MATHEMATICS IN THE WAR

Both world Wars I and II are the deadliest and most costly skirmishes to ever happen on the globe. During these periods, we understand it was a gruesome period to the world as it led to the loss of life, destroying of monumental towns all around the world as well as massive bombing.

However, during these same times filled with the popping of guns all over the globe, fleets of ships with tens of thousands of army men, and the whirring of aircraft’s wings aggressively pursuing towards the enemy territory, some men and women gathered and would conclusively come up with solutions that would help their countries have an edge over the enemy in the war.

These mathematicians and scholars did this to intercept enemy communication, understand the nature of the attack by the enemy, keep machinery safe, or at least be able to have machinery come back to base after an attack. Despite losing talented researchers and students, in the long run, this became the catalyst for the regeneration of new mathematicians, leading to the inception of new ideas and theories and for the creation of new and outstanding scientific schools.

We will look at two theorems that came up in the allies camps during the second world war.

One of the algorithms is

1. German Tank Theory

Statistics and mathematics were some of the weapons in the War that when utilized, assisted to have a big edge over the enemy. German Tank theorem is one of these algorithms. Coming up with the story we are about to read would have been one of the most creative abstracts of the war. But bearing that the German Tank Theory story is true, makes me think, whoever the scholars thought about it was even more brilliant.

By 1941, German troops had become a superior side in the world war and a huge contributor to that was the German tanks especially when they introduced the new Mark IV and V on their tanks.

Also, another problem for the allies was to know the speed of production of these German tanks. To get these numbers in an accurate format was almost impossible.
Asking intelligence to guess or captured troops to spill the truth were unsuccessful methods proving hard to decipher the quantity of production of German tanks. All the answers were contradictory hence statisticians needed to improve the accuracy.
It was discovered that all the captured tanks had Serial numbers on them. This led to a hypothesis that the German tanks were logically numbered in the same order they were produced which later was verified.

This meant, they could make conclusions of how many German tanks had been produced in the war. From a neutral perspective, this is somewhat genius. However, it is a deduction of statistics. Let’s take a look into it statistically.

A Point Estimate or an Estimator is a sample statistic that is used to estimate a population parameter.

In our case, we are trying to estimate the maximum number of serial tanks(population) based on sample tanks captured or seen in war.

For a point estimate to be considered a good one it has to be an unbiased estimator. An unbiased estimator is one that ofter multiple simulations, the statistic does not consistently overestimate or underestimate the true population maximum.

Some of the estimators that can be both biased and unbiased include the following.

· Mean + 2 Standard Deviations

· Maximum + 1 Standard Deviation

· Median *2

· Max + min

· Median + IQR

· Mean *2

· Maximum +(Maximum/ Sample size) — 1

The Allies Statisticians used the Estimator (Maximum)+ (Maximum/Sample size) -1 to make assumptions about the population size.

An example would be as follows.

Q. You capture 5 enemy tanks with serial numbers 2, 31, 43, 67,99 and 152. What is the likely population maximum?
Solution:
Step 1: Insert the given values into the formula. For this example, the sample size is 6 and the highest value in the sample is 152, so:

= 152 + (152 / 6) — 1


Step 2: Solve:

= 152 + (152 / 6) — 1
= 152 + 25–1
= 177–1
= 176
The population maximum is 176.

For us at this moment, it looks quite direct, however getting such a value after captured soldiers had said the number of German tanks is 3000 looks quite astonishing. However, the estimated value was the real number of German tanks made. I hope you now see why I thought this was quite interesting.

2. Survivorship Bias

In the thick of the second world war at around 1943, the Allies were cognizant that most of their aircraft and the pilots in them were being lost and did not make it back to the base after an air attack. Every time the resuming planes landed from the war in the air, the engineers needed to inspect and fortify the aircraft body.

Reinforcing of these planes had to be very tactical for the battle. They needed to be light and streamlined in the air for easier movement and have an impenetrable body to shield from any form of firepower. It’s a trade-off where you need the right balance of the two to ensure the plane fights the enemy with minimal air resistance and gets back with the pilot's safe. The principal concern was to know where to add armor and the quantity to add.

In this case, the allies involved data collection where each plane that made it back to the base was methodically inspected to know which parts of the plane were more hit by the enemy’s firearms. They collected data on the aircraft returning from the battlefield and charted the bullet holes. As it occurred, most of the hits were concentrated on the wings and the tail.

The chart talks about the pattern of bullet holes and their distribution across the aircraft. It means that those areas were frequently hit. The engineers decided to reinforce the most hit sections which were the wings and the tail.

However, Abraham Wald, a Hungarian/Jewish mathematician the writer of the famous ‘The Annals of Mathematical Statistics’ had a different opinion. His thinking insightfully pointed out the decision-making data was only collected on the planes that made it back and not the others that did not make it. Wald’s thinking conclusively meant that making inferences based only on aircraft that made it back to the allies’ camps was a bias.

His insights also perceived that planes that were heavily shot on the wings and the tail were able to make it back home. This in fact meant that the planes that did not make it home had been shot at other places rather than the wings and tail. Further investigations showed that the engine, a vital part of a plane was the least shot for the planes that made it back.
This meant, that the engine should be the part that is reinforced most so that all planes that are shot at this section can have a higher chance of making it back home.

This led to the birth of the cognitive bias the so-called survivorship bias. This means in all our conclusions we should not only look at the available data but all the data to make a conclusion without a bias. For instance, most people say that buildings from the past are way stronger than buildings in recent days.

However, this judgment is only being made from the buildings that were built then and still stand, and we do not cater to those that may have crashed down or been destroyed.

Hence in all our data analytics, let's make sure we account for all sections of the population when making a sample to use it to make an accurate conclusion of the population.

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