The German tank problem

[Alastair Thomas]

I thought this may interest forum members. It is heavily based upon the Wikipedia article of the same title. I have read somewhere that a book that includes an equation immediately loses the sale of 10,000 copies. However, never fear, although I have included an equation here, it is simple and can, nevertheless, be skipped over.
Logistics is an important arm of conflict. Napoleon’s army may have marched on its stomach but, nevertheless, resupply from France would be necessary for all manner of things. What resources the enemy has and will have is important information to a commander so considerable effort was expended in WWII to find out. One example of the techniques used was a statistical analysis of the serial numbers of parts of German tanks in order to estimate the rate of production. Of course intelligence was used as well and the two had to be compared to come up with the best guess.
Two mathematical methods were used: Bayesian and Frequentist. The former can be dispensed with quickly as it is not amenable to a simple description. It was invented by a Reverend Bayes in the 18th century. It is based upon estimating the chances of all sorts of things happening in order to arrive at the chance of a particular outcome. A famous example of its use was the finding of Air France 447, which crashed in the Atlantic.
The Frequentist approach is much easier to understand and both methods assume that the enemy has manufactured a series of tanks marked with consecutive whole numbers, beginning with serial number 1.
As captured and damaged tanks became available to the allies, the serial numbers on major items such as the chassis, engine, gearbox and wheels were noted.
What we want to know is “N” the number of tanks manufactured. What we do know is “k” the number of tanks/wheels we have examined and “m” the highest serial number recorded.
The Frequentist solution can be found from:
N (the number of tanks)=m(the highest number recorded)+m/(k(the number examined))-1
As an example, if four tanks were examined (k = 4) and they had serial numbers 19, 40, 42 and 60 (m = 60) then the above equation gives the answer N=74.
Statistical methods do not give exact numbers. They give the likeliest answer and then a spread of values around the answer (this is why statistics gets such a bad name). The fewer the number of samples (k) used, the wider the spread of numbers. It is for this reason that the wheels of the tank are interesting, there being 32 on each Panzer V (a tank of particular interest to the analysts prior to D-Day).
In parallel with the statistical methods, other methods were used. In the case of these wheels, investigation identified how many moulds were being used to cast them. Discussions with UK manufacturers indicated the likely maximum number of wheels that could be cast per month.
These methods arrived at an estimate of 270 tanks produced in February 1944. After the war German records were examined and the true answer was 276.
A comparison was made with the conventional intelligence approach and some results are shown below.

Month | Statistical estimate | Intelligence estimate | German records
June 1940 | 169 | 1,000 | 122
June 1941 | 244 | 1,550 | 271
August 1942 | 327 | 1,550 | 342

This analysis is used by most nations so attempts are made to encrypt the serial numbers. As you would expect great efforts are employed to crack the encryption.
Because all good posts include a picture here is a photo of my Father in a captured Fiat Ansaldo L6-40.
As the narrator of the famous record “Trains” signed off, “Ah well, back to the asylum”.
Alastair
F60S