The German tank problem is about the math that helped the Allies in WW2 to estimate the number of German tanks (panther) based on the ‘samples’, i.e., the ones captured. In a war, an accurate estimate of the maximum number of tanks on the enemy side helps estimate the size of the threat.
The Allies discovered that the components of the tanks had sequential serial numbers. Then they assumed that the probability of finding any tank from #1 to #N (the maximum) was equally distributed (uniform distribution) at 1/N. The serial numbers of the captured tanks then gave them the samples.
Imagine at some stage, the following five tanks were captured: 15, 47, 79, 28, 39. Organising them in increasing order, we get 15, 28, 39, 47, 79. We will consider them as random draws from the uniform distribution and calculate the gaps between them without taking the numbers themselves. They are 14, 12, 10, 7 and 31. The average of these gaps = 14.8. Add this correction factor to the maximum number 79 to get 94.
If m is the highest number, k is the number of tanks captured, and N is the (unknown) total number,
N = m + m/k – 1
Reference
German tank problem: Wiki
