One of the great joys in teaching probability, whether as a component of another course or as a course in itself, is access to the rich collection of “classic” problems that has accumulated over the years.
On the other hand, after you’ve taught a few sections of that course, the old standards tend to lose some of their freshness. Is the classic birthday problem (How many people must be gathered together so that there’s at least a 50% chance that two or more have the same birthday?) something that still amazes the uninitiated? Sure. Is it as exciting to teach it the tenth time as the first? Probably not so much.
With that in mind, Wolfgang Schwarz’s collection of 40 problems, hints for approaching them, and their solutions is a genuinely valuable resource for the experienced (and not-so-experienced) probability teacher. Among these is number 7, this variant on the birthday problem:
A worker’s legal code specifies as a holiday any day during which at least one worker in a certain factory has a birthday. All other days are working days. How many workers (n) must the factory employ so that the expected number of working man-days is maximized during the year?
Many of the same principles involved in the classic birthday problem are involved here (there are, for example, still many fractions with denominator 365), but there’s the added twist of an optimization question together with a fresh setting that has the potential to invigorate — or reinvigorate — a discussion of calendar probabilities.
There are 39 other such problems, all potentially useful to anyone teaching probability and mathematical statistics. The difficulty level ranges nicely from elementary to sophisticated, so most readers will find something challenging here.
Mark Bollman (email@example.com) 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.