors' clear treatment of the Bayes' Theorem and the fascinating Historical Remarks that accompany many chapters. A truly wonderful book! The best textbook I have ever used!

Five stars.


View all my reviews