The Art of War 孙子兵法 was a treatise written by Sun Tzu who was a contemporary of Confucius and Socrates: Gautama Buddha used to be considered coeval but modern scholarship now puts him a couple of generations younger. The art of war is mostly about putting one over on your enemy. It would be no fun for boys if you could push a button and the host across the valley were vaporised. Getting to a key position firstest with the mostest could win the battle even if it meant embracing risk by stripping soldiers from another part of your front. Generals, famously Patton, read about ancient battles to better understand the psychology of opposing generals: it didn't matter if their soldiers carried spears or carbines.
But all generals acknowledged that they needed quants to run the commissariat: no food at the front, no battle; archers need a lot of arrows; tanks need diesel. Not only must you be confident that you have sufficient materiel at your disposal, you also have to have some idea of how much the other chap has. The Irish word faisnéis, at the root of bithaisnéisíochta [bioinformatics] originally meant military intelligence - knowing whether there was an army the other side of that hill. And not only the existence of an army but how many regiments. In Sun Tzu's time and for the next 2500 years,
faisnéis was obtained by sending out a well-connected young chap, in a very smart uniform, on a horse to count camp-fires
In the run up to D-Day in WWII, one of the logistical problems that needed urgent solution was an estimate of the number of tanks the Germans had at their disposal. Not only the gross number but a strong indication of what types: Tigers I & II were not as worrying as the new Panzer V. It's a problem that could be given over to 'intelligence' but it turned out that good statistics got a far more accurate account. The base data was serial numbers from tanks, destroyed or captured, that were accessible to the Allies in other theatres of the war: notably North Africa and Italy. The German war-machine was obsessive in its accounting and numbered each gearbox and each chassis sequentially. The Allied quants had, therefore, long strings of numbers with a lot of gaps and were required to estimate the total size of the tank 'population' that had come off the assembly-lines in The Ruhr valley. They also needed to know how many would be available to the opposition, in 3, 6, 12 months time. So the key parameter was how many tanks could be produced in the Panzer factories each month. Clearly it was unlikely that the highest serial number available was the sum total of tanks up to the day that tank was destroyed. Nevertheless that was a minimum number.
Q. How much bigger was the true total?
A [simple]. N ~= m + m/k -1; where N is the total number estimated from m, the highest serial number and k the number of tanks in the sample.
A [more complicated Bayesian solution]:
Abraham Wald, not to mention Alan Turing, a bit of insight and a lot of maths solved the Tank Problem goodo.