The Mathematics Used to Solve Crime

this video was sponsored by brilliant imagine someone is committing crimes throughout the LA area on a regular basis maybe they've committed five crimes in the last month at these locations how could we go about catching this person one thing we can do is try finding a pattern in crime locations in order to make a prediction about the next one that can be tricky though since there's likely a huge element of randomness but decades ago Kim Ross mo a PhD criminologist had another idea he tried to find a formula that instead could find where the criminal likely lives based on past data he knows that criminals often don't commit crimes right by their own home but also they don't go too far away so from the data you can determine a quote hot zone which isn't too close or too far from the crime scenes it has a high probability of the person living there this is his equation for determining those probabilities I know it looks complex but it's actually not as bad as you think like take this P I comma J part if we go to our map with the crime scenes and put a grid over it any given Square will be in some row we'll label I and column will label J P I comma J is the probability that the criminal lives in that square how you calculate that value for any square is with this right side of the equation now take this denominator this first part just means take the arbitrary grid you're analyzing and one of the crime scenes and subtract there X Corps Donets which gives you this distance the absolute value just ensures that it's positive then this part just says do the same thing with the Y's which gives us this length add those up and we get the distance between the grid and the crime scene no it's not the straight-line distance but it is the distance if you cannot move to agonal and as we saw this term is in the denominator which means as that distance goes up the entire fraction and thus the probability go down this is expected because like I said these criminals usually don't go super far to commit a crime so a larger distance between our Square and the crime scene means a smaller probability the criminal lives there at least for this laughter but remember criminals don't commit these crimes close to their homes either and that's where this side of the equation comes in notice you have that same distance down here subtracted from something known as a buffer zone which is just a constant determined by what works best with known data or past crimes so as distance goes up this entire denominator actually decreases making the whole fraction and as the probability go up so physically if you're too close to the crime scene probability is low that the criminal lives there but as you get further the probability increases that is of course until you pass a certain point which is where the left side of the formula comes in again alter the distance and these two fractions change in opposite ways which is essentially the balancing act that creeps the hot zone of high probability that isn't too close and isn't too far from the crime scenes fee is sort of a constant that I'm not going to go into and that G F and B are constants that just make certain parts of this equation matter more than others or they add more weights to certain parts then lastly this part I'm sure you guys know means we calculate those two fractions for every single crime committed and add the results do this for every square in our grid and we create a heat map of probability what you're seeing here actually is the equations output based on real crimes of a serial killer from the 70s named Richard chase you can see the crime locations in green and the formula predicts his residence to be somewhere in this dark region his actual residence is plotted here in purple exactly as expected so at least in this case Rosco's formula works his formula was put to use in the 90s and successfully caught a serial rapist making Roz mo a celebrity in the crime-fighting world then for anyone who's a fan of the show numbers you may recognize all this in the first episode there's a scene where Charlie Eppes a mathematician looks out of the window at a sprinkler and says although he couldn't determine where the next drop will land he could figure out where the sprinkler head is located or the point of origin he then tells his brother who worked for the FBI that he can determine where serial killer lives using the same analysis later in the episode you see him working with Ross Moe's exact formula and the heat map that had generated Charlie now the fact that crime deals a lot with humans and their decisions can make things tough to solve or predict which is why we have to use probability like we saw earlier but something that has helped in the past is the fact that humans are not very good at being random well that's not totally true drunk texting our exes randomly at 3:00 a.m. humans are good at that and you shouldn't mess with us still no response unbelievable come on I even sent the winking tongue out emoji on the other hand being random number generators we are not so good at if we had a computer randomly generate let's say a hundred numbers that were all 1 through 20 and you did the same thing it's very likely that analysts would be able to figure out which list is the computers and which list is ours because again humans are not as random as they think in the study they found that people listed 7 and 17 way more times than the computer did and even if you're aware of that things like how often you repeat a number twice in a row or the lack of numbers that end in 5 or zero because they don't seem random to us could also provide evidence that a human generated the list in fact in 2009 Iran held a presidential election in which one can and won by a large margin which prompted people to think the results were fixed and thus to graduate students looked at the total amount of votes that each candidate won in each province now this would be over a hundred numbers in total if you looked at all provinces which I obviously cannot include on the screen but if we did we'd expect the last digit of those numbers to be random with an even distribution of 0 through 9 but that's not what happened number seven appeared twice as much as it should've as if the numbers were written by a human trying to be random this wasn't proof but it did point in the direction of some type of fraud going on or if I asked you to put a hundred dots on a piece of paper but make it look random this might look like what you create and it would be an immediate indicator to an analyst that likely a human was responsible for its creation not a random process when dealing with truly random processes clusters tend to form actually this is obvious for example when observing stars in the night sky or if you threw a bunch of coins in the air and note where they fell the problem is humans tend to view uniformity as random during World War two a statistician named Rd Clarke analyzed the pattern of bombs that are being dropped on London to determine whether they were random or targeted attacks since there was a high concentration of bombings in certain areas well using Poisson distribution the statistician determined the expected number of bombings to happen in a certain Square area and found it to be extremely close to the actual number those clusters did not indicate targeted attacks at all like human intuition may assume those clusters should have appeared if everything was random now let's see a very real world applications if we look at any given school all the students are connected in one way or another whether it be through social media classes they share close friendships and so on and by analyzing these connections we can thus make a graph which is what you're seeing here now these aren't the graphs you're used to from school but they have a ton of applications they can be used to represent cities and freeways that connect them they can represent Facebook connections and these graphs are actually used a lot when it comes to solving or preventing crime in fact through FBI investigations years ago a graph was created that include the terrorists involved in the 9/11 attacks and how they were connected what they did and what we're about to do is mathematically determine who the key people in this network are and who should have been prioritized by law enforcement let's go back to the school example though for any two people will say they're connected if they are close friends or in constant contact with each other for our hypothetical school let's say there are 20 people and this is how they're connected so there's a lot to take away from this already like these clusters we could interpret as friend groups or cliques at this school such as jocks bang kids and so on then we could probably consider this student to be the popular kid at the school since they have the most edges leaving their node aka there are close friends with more people than anyone else the amount of the edges connected to a node or seven in this case is known as the degree of the node but this is not the only important person in this network like what do you take of this person here that we'll call John I wouldn't call John a popular student since his node only has a degree of four as do multiple other students but he does stay now on this graph and that's because he has a close friend in all the cliques at this school without him the school would be disconnected to show his importance let's say a rumor spreading throughout the school and the faculty decides to investigate it they find several people have heard the rumor but they want to find who started it so who would you go to this is where John is a key player in our hypothetical wants a rumor start somewhere it will quickly spread throughout that friend group of course but it can't spread across cliques until John hears it and spreads it himself so if you can find out who john heard it from then it becomes easier to backtrack and narrow down who is likely the person that started it all john snow does not have a high degree but it does have the highest between this score to see what that means mathematically take any two points on the graph like these to our label a and b and find the shortest way for a rumor to go from one to another in this case there's two paths it can travel this path which requires one two three edges but it can also travel this path which is also three edges so first you count how many of those shortest paths go through john and we saw that both of them did and then you divide that by the total number of shortest paths which was also two leaving us with a value of 1 this is the betweenness of john as a link just from A to B but if maybe these two people became friends creating an edge that would create a third shortest path also a distance of 3 still only 2 of those go through John though so as between the score goes down to 2/3 since he's not as required to get a rumor from A to B if we calculated that value for every single pair of two nodes finding the shortest paths and how many of those go through John we have John's total between this score which would be higher than anyone elses then there's one last measurement I'll discuss quickly known as the closeness centrality if we want to calculate this for mr. popular all we do is give them a point for every direct friend they have or seven just like with the degree score but now you also give them two points for every person two away or direct friends of friends which would be three people you give them three points for every node three away which is three in this case and you just keep going in our case the last six nodes our distance of four away then you put all that under 19 or the number of nodes in our network minus one which just normalizes this value and this leaves us with 0.4 one three now if we look at these four nodes they all have the same degree score or the same amount of close friends yet they all have different closeness scores with John being the winner even though he's no more popular so this score gets credit for how close you are to everyone overall not just direct friends and close the scores are actually a good indication of how quickly information can spread through a network so now let's pull up that graph of those involved in the 9/11 attacks again it might be hard to see who law enforcement should prioritize but if we calculate those three scores for every node and rank the results we find what our eyes could probably not that one person shows up the top of each list and his node can be found here and it turns out this person was one of the ringleaders of the entire 9/11 plot although this was unfortunately found out after the fact graph theory still singled out one person mathematically who would have provided a lot of information that US intelligence had he been caught beforehand and whether it be terrorist organization street gangs or a bioterrorist attack and how a disease will spread graph theories being used by law enforcement criminologist mathematicians and so on every single day to track and prevent crime next up another application of mathematics in crime is blood stain pattern analysis in fact there's a lot to take away from analyzing blood patterns like blood that drops straight down from we'll create a circular stain where as blood that hits the surface at some angle maybe from a gunshot we'll have a more elliptical pattern something you may not realize though is that the dimensions bat ellipse are dependent on the incident angle not the speed of the fluid so different speeds will yield the same pattern if their angle of incidence is the same but using this we could still work backwards to recreate a crime scene in 1995 a man named Warren hornik drunkenly called the police saying his wife just committed suicide by shooting herself upon arrival there were bloodstains on the man's shirt that he said Gotha when he tried to give his wife CPR everyone believed he was innocent except for one analyst who said the tiny size of blood spots had to originate from high-velocity occurrences not due to giving CPR after the fact that same analyst then backed away from that claim and eventually said it was the lack of a blood splatter trail that proved it was likely a murder as of last year Warren still resides in prison due to this finding from one analyst yet many people think this is not proof and it is possible they sent an innocent man to prison but no one seems to be sure and in 2011 a woman named Brittany Norwood murdered one of her co-workers in the store they worked at she even tried to stage the event by tying herself up afterwards claiming intruders had raped both women eventually it was determined she was lying and then prosecution had to prove it was first-degree murder upon further analysis they found bloodstains under a nearby bookshelf which could not have gotten there from a standing fight do the required angle of incidence this proved that Brittany Norwood beat the victim as she lay unconscious on the ground and it was evidence like this I got Norwood life in prison now in the United States the largest employer of mathematicians is actually the NSA or National Security Agency who need mathematicians for breaking encrypted messages doing signal analysis and more like what you saw in this video even university professors and mathematics get contracting deals with the NSA when they need help with specific cases so for anyone going into math as a career this could be something you end up doing but for anyone looking to learn more about real-world applications of mathematics including its use and criminal analysis then I highly recommend checking out brilliant who I'd like to thank for sponsoring this video brilliant is an online educational platform that a wide variety of math and science courses ranging from basic algebra to vector calculus and quantum computing their courses take you through the technical material you need to know for these subjects but they also constantly test you along the way to ensure you understand the information the example problems you'll come across often have unique real world applications they'll help you understand the material on a deeper level and make you think more like a mathematician as you can see here in their physics of the everyday course you can even learn more about this criminal analysis like the math and physics of blood splatters or how collisions can be reconstructed and if you're really trying to keep your math and science skills sharp they even have daily problems that make learning a habit these provide you with the context and framework need to tackle a wide variety of problems on a daily basis from what happens when you cut a mobius strip in half to logic puzzles and how to uncover the truth in complex situations and if anything confuses you they have a large community of thousands of learners discussing these every single day so if you want to still be productive while procrastinating that really boring essay you currently have to write or you just want to learn something new then head on over to brilliant org slash major prep or click the link below also the first 200 of you to do so will get 20% off the annual subscription giving you full access to all courses and problems within the archives and with that I'm going to end that video there if you guys enjoyed be sure to LIKE and subscribe be sure to follow me on Twitter and join the mid-foot Facebook group for updates on everything hit that bell if you're not being notified and I'll see you all in the next video

Channel: Zach Star

Views: 294,804

Keywords: majorprep, major prep, mathematics of crime, math used in crime, math needed for solving crime, mathematics used to solve crime, how math is used to solve crime, math used in numbers, math used in numb3rs, applications of mathematics, applied math, applied mathematics, blood stain analysis, graph theory, geographic profiling, kim rossmo

Id: -cXBgHgX5UE

Length: 15min 58sec (958 seconds)

Published: Wed Feb 20 2019

Reddit Comments

[u/slainbyvatra]

Why don't they anymore?