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• Hi, I'm Adriene Hill, and Welcome back to Crash Course Statistics.

• Statistics and probability have been used in applications beyond the ones we usually

• think of like research science and business analytics.

• One of the most consequential applications of statistics is helping countries survive

• and win wars.

• Today, we'll talk about how people applied statistics to break codes, locate sunken submarines,

• and even predict the next big war.

• INTRO

• Our first story is pretty well known in the fields of computer science and statistics.

• In World War II, the Germans used what looked like a complicated typewriter to encode their

• messages.

• These machines, called Enigmas, allowed the Germans to type in messages and receive encoded

• ones back.

• You may have done some simple encoding in your childhood.

• Like, if you wanted to send a message to your friend that says “I like, like Alexbut

• you don't want Alex or anyone else for that matter to be able to read the message.

• You could create a key so that each letter is represented by another letter like this:

• If you wanted to write “I likeyou'd find those letters in the top row, and write

• down their counterparts.

• “I likebecomes “Y tymfwhich makes no sense...unless you have the key.

• Your entire message would go from this:

• “I like, like Alex.”

• To this:

• “Y tymf, tymf ptfw.”

• So you're safe to deliver your message.

• But sometimes decoding messages has much higher stakes than protecting crushes.

• Like when there's a war going on.

• So by necessity, the keys--or methods of encryption--are much more complex.

• During the Enigma Encryption process, a letter was sent through three rounds of encoding--similar

• to how we encoded our message about Alex.

• But the enigma had three Rotors, or wheels, doing the encoding.

• And the enigma machines would rotate the wheels systematically after EVERY LETTER.

• A letter that appeared in the original message twice could get encoded as two totally different

• letters.

• There were 26 settings on each wheel, one setting for every letter in the alphabet.

• So there were 17,576 possible starting settings (just for the wheels!) making it impossible

• to figure out a message by manually trying each start point.

• If you wanted to decode a message, you needed to know how those wheels were set.

• The Germans also had multiple wheel options AND plugboards, making things even more complicated.

• Alan Turing and his team developed a technique called Banburismus for deciphering messages

• from the German Navy - which exploited the fact that sometimes pairs of messages would

• have chunks of text within them that had been encoded with the same settings.

• They used a very time-consuming method to find these pairs.

• Every intercepted message got hole punched, in order, into paper that was lined with horizontal

• alphabets.

• Then, one message was placed on top of another message, so a person could see how often holes

• overlapped.

• Why?

• Well, two messages that were encoded with different Enigma settings would only have

• letters that matched by random chance.

• The German navy had a primary Enigma that they were using known asDolphin.”

• Two messages encoded by the same Dolphin settings had a 1/17 chance of having randomly matching

• letters.

• If two messages were encoded using DIFFERENT settings, there's a 1/26 change of having

• randomly matching letters.

• So, more matches than 1/26 would be increasing evidence that the messages were encoded using

• the same settings.

• The Enigma codebreakers used that knowledge to determine whether two intercepted messages

• were more likely to be encoded using the same or different settings.

• They were also able to use other knowledge in the decoding process.

• Like, the team already knew that 90% of Enigma messages contained the German wordein,”

• which can meanone,” “a,” oran.”

• Plus, there were phrases about the weather that were getting repeated often in messages.

• 'Cuz, boats.

• When Turing and his team determined it was 50 times more likely that the messages were

• encoded with the same settings than not, they considered it almost certain they'd found

• a match.

• They had a machine calledthe bombethat could automatically cycle through a bunch

• of those wheel settings in order to decode messages.

• But it took a LONG time to go through all of the possibilities, so being able to narrow

• them down was a necessary step.

• As Mike Lee and Benedict King put it in their article in The Conversation, “Turing's

• crucial Bayesian insight was that certain messages were much more likely than other

• messages.”

• All this knowledge helped the team figure out how the Enigma's wheels were set when

• it encoded a given message.

• Using Bayesian reasoning helped Turing's team crack the Enigma code, and limited the

• amount of settings they had to test by hand.

• Some historians think cracking the Enigma may have shortened the War by 2-3 years, saving

• millions of lives.

• In WWII, German U-boats were systematically taking down Allied ships, including unarmed

• merchant ships with supplies.

• While some ships escaped unharmed like the Empress of Scotland which carried Turing from

• New York back to Europe the Allied forces suffered many losses.

• Locating the U-boats was not an easy task, but the mathematician B.O.

• Koopman used Bayesian reasoning.

• Koopman would first ask experts where the U-boat was likely headed.

• With limited time and resources, prior information and beliefs about the U-boat were important.

• Koopman commented that: “Police will patrol localities of high incidence of crime.

• Public health officials will have ideas in advance of the likely sources of infection

• and will examine them first.”

• And he wanted to do the same with the German U-boats.

• Using signals from the ship, Koopman was able to target a 236 mile radius for planes to

• search.

• But that's still big.

• He would assign a 50-50 probability that the U-boat was inside the circle, then he would

• use all of the military information that he had access to in order to update those beliefs.

• That way he could make the best decisions with whatever information he currently had.

• You could plot out a grid that represented your apartment, and you could assign a probability

• that your keys are in each 1 foot by 1 foot square based on the likelihood of possible

• ways you misplaced them.

• So maybe your keys fell out of your bag, which would put them somewhere in this square.

• Or maybe your cat got into your bag and dumped its contents onto the floor.

• Then they'd be in this square.

• Or maybe you left them in your jacket pocket.

• Then they'd be here.

• Based on how likely you think these scenarios are...and the knowledge that your cat loves

• to push things off of tables... the best guess is that the cat knocked over your bag again...you

• can use Bayesian reasoning to create a probability map of where your keys are most likely to

• be.

• You could also include information about how likely you are to find your keys if you searched

• for them in that square.

• Keys that fell behind the refrigerator might be hard to find even if you did search there.

• It'd also be really hard to find your keys if they went down a drain outside your door.

• Combining all this information would leave you with a map of your apartment that tells

• you the best places to search.

• This same theory--called Bayesian Search Theory-- was also applied by John Craven to find a

• missing nuclear submarine in 1968.

• Craven collected experts' opinions on what happened to the USS Scorpion, and used Bayesian

• Search Theory to create a map of where the sub would likely be found.

• And it worked! Craven found the sub right next to where he expected it.

• Often in war it's also essential to know approximately how MANY of these vehicles exist.

• During WWII, Allied forces used traditional techniques such as spying and interrogating

• captured German soldiers and estimated that the Germans were producing about 1,400 tanks

• a MONTH.

• But that seemed high.

• Luckily, the Allies had already captured some tanks with serial numbers on them.

• So they used some clever math to estimate the actual total number of tanks.

• Assuming that the tanks' serial numbers went in order which was a reasonable assumption

• they could use the range of the serial numbers to estimate how many there were.

• For example, if we found 4 tanks with the serial numbers 7, 17, 47, and 65.

• We'd know there are at LEAST 65 tanks.

• But it's possible there are 67 tanks.

• Or 102 tanks.

• Or 500 tanks.

• We need a way to estimate what the most likely maximum is.

• There are many ways to do this, but one simple one is to use this formula, where m is the

• maximum serial number you observed ours is 65 and n is the number of observations you

• So our best guess at how many tanks there are based on the data we collected is 80.25

• we'll round that to 80.

• Because you can't have .25 tanks.

• When the Allies used similar techniques, they estimated that there were 256 tanks being

• A much more accurate estimate.

• The Germans were actually making about 255 tanks a month at the time.

• And note to self: when fighting a war, do not use sequential serial numbers - unless

• you're fighting raccoons - they can't read.

• Jumping forward in time, to today.

• Some researchers use statistical models to predict when the next big war will be.

• It has been a long time since the last major World War.

• Aaron Clauset of the University of Colorado has set out to examine other stretches of

• peace.

• And this isn't as simple as just counting the years between major wars and calculating

• the average time of peace.

• Clauset looked for trends and correlations that might predict the number of years between

• major conflicts..

• He found that across history, huge stretches of peace were not unusual.

• In fact it was downright common to see 100 to 140 years of peace following a large scale

• war.

• This long stretch of time without large-scale world war is more rule than exception.

• Statistics has many important applications.

• War being one particularly high stakes application.

• Mathematicians and Statisticians played a huge role in WWII, and they continue to be

• a part of defense departments and military planning to this day.

• Out of necessity, we often make huge strides in the fields of math and statistics during

• wars.

• They force us to solve problems we may not have needed to solve in times of peace.

• Things like the Bayesian Search Theory that Koopman worked is also used in times of peace

• like in helping us find missing planes.

• And the code breaking done by Turing and his team was not only important in introducing

• statisticians to Bayesian inference, but it provided foundations for future code breaking

• and encryption work that's being done today.

• Thanks for watching, I'll see you next time.

Hi, I'm Adriene Hill, and Welcome back to Crash Course Statistics.

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War: Crash Course Statistics #42

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林宜悉 posted on 2020/03/30
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