The German Tank Problem: Math/Stats At War!

During World War II the German army used tanks to devastating advantage. 
The Allies needed accurate estimates of their tank production and 
deployment. They used two approaches to find these values: spies, and 
statistics. In this talk we describe the statistical approach and its 
generalization. Assuming the tanks are labeled consecutively starting at 
1, if we observe $k$ serial numbers from an unknown number $N$ of tanks, 
with the maximum observed value $m$, what is the best estimate for $N$? 
This is now known as the German Tank Problem, and is a terrific example of 
the applicability of mathematics and statistics in the real world. We 
quickly review some needed combinatorial identities (which is why we are 
able to obtain clean, closed form expressions), give the proof for the 
standard problem, discuss the generalization, and show how if we were 
unable to do the algebra we could guess the formula by an application of 
linear regression, thus highlighting its power and applicability. Most of 
the talk only uses basic algebra and elementary knowledge of WWII.