How statistics defeated Hitler and won the war

The Guardian is running a short article on how statisticians helped win World War Two for the Allies.

Here is a story about mathematical deduction that I love, mainly because it is said to be true, and because it had an impact (albeit small) on the outcome of the second world war. It is the story of how a simple statistical formula successfully estimated the number of tanks the enemy was producing, at a time when this could not be directly observed by the allied spy network.

I'd question the "albeit small" qualification -- had the Allies not trusted the statisticians' estimates, they could easily have delayed the invasion of Europe. Who knows what effect that would have had? Soviet tanks on the French border?

What is impressive about the maths is that the statisticians were virtually exactly correct in their estimate: they estimated that Germany was producing 246 Panzer Mark IV and V tanks per month, compared to the estimates by intelligence forces of 1,400. After the war, captured German production records showed that the actual number of tanks produced was 245 per month.

But what is really impressive is that the Guardian hasn't dumbed the article down to the point of uselessness: they actually give the statisticians' formula, and use the correct terminology:

Given a sample size S and maximum serial number M, a good estimator of the number of tanks made would be (M-1)(S+1)/S.

Now, all we need is to see how they derived that formula, and my day will be complete.