German tank problem

Sometimes science is applied in really amazing ways, one of examples is German tank problem. During World War II, the Allies observed that the Nazis created a new model of tank. Total number of manufactured units was unknown, but the Allies, know serial numbers of some tanks. In this case, the first produced tank has serial number starts = 1, the second has serial number = 2, etc. How the Allies could estimate total amount of manufactured tanks?

Answer may be estimated by using simple equitation. Let's assume that N = estimated total amount of enemy tanks, k = amount of observed tanks, m = maximum observed serial number. Then:

N = m - 1 + m/k


We observed enemy tanks with serial numbers: 20, 23, 45, 47, 48. In this case, m=48, k=4, answer can be computed on calculator or for example in R language (I really like to use it as a powerful "calculator"):

> 48 - 1 + 48/4
[1] 59

Other use cases

The same as used for German tank problem was used to estimate amount of sold Apple's products. The customers were asked to upload their serial numbers.