Math Adventures/German Tank Problem

The German tank problem is a set of statistics problems named for an important historical application.

During World War II, production of German tanks such as the Panther was accurately estimated by Allied intelligence using statistical methods.

During the course of the Second World War, the Western Allies made sustained efforts to determine the extent of German production and approached this in two major ways: conventional intelligence gathering and statistical estimation. In many cases, statistical analysis substantially improved on conventional intelligence. In some cases, conventional intelligence was used in conjunction with statistical methods, as was the case in estimation of Panther tank production just prior to D-Day.

How can a sample of serial numbers, obtained from equipment (tanks in the historical case) observed in the field, be used to estimate the number of units being manufactured?

Describe your proposed solution before reading about various expert solutions.