As a number theorist, I have of course heard of Khinchin (consider, e.g., his famous “three pearls” and his book on continued fractions). But it is in fact the case that he was also a major player in the former Soviet Union in the area of probability. His coauthor for the book under review, Gnedenko, was a student of Kolmogorov and was therefore only one degree of separation away from one of the very founders of the modern theory of probability. Thus, it comes as no surprise that An Elementary Introduction to the Theory of Probability is exceedingly useful at the same time that it is both idiosyncratic and even anachronistic (or so one would hope).
The anachronism is largely contained in the nature of the examples chosen to introduce and illustrate certain themes in the book. For example, already on p. 3, the first example concerns the success rate of a marksman, while the second example, half a page down, concerns inspections at a factory. Then, on p. 9, two marksmen enter the arena, and on p. 11, at the start of the second chapter, we encounter a competition of … marksmen. All a bit different from the slew of examples I have encountered in introductory texts on probability that I have used over the last twenty years in courses aimed at a certain part of the mathematically hostile undergraduate population at my university, namely the psychology majors.
But this is not to say that I was seriously considering Gnedenko and Khinchin’s book as a candidate for primary text for my next outing in “Prob/Stat’s for Psych Majors,” of course, even if the book’s Foreword to the First Edition (1945) advertises it as meant for “leaders (and occasionally also rank and file workers) in the military, industry, agricultural economy, economy, etc., whose mathematical training is very limited.” Indeed, even though Gnedenko and Khinchin go on to say that their little book “is completely accessible to all those who have completed the 10year secondary school (ages 7–17),” I would not foist it on an American undergraduate populace. It is not that the material presented is inaccessible; quite the contrary, it is eminently accessible and very well crafted. My reasons for classing An Elementary Introduction to the Theory of Probability as unsuitable for us today is that it is entirely devoid of exercises, and, more importantly, its style is truly aimed at the Soviet proletariat, not the (post?)modern Western youths we try to teach some probability to as part of our service charter — youths headed for careers in, say clinical psychology or medicine, instead of service to the Soviet state. In other words, it just wouldn’t fly.
On the other hand, every one who does duty in the trenches of “Prob/Stat’s for Psych Majors” is likely to find this book hugely valuable as a pedagogical resource. For the authors, who are indeed past masters of the subject, certainly do a wonderful job introducing elementary probability themes via very well chosen (if idiosyncratic) examples, worked out in great detail. I, who never had any formal training in prob/stat, will be served greatly by these discussions not only in order to bolster my own limited understanding but to provide me with material for motivation and examples during my lectures — making certain changes, perhaps, although I must say that I find the whole business of destroying enemy planes with rifle shots fascinating (cf. p.24). (Was it G. H. Hardy or David Hilbert or Edmund Landau who said about World War I that his experience was essentially that of his counterpart on the other side “except for trivial changes of sign”? Is a MosinNagant to an ME 109 like a LeeEnfield to a Stuka? Oh, well… Godwilling these are now rhetorical questions.)
And there is an additional reason to get hold of this little (and cheap!) book: it’s got proofs! Bernoulli’s Theorem is dealt with starting on p.50, and the proof of the law of large numbers starts on p.97 (preceded by a critical discussion).
So, I’m taking this book to my office and will place it right next to the text I use for “Prob/Stat’s for Psych Majors,” which I’ll doubtless teach again before too long.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.
