Lessons from the German Tank Problem

TL;DR

This paper discusses the historical German Tank Problem, deriving statistical estimates for tank production, and introduces a new generalization for cases where serial numbers don't start at 1, illustrating the practical application of statistical methods.

Contribution

It reproduces known estimates for the German Tank Problem and introduces a novel generalization for non-starting serial numbers, with educational insights on regression and functional relationships.

Findings

Derived the classic estimate for total tanks using observed serial numbers.

Compared the statistical estimate's effectiveness to intelligence gathered by spies.

Presented a new generalization for cases with arbitrary starting serial numbers.

Abstract

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. This note describes the statistical approach. 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 $m$, then the best estimate for $N$ is $m(1 + 1/k) - 1$. 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. The first part of the paper reproduces known results, specifically deriving this estimate and comparing its effectiveness to that of the spies. The second part presents a result we have not found in print elsewhere, the generalization to the case where the smallest value is not necessarily 1. We emphasize in detail why we are able to obtain such clean, closed-form expressions for the estimates, and conclude with an appendix highlighting how to use this problem to teach regression and how statistics can help us find functional relationships.