What does the mess look like or did the fascists have homing missiles

Original author: Aatish Bhatia
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June 13, 1944, a week after the Allied invasion of Normandy, a loud buzzing sound thundered in the sky of a battered London. The sound source was the newly developed German weapon of war: the V-1 air bomb. Being the forerunner of cruise missiles, the V-1 was a self-propelled bomb controlled by gyroscopes, powered by a simple pulsating jet engine that absorbed air and ignited fuel 50 times per second. Such a high pulsation frequency gave the bomb a characteristic sound , earning her the nickname "buzzing bomb" (in the original - "buzz bomb" - approx. Transl.).

From June to October 1944, the Germans launched 9,521 buzzing bombs from the shores of France and the Netherlands, 2,419 of which reached their goals in London. The British were very worried about the real accuracy of these aerial drones. Did they fall on the city randomly, or did they hit the intended targets? Did the Germans really develop accurate homing bombs?

Fortunately, they were scrupulous in maintaining statistics on the place and time of the fall of almost every bomb that was dropped on London during World War II. Using this data, they could statistically find out if bombs accidentally fell on a city or if they were accurately aimed. It was a simple math question with very important consequences.

Imagine for a moment that you are working for British intelligence, and you are tasked with solving this problem. Someone hands you a piece of paper with a cloud of dots on it, and your task is to find out which of the models is random.

Let's do it with an example. Here are two models from Stephen Pinker’s book “The Best in Us” (“ The better angels of our nature ” in the original - approx. Transl.). One of the models is randomly generated, the other imitates a drawing from nature. Can you say what is what?

Reflected?

Here is Pinker's explanation:
The one on the left (with clots, threads, voids and fibers) is an array that was built by chance - these are stars. The right pattern, in which the system seems to be absent - this is a system whose positions were formed by shocks from each other - these are fireflies.

That's right - fireflies. The dots in the right figure indicate the position of the fireflies on the ceiling of Waitomo Cave in New Zealand. These fireflies are not randomly located; they compete for food and repel each other. They are very interested not to stick together.

Try to evenly sprinkle sand on the surface, and it should look like the right drawing. You instinctively avoid places where you have already poured sand. Random processes do not have such prejudices, grains of sand just fall where they should fall and thicken together. It is more like scattering of sand with your eyes closed. The key difference here is that randomness is not the same as uniformity. A real accident can come with clusters of constellations that are painted in the night sky.

Here is another example.
Imagine that the professor asked his students to flip a coin 100 times. One student diligently completed the task and recorded the results. Another student was a little slacker, and he decided to fake the throwing results, instead of conducting an experiment. Can you identify which student is a loafer?

Student 1:
ROOORORRRRORRORRROORORROR
OOOROROORORROORRRRORRRORO
RROORRRRRRRROROOOOORORORO
ROROROOOOOROORRRRRORROORO

Student 2:
ORRORROROORRORORORROORORR
ORROOORRORROROROROORRORRO
ROROROROOORROROROROORORRR
OROOROROROROORRORORORROOR

Pause, think.

The data of the first student are long clusters, up to eight elements in a row. Surprisingly, this is actually what comes from random coin tosses (I know - I made a hundred coin tosses to get such data!). Suspiciously absent clusters in the data of the second student. In fact, for a hundred coin tosses, he did not receive a row of four or more eagles or tails in a row. The chance that this will ever happen is about 0.1%.

Trying to find out if a set of numbers is random is like mysterious math games, but it's not far from the truth. The study of random fluctuations has its roots in nineteenth-century French criminal statistics. France was on the path of rapid urbanization, population density in cities began to increase, crime and poverty became acute social problems.

In 1825, France began to collect statistical data on criminal cases, which was perhaps the first time that statistical analysis was used to study social problems. Adolf Ketle was a Belgian mathematician and one of the pioneers in the field of social sciences. His controversial goal was to apply the ideas of probabilities used in astronomy to understand the laws that govern people.

Co wordsMichael Maltz :
When searching for patterns in criminal statistics that was found in astronomical observations, he insisted that just as there is a real star location, there is also a real crime rate. Ketle argued that the “average person” had a statistically constant “penchant for crime,” which would create “social physics”.

Quetelet noted that the number of convictions was slowly falling over time, and concluded that there was a tendency toward a decrease in “penalties for crime” among French citizens. There were some problems with the data that he used, but the main errors in his method were discovered by the brilliant French scholar and scientist Simeon Denis Poisson .

Poisson's idea was brilliant and surprisingly modern. Speaking in modern language, he argued that Kettle lacked a model for his data, he did not explain how the jury actually came to their decisions. According to Poisson, the jury was simply wrong. The data we observe is about changing beliefs, but what we want to know- this is the probability that the defendant is guilty. These two quantities are not the same, but they can be related. As a result, when we take the whole process into account, a certain number of variables appear inherent in the convictions, and the result of this is what we see in the criminal statistics of France.

In 1837, Poisson published this result in his work “ Investigations of the theory of probability of court decisions in criminal and civil cases ”. In this work, he introduced the formula, which we now call the Poisson distribution. She explains to us how the chances of a large number of rare events turn into a concrete result (how most French jurors make the wrong decision). Suppose that an average of 45 people a year are struck by lightning. Substitute this in the Poisson formula along with the population, and it will give out what is the probability that 10 people a year will be struck by lightning, or 50, or 100. It is assumed that lightning strikes are independent, rare events that can also occur in any time. In other words, the Poisson formula can show you how likely it is to receive a rare event at random .

One of the first applications of the Poisson formula came from an unlikely place. We jump sixty years ahead, through the Franco-Prussian war, and find ourselves in Prussia in 1898. Vladislav Bortkevich , a Russian statistician of Polish descent, tried to understand why in some years an unusually large number of soldiers in the Prussian army were killed under the hooves of horses. Sometimes in one unit there were 4 such deaths in one year. Was it just a coincidence?

A constant frequency of death from a horse is unlikely. Bortkevich realized that he could use the Poisson formula to find out how many deaths we expect to see. Here is a prediction, in comparison with real data.
Horse DeathsPredicted Poisson CasesObserved cases
0108.67109
166.2965
220.2222
34.113
40.631
50.080
60.010

See how well they fit? Clusters of horse-related deaths are what one would expect if we considered horse strikes as hooves a purely random process. Accident comes with clusters.

I decided to check it out on my own. I searched for publicly available data on deaths due to rare events, and stumbled upon the “ International Shark Attack File, ” which counts cases of shark attacks on people around the world. Here is evidence of shark attacks in South Africa.
YearNumber of Shark Attacks in South Africa
20004
20013
20023
20032
20045
20054
20064
20072
20080
20096
20107
20115

These numbers are quite small, an average of 3.75. But let's compare 2008 and 2009. In one year there were 0 shark attacks, and the next as many as 6. And in 2010 there were 7. You can already imagine how the headlines shout: “ Sharks are attacking! ". But is this really a rebellion of sharks? To find out, I compared Mr. Poisson's forecast data.

The horizontal number of attacks indicates the vertical number of years. For example, the longest blue bar indicates that there were 4 attacks for 3 years (2000, 2005 and 2006). The red dotted line is the Poisson distribution, and it represents the results that would be expected if the shark attack were accepted as a purely random process. This fits well with the data, and I'm afraid it rules out the big South African shark uprising of 2010. The lesson, again, is that chance does not mean uniformity .

And that brings us back to the buzzing bombs. Here is a visualization of the number of bombs dropped over various parts of the city:

This is far from uniform distribution, but is this evidence of accurate targeting? At the moment, you can guess how to answer this question. In a report titled “Application of the Poisson Distribution,” the British statistician named after R.D. Clark wrote:
During the aerial bombardment of London, it was widely believed that bomb targeting points tend to form clusters. Therefore, it was decided to apply a statistical test to find out how correct this statement is.

Clark took an area of ​​12 km x 12 km, which was heavily bombed, and divided it with a grid. As a result, he got 576 squares, each the size of 25 city blocks. Then he counted the number of squares from 0 bombs dropped, with the 1st bomb dropped, with 2 and so on.

A total of 537 bombs fell on these 576 squares. This is about one bomb per square on average. He substituted these numbers in the Poisson formula to find out how many clusters are expected to be obtained randomly. Here is the corresponding table from his article:

Compare the two columns, and you can see how incredibly accurately the prediction corresponds to reality. There are 7 squares that were hit by 4 bombs each - this is what you would expect due to chance.Bombs didn’t hit most of London. They collapsed at random, in a destructive city-wide game of Russian roulette.

Poisson's distribution has the habit of sneaking up in all sorts of places, in some cases inconsequential, and in others changing your life. The number of mutations in DNA and the age of your cells; the number of cars in front of you at the traffic lights, or patients in line in front of you to the emergency department; the number of typos in each of my blog posts; the number of patients with leukemia in a given city; the number of births and deaths, marriages and divorces, or suicides and killings in a given year; number of fleas on your dog.

These Victorian scientists have taught us that from worldly moments to life and death, chance plays a stronger role in our lives than we are willing to admit. Unfortunately, this fact does not give comfort when the cards in the waterfall of life are not distributed in our favor.

“So much life, it seems to me, is determined by pure chance.” - Sidney Poitiers