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Basic Probability Theory

Robert B. Ash
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Mark Bollman
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In reading through this 2008 Dover reissue of a 1970 book, I was again reminded that a lot of good textbooks predate the reform movement in both calculus and statistics. Robert Ash’s rigorous look at an important subject is well worthy of its re-release.

Probability occasionally suffers from a reputation as less-than-serious mathematics, but this book, with its liberal use of Borel sets and multidimensional integrals, stands as a solid counterargument to that claim. At the same time, this treatment is not so serious as to be completely removed from the more entertaining side of the subject. In giving the example that the number of raisins in a cookie may be approximated by a Poisson random variable (p. 99), we find the following:

Here the assumptions are not entirely clear. Perhaps what is envisioned is that the dough is bombarded by a raisin gun at some stage in the cookie-making process.

I rather like that idea. The image of a raisin gun will, I suspect, be in the back of my mind whenever I describe the Poisson random variable to a statistics class.

One typographical suggestion, intended for blackboard use, caught my attention immediately: the use of “R” to denote a random variable and thus avoid the difficulty of keeping capital and lower-case X’s distinct in an expression like P(X = x). Though the textbook I am currently teaching from uses Y’s rather than X’s in an effort to address this concern, the choice of R is much better. Notions like these are studded throughout the text — much like the raisins in that hypothetical cookie — and combine with the mathematics for excellent coverage of probability and a brief nod to statistics.

Mark Bollman ( is an associate professor of mathematics at Albion College in Michigan. His mathematical interests include number theory, probability, and geometry. His claim to be the only Project NExT fellow (Forest dot, 2002) who has taught both English composition and organic chemistry to college students has not, to his knowledge, been successfully contradicted. If it ever is, he is sure that his experience teaching introductory geology will break the deadlock.


1. Basic Concepts
2. Random Variables
3. Expectation
4. Conditional Probability and Expectation
5. Characteristic Functions
6. Infinite Sequences of Random Variables
7. Markov Chains
8. Introduction to Statistics
A Brief Bibliography
Solutions to Problems