How Not to Be Wrong: The Power of Mathematical Thinking – with Jordan Ellenberg


Well, thank you so much everybody for coming out and being here. As a math professor I’m… it’s always a treat when people are coming to listen to me talk about math And they’re not required to, as they are… as they are in my class. So, let’s see, um The book “How not to be wrong […]”, and in some sense it is a book of stories — stories about math And I’m gonna just tell a couple of stories tonight, one short and one long. And let me start with the short story It’s just one slide long It’s a story about a mathematician named Abraham Wald who was a mathematician — he was born in Hungary, Works in Austria, has to flee Austria when the Germans take over and ends up finally uh in New York City, a professor at Columbia. And during the war he participates in an outfit called the Statistical Research Group, the SRG… I actually had never heard of this before I learned about it for writing this book, but it was a top-secret installation where some of the top Mathematicians and statisticians in the United States Were working in the neighborhood of Columbia University in New York were working on problems related to the conduct of the war, problems of a mathematical nature So it was kind of like the Manhattan project except it was actually in Manhattan and one day One day a group of generals came to the SRG and they came to Abraham Wald with a question They said “We have a math problem for you We have noticed that when the planes, when our bombers and our fighters come back from flying missions over Germany They kind of look like this. They’re riddled with bullet holes But the bullet holes are not arrayed evenly over the planes that are coming back. That’s what we’ve noticed There’s more holes in the fuselage of the plane, there’s fewer holes in the engine, different parts of the plane have different number of bullets and Here’s what we want to know: We need to figure out how to put the armor on the planes.” This is a serious issue, right? Because if you put too much armor, the plane doesn’t fly, and that’s bad. If you put too little armor, the plane gets shot down, and that’s bad. So they came to the SRG, the generals did, and said “We need to know How much more armor should we be putting on the parts of the plane that are getting hit more? Is there some kind of formula for this? Is there some kind of math equation that you math guys can do for us to solve this problem?” And here’s how the story ends. Wald says “No.” He says “You have it completely wrong, You have to put the armor where the bullet holes are not. You have to armor the engines, not the fuselage. Why? Because it’s not that the Germans can’t hit your planes on the engines; It’s that the planes that got hit on the engines are not the ones that are coming back from the missions.” So is this a math story or not? I say that it is, not just because Abraham Wald was a mathematician But because I think in some fundamental way he was really thinking mathematically And the story is there to tell a lesson, a lesson that to be a mathematician is Not just to compute a formula, right? A machine can do that, even in the 40s, like, machines could do that… The purpose of the mathematical way of thinking is not only to answer numerical questions, but also to ask the right questions to interrogate a question that we’re faced with and ask what assumptions underlie it and if those assumptions are reasonable. And sometimes to overturn the question entirely, as Abraham Wald did That’s part of the case for mathematical thinking Now that story it’s kind of like a Little tiny story with a punch, right? I could sort of tell it in five minutes I tell it in a few pages in the book. One of the great things about writing a long book And it is kind of long– Maybe I shouldn’t admit that… Penguin did an amazing job actually. I was just talking with Eddie about this of, like, if anybody has it They made it look like it’s not very long. It’s very, it’s a very impressive piece of book production. It is kind of long actually Anyway, one of the good things about writing a long book as opposed to Writing, let’s say, a thousand words in a newspaper or magazine is that it allows you to stretch out a little bit and draw out connections because The fact is that’s the way mathematics is Right, I mean, some things in mathematics are, like, these little instant punches, a moment of insight, an a-ha moment But most of the body of mathematics, what’s exciting about it is the way that a lot of different ideas are connected under the skin and if you really want to follow The connections out wherever they lead You need a little more space, so I’m going to tell a longer story, one that Actually, I’m going to probably take about 40 minutes to tell a st–, to tell maybe, like, a third of a very long story That’s uh, that’s in the book And the story has to do with the lottery in the U.S. state of Massachusetts What you’re looking at here is a picture from the very last drawing of a lottery game called Cash Windfall in Massachusetts This is a picture from January 2012, um, and the point of this story is to explain Why this was the last drawing. in order, but in order to do that, let me start with a little kind of a starter about how lotteries work in general. Okay, so this is how a lottery works: You pay a small amount of money Let’s say $2 and I’m going to give a little simplified version to start with You pay a small amount of money for a small chance of winning a large amount of money, so for instance Let’s say a lottery ticket cost $2 and maybe there’s a one in two hundred chance that you win $300 back or should I have changed the currency for this? That would have been Thoughtful… Sorry, I didn’t think to do that. Okay. Well the story takes place in the United States So if you play this game a thousand times… And people who play the lottery do play a thousand times Oh people really like to play the lottery a lot. If you play this game a lot, if you play it a thousand times Well, how many times are you going to win? Well, of course, There’s variation in that, it’s random but if there is a one in two hundred chance of winning in your thousand plays you’ll probably win about five times, right? So that means you win five of those $300 prizes or $1,500 Sounds pretty good… Until you think about the fact that you spent two thousand dollars in tickets To get that $1500 in winning… So the sort of term of art that mathematicians use to talk about this kind of computation is “expected value” so We would say the expected value of this ticket is a dollar and fifty cents. That is how much you’re going to win, on average, per play So that’s the mathematical term. I got to say though, that It’s sort of a terrible term, it’s one of those that we wish we could take back We can’t, the notation is what it is, but It’s a bad piece of terminology because the expected value Whatever it means, it certainly does not mean the value we expect that ticket to have. In fact that’s not even a possible value for the ticket to have, right? That ticket is either worth nothing or it’s worth $300 But it is definitely not worth a dollar and fifty cents, so somehow if we had to do it all over again We would probably call this the average value But it’s a much more reasonable summation of what we’re actually trying to describe. A dollar fifty is how much the average ticket is worth. I’m going to have the average ticket cost two dollars. In fact, all the tickets cost two dollars and a fundamental rule of thumb Is that you shouldn’t pay two dollars for something that’s worth a dollar fifty and here you have, in a nutshell, the mathematical case against playing the lottery. And now I’ll complicate that a little bit so what you’re looking at here is an actual list of payoffs for a lottery game in Massachusetts, the regular state lottery. The exact numbers are not important but I want you to look at this computation at the bottom you don’t have to check it for yourself but what I want to point out is that The example I gave at the beginning, the simplified example was unrealistic in a couple of ways: One way it was unrealistic is that there was only two classes of prize, real lotteries are not like this, real lotteries Have a lot of different prizes. They have a big jackpot that you get if you get all the numbers correct But that jackpot is really hard to win and it’s kind of demoralizing for people If there were only the jackpot people probably wouldn’t play, right? Because they probably wouldn’t feel like they could win so Jack said real lotteries have, like, a whole sequence of Lower-tier prizes, some of which, like matching three out of the six numbers in the Massachusetts lottery, are really not that hard to win at all, the payoff is kind of low, only $5 But you have a one in forty-seven chance of winning. That means if you play a lot, if you play every day, probably, every so often, you’re going to win And it’s quite common for like a friend of yours to win or somebody you know to win So it keeps people playing, right? Running a lottery has a lot to do with psychology So that’s one way that my simplified example was unrealistic: too few tiers of prizes; and the other way my My simplified model was unrealistic is that it was incredibly generous to the players No real lottery pays back a dollar fifty For every… as the average value of a $2 ticket. In this Massachusetts lottery the expected value of a $2 ticket was just 80 cents That’s a lot lower. In fact, I was looking, actually, at some lotteries that are played here. Has anybody played EuroMillions? That’s a 40 cent expected value on a 2… Rough, well 0.4 of a Euro expected value on a 2 euro ticket. That’s insanity! That’s what, this would never be tolerated the United States. I just want to tell you guys that. Even American lottery players would be, like, “That is a stinker of a game.” people who, like, play this every day would, like, not play EuroMillions. Ok, so as I said, the the reason you don’t have a jackpot– don’t have just a jackpot is because it’s demoralizing when nobody wins; if people are not winning the jackpot, people start to get depressed and people stop playing. And this is what happened in the state of Massachusetts in the year of of about 2004 / 2005… A whole year went by without anybody winning the jackpot and they could see at the lottery commission that people were stopping playing the game People were depressed, they didn’t feel like there was a chance, and it wasn’t working. So they said we got to make a change, we got to do something to goose interest in our game, so They instituted a new rule, a rule called the roll down rule. Let me explain how it works They said instead of just letting that money pile up in the jackpot, right? Because if nobody wins the Jackpot the Jackpot pool gets bigger. That money that is not given out in prizes. just kind of makes the jackpot bigger and bigger and bigger They said okay that is not satisfying people, because they feel like that jackpot may get bigger and bigger and bigger But I’m never going to win. They said let’s make a new rule if that jackpot goes over two million dollars, and nobody wins the Jackpot that drawing then it’s going to roll down All that money is going to roll down into the lower tier prizes and make them bigger That’s exciting. That’s maybe a good way to get people interested So they were trying to design a game that looked like a better deal for the player And in fact they did their job a little bit too well I can always tell how Mathy an audience is by how big a laugh I can get with a table. That’s always um So this is what the payoff Matrix for um for the Massachusetts cash Windfall lottery looked like on February 7th 2005 um so for instance that four of six prize… There’s a one in eight hundred chance of winning– you remember in a usual drawing that was a doll $150.00 prize… On this roll down day, in which no one won the jackpot, that prize was actually worth almost $2,400 so stop to think about that There’s a one in eight hundred chance of winning and the prize is worth twenty-four hundred dollars So if you bought eight hundred tickets you were probably going to win twenty-four hundred dollars right there For the sixteen hundred dollars you spent buying eight hundred tickets, and that’s just the four out of six prizes right there’s other prizes too, each of which has some value and when you add it all up You find that the average value of a $2.00 lottery ticket sold on this day Was five dollars and 53 cents So that is not a bad investment! So how do I know, by the way, exactly what the payoffs were, for this– why do I know what payoffs were for this particular day of the Massachusetts State Lottery? I know it because I read about it in the following document. Which you shouldn’t be able to like read from where you sit, but let me tell you what this is This is a 25 page letter from the inspector general of the state of Massachusetts to the state treasurer Trying to explain what had happened to the state lottery. And I got to tell you guys… I feel safe and saying this is the only Fiscal oversight document by a municipal official that you will ever read that makes you wonder if somebody has the movie rights to it! It really is kind of a crazy story, which, again, in the book I tell at length; here I’m going to tell it to you somewhat briefly um What happened? Well, what happened is that On February 7th 2005 um The state lottery started getting phone calls, they got a phone call from a Star market in Cambridge, Massachusetts. (which was like a convenience store) Saying some college kids just came in and want to buy 5000 lottery tickets. Is that okay? So there’s a rule, you know… if somebody who as a single buyer wants to buy a lot of lottery tickets They have to call the state lottery and make and get a special waiver But this this is granted and by the way this was not the only place there was similar large buys In several places around the state, but what was going on? What was going on is that There are a lot of people who could make a table like the one that I just showed you… And some people did. so for instance one of the main players of the story are two– It’s a guy from MIT called James Harvey. He was a senior at MIT at the time and And as it happened He was doing an independent study project in January 2005 on the expected value of State lottery tickets… He was a lucky guy! And as part of his project, um he computed the value of like all currently running, Massachusetts lottery games and Presumably he drew a table very much like the one I just showed you, and the first thing he did was go around to all of his friends in his dorm at MIT and Say look you should really give me all the money that you have right now so I can go buy lottery tickets with it um and if you go to MIT I think Everybody in your dorm can also compute that table and see that that’s actually a wise idea and so they sort of coalesced their money and and bought– and bought all those tickets, in Cambridge. There was another group, called the Dr. John Lottery Club, which was based around biomedical researchers at Northeastern University, which is also in Boston and then maybe my favorite guy Was a guy called Jerry Selbee, who was a retired engineer, in Michigan… How did he ever hear about it? well… Where did Massachusetts get the idea for this roll down rule? They got it from a roll down game, in Michigan, that had just closed I’m not sure they asked Michigan why it closed… But Jerry Selbee knew why it closed, because he had made about two million dollars off the Michigan game Over the previous seven years or so So he was– he could not believe his ears when he saw that Massachusetts was opening the gates again So he immediately got in the car with his wife and drove like the nearest point in the state of Massachusetts to Michigan But that’s– that’s pretty far, guys I don’t know if everybody like knows the location of all the US states, but that’s probably about a 14 hour drive. I’d say And he made a big buy up in the Northwest corner of the state and kept on doing this; so um The story that is outlined in this long 25 white page document is the way that these three groups of high-volume players Continued to buy more and more lottery tickets taking their winnings and plowing them back into the investment scheme and buying yet more until, by the time this reached some kind of equilibrium, just to give you some sense the Inspector General estimates that, on a given roll down day, somewhere between 80 and 90 percent of all tickets sold in Cash Windfall were being sold to a member of one of these three groups um So how does this story end? um Well, it ends like this. This is the front page of the Boston Globe in summer 2011… At some point somebody figures out this is going on, the Globe gets tipped off. They run this story explaining what’s going on with the state lottery; and at this point the game is up. Right? Once people perceive that the game is not what it seems [uh] then people stop playing and then it doesn’t work anymore, then no more money flows into the system. So… in a way, that’s the end of the story but if you read it from a mathematical point of view the chronological end of the story Is not really the end, right? Because, as mathematicians, there are some puzzles that remain… At least they remained for me when I was like reading this story and I was trying to read it with a mathematician’s eye and try to understand what really went on here So I’m gonna spend the rest of our time together talking about two mathematical puzzles That I think we’re left with, having told the bare-bones version of this story… One is easy, one is hard… Let’s start with the easy one um the first puzzle is How could you actually get away with this? This is a little weird right let me remind you that the state knows who’s winning the lottery Right? Because they have to give you the money, so it’s not a secret The state knows that all the winning numbers are coming from the same three convenience stores again and again Let me remind you something else; if you guys that were paying attention to the dates When’s that first roll down? February 2005. When’s this article in the Boston Globe? July 2011. So there was time to figure out, but something was amiss. This is six years, we’re talking about um So this is puzzle one. How did the state not figure out what was going on? well… This is the reason this puzzle is easy– No, I like bureaucrats, okay… um So the reason this puzzle is easy is is the following Here’s the answer: the state did figure it out, and how do I know this I know it because it’s in the inspector General’s report and in fact I slightly lied to you I said that when James Harvey figured out the new Massachusetts lottery game had a positive expected value, I said the first thing he did was Get money from all of his friends in his dorm and go buy a lot of tickets But no, that is the second thing that he did. The first thing he did because kids who go to MIT are like Good kids who like play by the rules, and get good grades, right? The first thing he did was get on the subway and go to Braintree, Massachusetts And go to the state lottery headquarters and have a meeting with them And he said “Look, your new game has a positive expected value… I’m planning to buy thousands and thousands of tickets and make a lot of money… Is that legal?” And the Inspector general does not record exactly what response he got to this query But it must have been something like “Sure, knock yourself out” because the next thing that happened was what I just told you and then It went on happening for another six years Okay, so that’s the answer to that puzzle, but that answer kind of spawns another question as so often happens so the ques– the– it spawns the question why didn’t the state do anything about it? if the state knew from day one that this was going on? okay, so to answer this question I need to use a very sophisticated mathematical diagram which represents the limit of my powerpoint skills… so what’s actually going on here? “Random Strategies”, I should say, is the name of James Harvey and Yuran — James Harvey’s team It’s the name of his group of people um You might object that their strategy was really not very random at all. What’s ac– what’s actually going on here is that the dorm they lived in was called Random Hall Which is a place at MIT and that was where the money was coming from. So, um… How should you think of what’s going on here? Well I want to remind you one more very important thing about how the lottery works, which is this: when a lottery ticket is sold for $2 Massachusetts takes 80 cents of that money and that’s state revenue, right? That’s what goes to pay police officers and pave the streets and keep the lights on and do all the things that the lottery is intended to do And then the rest of that money is eventually going to get disbursed in prizes in one form or another So what does that mean? That means that from the point of view of the state The amount of money it makes is 80 cents times the number of tickets sold; the state does not care who wins the lottery! The state only cares how many tickets are sold. so this is a crucial point because when this story came out of the newspaper, I think was presented as That these folks had somehow Cheated the state out of a lot of money; in fact the inspector general estimates that the state of Massachusetts took in somewhere between ten and fifteen million Dollars extra revenue above what they would have, had these three large groups of betters not existed… So I think it’s safe to say that somehow if you come away with an eight-figure win You are not the person who got scammed So what’s going on? Where was this money coming from? Well, of course, the money was coming from the people who were playing the lottery on the non roll-down days so that’s what this [uh] That’s what this figure is meant to emphasize You should think of what was happening as a movement of money from all the regular players To these groups of people who were playing only on the roll down days um With Massachusetts getting 8– 80 cents every time a ticket is sold Maybe a good analogy is like this… um Again when this story came out of the newspapers it was sort of at the same time actually that there was a big story about MIT students um winning a lot of money at blackjack (does anybody remember the story?) in Las Vegas casinos And so it was sort of– that these two stories were talked about in the breath They said how did the kids at MIT figure out how to beat the house? Okay, let me explain why that’s wrong. What were the kids at MIT doing? They were making a lot of bets, right? They were buying just to give you the scale, about 200,000 tickets every roll down day They were making a lot of bets, each one of which had a small positive expected value I should also say by the way that once a lot of people were playing It didn’t stay like five dollars and fifty cents for $2 ticket was more like a 15% profit on average. That’s still pretty good So they’re making a small– and so some of those bets are going to win, some of them are going to lose But if each one is slightly tilted towards the MIT kids then on the whole they’re very likely going to make money so if that’s your strategy You’re making a lot of bets, which are slightly tilted in your favor… You are not beating the house, you are the house! I mean that is what the house does. And so I think a productive way to think about what was actually going on, from a mathematical point of view, is to compare it to the following diagram Which is exactly the same. The kids from MIT and the other high volume betters were playing the role of the casino the regular lottery players were playing the role of the regular betters who come to the casino and bet and they in the aggregate make Lots of bets which overall have a slightly negative expected value and money is flowing away from them; and every time it does every time the money flows in Las Vegas the state of Nevada reaches in and takes a cut Right? Because States don’t like to gamble, States like to collect taxes That’s their skill set. That’s what they’re good at, and that’s what they do And that’s what they were doing in Massachusetts. In other words what you should think of as having happened is that the state of Massachusetts… (i still don’t know whether on purpose or sort of stumbling into it) had licensed a gigantic under-publicized virtual casino on which they collected lots of taxes and made a good profit And which carried on until people found out about it… So that, in the end, I think, is the answer. I think that’s a satisfying answer to the first question, of How did this go on for so long? But now I want to turn to the second question, which turns out to have quite a bit more mathematical heft. You see, I’ve been talking about these three groups of betters as if they were all the same, but that’s not quite true There’s one very interesting difference between the three groups, that the Inspector General’s pointed out; which is that Jerry Selbee and Dr. John used what’s called the quick pick machine. I don’t know if there’s an analogue to that in the UK… So, what is this? This is a machine that picks random numbers for you to play And that seems like a good idea, right? Because we all know that you can’t predict what numbers the lottery’s going to come up with; any number is as good as other… if you’re going to buy two hundred thousand tickets, It certainly seems like it would save a lot of time and cost you nothing to have those tickets printed out randomly for you by machine, but Random Strategies, contrary to their name, did not do this! They filled out their tickets by hand, 200 thousand of them… Why?! Why do this? This is a humongous pain, and you know, the inspector general’s report mentions that they did this, but didn’t say why. And I became kind of obsessed with this, because I was like these people are smart, they know what they’re doing, they know math They know that the expected value of each ticket is the same, why would they care which tickets they had? That’s what I want to spend the rest of our time together talking about so as mathematicians When we’re faced with a problem we don’t understand, the first thing we do is we try to make it simpler. We try to replace our problem with a simpler problem. Hopefully which has the same features Enough of the same features the original one that we can use it to gain some insight, so that’s exactly what I’m going to do… Let me replace this lottery with a smaller game which, for reasons that are lost in history, is sometimes called the Transylvanian lottery… And here, instead of having 46 different numbers like in the actual Massachusetts lottery There are only 7. Only 7 balls in the cage. And instead of picking six of those, you’re only going to pick three. And the reason I do that is because it means that the number of Jackpots is now so small that I can list them all on one slide Here they are, all the different ways of picking three numbers out of seven. For the combinatorics fans in the audience The number of these is 35, which is “7 choose 3” and that’s called 7 choose 3 because it’s the number of ways of choosing 3 things out of 7 But it doesn’t matter if you know that… just matters that you believe me that these are all the possible combinations. And now we can start to say well, what if the game were this small? Let’s see if we can understand what would be the benefit of choosing the numbers yourself as opposed to picking them randomly. Well, first of all I have to sort of tell you what the rules of the game are, now that I’ve shrunk it a little bit… Again, let’s simplify. Let’s not have like six different tiers of prize. Let’s only have two so in this simplified game There’s two kinds of prizes: you get a jackpot, which is worth six dollars if you get all three numbers right… Order doesn’t matter by the way I can emphasize that; if you get two out of the three numbers right you have a smaller prize which let’s call it deuce which is worth two dollars; and if you get one or zero of the numbers right then you get nothing. So, okay, this is very simple But it has some of the features of the original game, right? it has multiple tiers of prizes… Well, only two… And it sort of has the same form, but at a much smaller scale. And… Now in this game… what is a high volume better? You don’t have to buy 200,000 tickets to be a high volume better in this game I mean there are only 35 different tickets to buy! So for us let’s discuss the problem of buying seven tickets. That’s a lot, that’s a fifth of all possible tickets, right? So that’s a pretty big purchase. And what I want to show you is What happens if you pick seven tickets at random? And I’ve computed this for you, showing you– I haven’t actually shown you the jackpot probabilities But I’ve shown you how many deuces you can expect to get… It turns out that the expected number of deuces, if you buy seven tickets, is 2.4 And so it’s not surprising… that the most likely number of deuces to get are two or three… There’s a 30% chance of getting two deuces, a tw– about a 26 percent chance of getting three… and there’s some probability of getting fewer, and there’s some probability of getting more. In fact, you can easily– you can, given this, you can compute the expected value of this ticket Of your 7 tickets you expect to get two point four deuces The expected number of Jackpots… that’s 1/5th. Why is it 1/5th? Because well, you’ve got one fifth of the tickets, so there’s a 1/5– a one in five chance that the Jackpot is among them… So let’s see: 2.4 times $2 is 4.80 and then one fifth of that six dollar prize is a dollar twenty… put that together, and the expected value of your seven tickets is $6. And again, I want to remind you that um That every ticket has the same expected value, so it actually doesn’t matter which seven tickets. I’ve written down… (this is sort of a fun exercise to do yourself) any seven tickets I chose would have an expected value of six dollars This is the source of our intuition that it shouldn’t matter which seven you pick so Here’s what we’re going to do… Ready? You are going to think and write down seven tickets that you want to buy… So a ticket is just a choice of three numbers from 1 through 7 And here’s the game we’re going to play: I told you that on average We can expect to win six dollars every time, so what we’re going to do is we’re going to pick some random numbers We’re going to play the game every one of you who has done it is gonna like see how much money you won on your seven tickets… You got to be honest; and here’s how it’s gonna work It’s like an elimination game: if you get less than six dollars, you’re out. And I’m playing this game too, by the way, so I’m gonna show you my tickets… Everybody picked? Okay, no copying! All right um Okay, and there’s– and here’s mine Okay, now I do need one more thing, which is I need somebody to be the random ball cage Is there anybody really random here? Okay, this kid. okay, so um Okay, so try, if you can, not to look at mine, and not to look at your friends’ there, and just like, call out three random numbers between one and seven… They should be three different numbers, right? cuz when they come out of the ball cage they’re all different… And then we’re all gonna score. Okay… Okay, so five– so I’m going to order that as five six seven just so we’ll have them in order, okay? So everybody gets what we’re doing. The jackpot is five six seven you guys are scoring your seven tickets… if you have five six seven you get six dollars just for that, if you’d– and, if you have a five and a seven you get two dollars for that, if you have a five and a six you get two dollars for that If you have a 6 and a 7 you get two dollars for that. Let’s see how I did. Maybe that– so So I’ve got 167 that gets me $2 I don’t have a jackpot there. I have two five six that gets me another $2 And I have three five seven and that gives me another $2, right? because I have the five and the seven. So I have six, so I’m still in, I have six. Okay Okay, all right. Let’s do it again. Ball cage man, are you ready? Okay? One six seven. Okay. I’ve got a jackpot. I already have six… Let me see if I’ve got anything else. uh… five… six… no, and no Everybody count those up. Okay. Who’s still in? Okay… fewer, but still a fair number, okay? I’m ready. We’re gonna– two four five Okay, I do not have a jackpot. Okay. I’m gonna look I’ve got one four five that’s a deuce for me. I’ve got two four seven that’s a deuce for me, and I’ve got two five six And that’s it. So I got three deuces so I’m– so I got six dollars, so I’m still in. Alright. Let’s do– let’s do one more Three one four okay, otherwise known as one three four, okay… so one three four are the numbers when I’ve got two deuces on my first two right there… and then… and then I’ve got three four six okay, so I’ve got Three deuces okay, so I’m in with six dollars Okay, so let’s– let’s end it at four rounds I kind of like this game, but we can only do it for so long… okay so who– so I got– so I’m still in, I– I made my six every time, and it– who else got 6 or above every time? Okay, so a decent number… probably about like What do you guys think? Maybe like one in five of the people who are playing like some some a handful of people okay, so What did you guys notice, especially a kid about my winnings? Anybody notice anything? besides the fact that I’m still in, because I’m great at this game. I always got six, right? Not only did I never go under, I never went over… And that is no coincidence. In fact, the miraculous thing about these numbers that I’ve chosen Is that although it looks like I’m playing a gambling game I am not, because no matter what the jackpot is I will win six dollars, on the nose. So this is how the payoff matrix looks for my seven numbers Instead of this wide range of possibilities there are in fact only two things that can happen: four out of five times I will get three deuces and I will win six dollars And one out of five times I will have the jackpot, and I will win six dollars. And what that means is I cannot be eliminated from this game So these two bets have the same expected value, but they’re not the same bet, they are rather different; in the language of finance we would say that they have the same return, but the second one has a lot less risk Right? There’s no chance of getting one of these, like, very poor results like only two deuces are one deuce or no prize at all… That cannot happen. And most people Given the choice between two investments that have the same return will choose the one with the less risk I mean, don’t get me wrong. There’s downside to that too. I’m also giving up the possibility of winning a lot more, right? But that is a choice most people will make, especially If your business plan involves borrowing money from people… In finance we would call this “leverage” If the way you’re running your business is borrowing money from all your friends to buy lottery tickets and then you buy a lot of lottery tickets, and then you lose all their– all of their money And then if you come back and say, but by the expected value computation, if you can somehow find more money to give me, in the long run we’ll come out ahead That is usually not such a successful business strategy, right? If you’re playing with other people’s money you really don’t want to lose, and so a hedging strategy like this one… um where you eliminate the possibility of Loss, is quite attractive. The question is How would you come up with these numbers? How would you come up with these numbers that are somehow so perfectly arrayed so that you eliminate all possibility of Loss? well… They come from at least what to me was a rather surprising source… They come from Geometry So, what are we looking at here? We’re looking at very many things One thing we’re looking at is a kind of diagram of those seven tickets I just showed you if you see here there’s seven points here corresponding to the seven numbers, and there’s also seven little curves, right? There’s… and each of the Curves contains three of the points So if you look at that little edge along the bottom is one two three… That’s one of my tickets The circular thing is two five six. That’s another one of my tickets There’s a line That’s a line second that’s one four five is one that’s three five seven and if you go back to the previous slide you could those are exactly the tickets that I have. So this is a kind of diagram by which I keep track of what my seven tickets were So one thing it is this is a picture of my magic set of tickets, but another thing it is is I would say it’s a picture of the plane… A somewhat weird thing to say Doesn’t look like a plane. It looks like some kind of a funny triangle with a circle drawn in it But I’m going to say that this is a plane and the points are points and those little curved things are lines okay, that’s a weird thing to– that’s an even weirder thing to say, right? because For those taking Geometry even the ones that are lines they don’t look like lines They’re line segments, right? everybody remember this distinction Because they’re finite in extent And then there’s that one in the middle that doesn’t look like a line at all, looks like a circle! And yet, I demand the right to say that these are lines. Why? because to a modern mathematician we’re not tied to our usual geometric notions of points and lines when we talk about points and lines… To us, a point is a thing that behaves like a point and a line is a thing that behaves like a line Okay What does that mean?! well, what are our rules for how points and lines behave? They were given to us by Euclid, right? I mean, this is the rules of geometry and what I want to point out is that with this definition of points and lines the rules of geometry are obeyed It’s kind of fun to check this for yourself on the picture, that… every two lines intersect in a single point, and any two points determine a unique line. Just like they’re supposed to according to the rules of geometry Is anybody taking geometry right now by the way? I think everybody in the room is either too young or too old Because there is one way in which these are actually not Euclid’s rules… Everything I say here is true about this picture, but it’s a bit different from regular Geometry because in regular Geometry Lines can be parallel So you can have two lines that don’t intersect at a point. See, I consider that a problem Rules with exceptions are bad, rules without exceptions are much nicer so this geometry is actually much better than euclidian geometry it’s what’s called a projective geometry, in which there are no parallel lines… and if I had another hour to tell you this story I would tell you all about how the basic ideas of projective geometry were first developed not actually by Mathematicians but by painters. Because they were the ones who had to figure out how to depict a three-dimensional world on a two-dimensional canvas and if you’ve seen sort of one-point perspective painting you’re familiar with the fact that there are a lot fewer parallel lines in paintings than there are in the real world, right? the railroad tracks Okay, I know there weren’t railroad tracks in the 15th century but I mean that’s the idea… A parallel thing is like sort of seemed to come to a point on the canvas exactly for this reason that in projective geometry there’s no such thing as parallel lines… any two lines meet. So maybe I’ve sort of made some kind of convincing case that it’s not ridiculous to call these curvy looking things lines and to call these pointy looking things points What– what connection does it have with what I’m actually talking about? Well, here’s the deal: Why is it that this set of tickets has the properties I said it had? How did I know that when my friend over here said 1 4 5… oh, that was one that was the jackpot. Let me ch– pick a different one 3– 5 6 7 was another one that he said. Look– let’s look at 5, 6, and 7; that was the first one, right? How did I know I was going to have three deuces? well How did I know I was going to have a ticket with a six and a seven? I knew that because there is a line through the points six and seven; there it is: one six seven, that’s one of my tickets How did I know I was going to have a ticket that contained five and seven, giving me another deuce prize? because I know there’s a line through the points five and seven, and there it is: Three five seven; and finally, five and six… What goes through them? is the line two five six So the fact… the geometric fact that through any two points there’s exactly one line that Is exactly what is required to guarantee– I mean in sort of lottery language, that says that given any jackpot each pair of numbers in that jackpot will be on one of– exactly one of my tickets, and so I’ll get exactly three deuces. The only thing that wait, that can get a little messed up is if the three points, that my friend chose, were actually collinear, like one four five. Then I do have a ticket containing one and four, and I do have a ticket containing one and five, and I do have a ticket containing four and five as the Geometry guarantees, but they’re all the same ticket. And there you have it, those are the two things that can happen: the three points can either be collinear or not; I will either get the jackpot or I get three deuces. so This is quite mysterious this thing by the way, I should I should give credit where it’s due This is called the Fano plane It was developed at the very outset of axiomatic geometry, by Gino Fano, who was an Italian Geometer of late– of the late 19th century There’s one problem though. I mean I really feel like I’ve given a nice solution to how you should buy your seven tickets, to eliminate risk… The problem is that, of course, let’s not forget that I simplified the problem… There are not seven numbers in Cash Windfall. There are 46; you don’t pick three of them, you pick six of them, so maybe I was sort of clever and sort of found some nice picture I could draw from classical Geometry to handle this small problem, but what about the big one, so Here we have our toe sort of on the edge of a very beautiful large Old area of mathematics called the theory of combinatorial designs I’m not going to have the time to tell the whole story, though it’s a great story. Let me just say that there exists an entire long mathematical story of how to develop configurations like this in all manner of situations and actually I’ll just say because it’s kind of UK related This is Peter Keevash. He’s a professor at Oxford and In some sense last year he proved a truly remarkable theorem that essentially puts the cap on about 150 years of work in this area, proving that a design like the one I just showed you that those designs are ubiquitous; basically in every context in which you would expect there to exist such a thing, there really does exist such a thing. And this was quite– I think this was quite unexpected actually, that this problem was going to be solved so soon. it’s a really miraculous piece of work by Peter That being said, when I was thinking about this problem of Cash Windfall, I wasn’t thinking about the general case of good designs appearing in every possible context I was thinking in one– I was thinking about this one particular context of… of, um, six numbers chosen out of 46 balls Moreover, Peter’s proof is what’s called non-constructive, right? It doesn’t necessarily give you an easy quick way to actually write down such a design it sort of just proves that they exist That being said though there is as I said this very long literature about problems like this, and so I became kind of obsessive trying to figure out what these kids at MIT actually did… I became certain that they use some kind of combinatorial design strategy and then minimizing risk was the reason that they were choosing their own tickets… And I wanted to kind of reverse engineer to figure out what it was; and after some searching I found the following paper, also a UK product by the way. This is RHF Denniston for the University of Leicester who wrote a paper in 1976 Generating some kind of combinatorial designs actually these guys were actually on 48 balls not 46 but I tweaked it a little bit and I was able to come up with a configuration of about 230,000 tickets [um] which gives you a 2% chance of winning the jackpot, a 72 chance– percent chance of hitting six of those five out of six prizes, and a twenty four percent chance of winning five of those those five out of six prizes, so essentially guaranteeing that you would win at least five of those big prizes and essentially hedging away all risk in a way that actually looks rather different the risk of Loss would be much greater if you picked your 230,000 tickets randomly… And people always ask if I can draw a picture of this one, I cannot I’m sorry I wish there was like a beautiful picture like the picture of the Fano plane there– there is not… But it’s, in some sense, similar in spirit. In that it does have to do with finite projective geometry just like the Fano plane does But in a much more intricate way, which is why it was only developed about 100 years later So, I’ll just close by saying that I don’t actually know, because I was not able to get these guys to tell me if this is actually what they did. But if they didn’t, I think it’s what they should have done. Okay, I’ll stop there, and I’ll take questions. Thank you so much.

100 thoughts on “How Not to Be Wrong: The Power of Mathematical Thinking – with Jordan Ellenberg

  1. This was an excellent 30 minute talk, padded to 48 minutes with blank pauses, uumms, telling what he's about to say, and other aggravations. It hurts to watch this, but the good 30 minutes is worth the pain of the other 18.

  2. Hi,

    Can someone please explain how he got a 5.3% chance of getting no deuces with 7 tickets.

    Thank you.

    It's really bugging me because my calculations give me 2.53 %

  3. All numbers should be viewed as spins. No spin zero positive spin 1 and negative spin minus 1. And sum of all positive is inverted negative 12. All numbers are powers of some other.

  4. Please don’t hate on him. It’s a misunderstanding. The title doesn’t explain that the title is the name of his book, so it basically half clickbait. That’s why he is talking about his book. The book is actually great!

  5. Can one of the people that didn't find him annoying explain to me the last part cause i couldn't make it to the end. Fell asleep yesterday but now i still want to know the answer

  6. If you have no other words but "fuselage" for it (wtf), you deserve bullet holes in it. Get some sensible words of your own, English.

  7. @ 8:06 the expected value of the lottery ticket is not 79.8 cents, it's 59.1 cents.
    The error is in assuming that a 'free ticket' has the value of the ticket price, when the value is actually the expected value of the ticket.

    This produces an equation like:
    n = 1/9.3 + 4/39 + 15/80 + 5/47 + n/6.8
    n-n/6.8 = 1/9.3 + 4/39 + 15/80 + 5/47
    5.8n = 6.8x(1/9.3 + 4/39 + 15/80 + 5/47)
    n = (6.8/5.8)x(1/9.3 + 4/39 + 15/80 + 5/47)
    which works out to be $0.5908…

  8. One of the other speakers used the airplane story… can't remember the other one… watched too many so far…

  9. I hope they left all the umms out of his book. If I had a dollar for every umm or uhh in this lecture I could buy a 1000 lottery tickets.

  10. My observation has been that when using “quick picks,” the lottery appears to be set up to “share” by giving several people the same combination, thus spreading out a large winning and creating more winners with smaller amounts. Many large totals are split among several people who admit they used “quick picks.”

  11. This reminds me of the Chinese fortune cookie problem. In Canada they have a lotto called 6-49. Or in combinatronics 49 choose 6. Or 13.9 million combinations

    In many Chinese restaurants they give out fortune cookies with your bill. People would use these numbers for picking lotto numbers. Of course there was no better chance to win then using any other 6 numbers then the 6 that was on the fortune cookie. What people did not realize was the printer of the fortune cookies was lazy and they did not have a random number generator for the six numbers. So the printer printed a few thousand fortune cookies using the same 6 numbers. So purely by chance of course after many years the lotto number matching the fortune cookie came up. The lotto commission discovered many people won using the same 6 numbers and people were disappointed to share the jackpot with many others. At first the commission suspected fraud but soon found out that people were using the numbers on the fortune cookie. The commission talked to the printer and asked them to use more random numbers when printing fortune cookies. So the advise in Canada is to NOT use numbers from fortune cookies because it decreases your chance of sharing the jackpot.

  12. If he stops and starts and umms and ahhs it is probably because he is slowing down his thinking, he has a tested IQ of 180.

  13. How about getting the number as "111" in that case you have all deuce..I hope lottery example outcome is 10P7 not 10C7. Considering repetition is allowed.

  14. In the calculation of the expected value for the Mass Lottery, you used $2/6.8 for the contribution of winning a free ticket. But a free ticket isn't worth $2. As a first order estimate – as you point out – the ticket is worth 80 cents. Using simple feedback loop reveals the true value of a ticket as $0.59.

  15. Badly explained. I very very much doubt that on roll down there was fix returns, I very very much doubt that you would know in advance exactly how much you would get for 4/5/6 matches. Rather, like for the Euro Millions, the money "not won on the jacpkot" in the previous runs is redistributed. This means that the state is not loosing money, the more people play, the more they gain. Even on those roll-down days. It stops when people stops playing on the "regular" weeks because they feel cheated.

  16. It would appear that the 200 or so thumbs down people are all great speakers but……not too intelligent. If you want to learn things in life sometimes you have to forget the medium and focus on the information being transmitted. Great talk. Thank you.

  17. I think that you are making an error in your presentation. The new odds on the roll down day were calculated on the basis that if the same number of tickets were sold as the previous non roll down day then the number of small prizes would equal the jackpot that had not gone down.

    Note that in a normal game if the jackpot goes off multiple times in the same game then the winners have to share the jackpot.

    This is not the case when there are smaller prizes.

    So the reason why the state could not see why the roll down days lost money is that under normal circumstances the number of people that play the lottery is static.

    But with the influx of money from the clubs then assumptions go out the window.

  18. Here's a better idea – Only play the lottery if you think you will lose ! ie Dont play the lottery if you think you will win !

  19. I'm a mathematician and I gave up on this after fewer than 15 minutes. it doesn't help that he says "math" or that he stopped talking about dollars only to replace them with Euros, which we don't have in the UK either, whether we're playing Euro Millions or not. Whatever his point was, I wish he'd have got to it quickly.

  20. Half way in to this video and this guy is still droning on about a lottery in massechusets and still has not addressed the title of his presentation.

    Why are americans always so manipulative.

  21. My favorite Econ teacher would say Ummmm all the time. He was an amazing genius. As a result …Ummmmm…..I am in the top 5% of earners in the USA. i only wish I could be in the top 1% but I grew up in section 8 housing.

  22. At 8:08, the prize for getting 2 matches is a "Free ticket" being sold at $2, and not $2 in cash. The actual worth of the ticket for the winner should be the expected value of the ticket. So the equation should read:
    x = ($ 1 million/9.3 million) + ($4000/39,000) + ($150/800) + ($5/47) + (x/6.8) where x is the expected value..
    Solving for x gives the expected value as 59 cents.
    So much for "Mathematical Thinking"!!

  23. I recently saw an interview on 60 MInutes on CBS and it was on a rather clever gentleman, and his wife, who figured all this out and made many millions of dollars, over the years–and they are making a movie out of it! Coming soon…

  24. Apparently none of the people in the comments has come across his book "How Not To Be Wrong: The Power Of Mathematical Thinking".. Try reading it and it'll remind you how ignorant you are.

  25. Great book… very entertaining read. Got it from the library, but when I finished I had to go buy a copy just to have, cus I love having cool books (especially cool math books). This is a cool one.

  26. better at 1.5x… he already has a chipmunk voice so it seems a bit more natural somehow…Ummm, ya

  27. At around 3:50, I understood that he was actually talking about "rational intuition" : intellectual constructs leading to science and philosophy (real philosophy, not Oprah's "feel good" psychology for lemmings). Mathematics is a "hardcore" consequence of such constructs, a tool that we've invented to aid us for such endeavor in "materialistic" science.

  28. The talk is good. The book is great. I am on the 2nd read and find it to be one of the better maths books I have read. Any engineer or engineering student should get much from this book simply from the exposure to applied statistical mathematics. Thank you for a wonderful book.

  29. Would've been nice if one of the regular players that lost money on that had sued the state for arbitrage

  30. Expected, value IS what one would expect, however on average. So why not just use the term average expected value instead?

  31. He made his explanation of why/how he chose his numbers much more complicated than it really had to be. His picks were simply the set of all 3 digit combinations that include a unique pair of 2 numbers. You will see that no combination will include the same 2 numbers (i.e you wouldn't use 1,2,3 and 2,3,4). There are 7 possible combinations which is why he said to select 7 tickets. Since you only need 2 number correct in order to win, you always win. So I would imagine that the MIT students did the same thing but scaled up and that the number of possible combinations using this method matched the number of tickets the MIT group was purchasing.

  32. Why do people fail to recognise that the abbreviation of the plural word mathematics is "maths".  How can this be so difficult ??    Physics, another plural word has an "s". Do some simple learning !!!   [Aussie in BC]

  33. this guy gave them literally one minute (while still talking) to pick 7 sets of 3 numbers from 1 to 7 before asking for someone to call out. and flashed a set as well. I still didn't understand the transylvanian lottery until he'd gamed it once. Way to ruin an opportunity for fun interactivity.

  34. This is what makes thi "mathematical" nonsense so boooooooriinnng.: the fact that its all done in order to figure out how to win at idiot games. My solution : Who Cares ? Maybe if he applied it to something interesting, it might be fun to listen. I'm leaving.

  35. TLDR; "I'll tell you a half story about how young kids at MIT won the lottery several times and you have to buy my book to know how it ended"

  36. I consider myself VERY fortunate to have been exposed to formal logic in an 'advanced' 9th grade class, as a subset of… GREEK PHILOSOPHY?!? This stuff should be taught to EVERYONE, starting in the third grade, until everyone 'gets it.' The only problem would be, it would kill off the advertising industry, people would have to make products that WORKED right, not sound attractive. The churches would take a big hit, too. I went on to study logic more deeply, out of interest, but jeepers, you can really get the gist of it spending several minutes reading up on 'logical fallacies' over at Wikipedia, then train yourself to APPLY it. Oh
    wait… what he's talking about the lottery is the tip of the iceberg, our entire economy depends on a whole bunch of people acting really, really stupid.

  37. "There's" is a contraction of "there is" and refers to a singular person, place or thing. Your grammar indicates that you do not understand the notion of "singular" and "plural". "Plural" refers to more than one person, place or thing. "They're" or "there are" is the word that you seem unaware of. "They're more than you can imagine…".

  38. You don't need any weird geometry to hedge your bets on the lottery tickets, this overcomplicates things dramatically. You just need to avoid picking the same numbers too much and not picking other numbers enough. As a programmer, I find this natural… as you generate tickets, just keep track of how many times you've picked each number, and always pick from the least-picked ones.

  39. At around 10:10 he says the European lottery has a 0.40 expected value so its worse than the Mass lottey, but 0.79 cents on a 2$ ticket is also 0.40 expected return… Not very good math for a math guy (still enjoyed the video)

  40. An interesting talk.
    I wonder what would be the shape of the connected lottery number geometry be capable of being represented in 2d?
    Always been fascinated by geometry and the nature of geometry. It's good to know new insights in geometry are being made all the time.

  41. Just confirms the saying that lottery money transfers money from the poor to the rich, not vice versa.

  42. To Ask, Is the Riemann Hypothesis true? Is akin to asking, is the Prime N2, the only even Prime out to Infinity?! Of course the answer to both questions is YES! And we prove it by Intuition, Logic & Common Sence. -1+0^0/2+1. Qed.RH.TM PROVEN!!!

  43. Lovely stories. Making logic and math interesting. He knows and loves his subject. The negative comments must be some form of jealousy….

  44. lol you may not download academic articles on a guest account, but we don't mind that you scam people out of millions of dollars of lottery investments
    mit logic

  45. google is the same thing. fist they put their friends at the top.. so your business gets cheated., then the monopolynsotes rais prices everywhere so you pay AGAIN.. then they demand everybody do SEO optomization, whichngets passed on to YOU again..
    but heres the mathematical insight.. youre not just paying thee times..
    THE MARKET BELONGS TO THE PEOPLE.. somthat money should go,to the state.. so,younhave to pay again Three more times.. the missing taxes must be made up by you. Google is theft SIX times over. totally rapeing the u.s. marketplace. ..

  46. WAIT!! So why does it make sense to fill in lots of tickets by hand and not automatically?? I missed that part. Did someone invent an algorithm to pick random numbers that is more likely to win a lottery than mere chance?

  47. Philosopher: Is it right to make weapons of mass destruction?
    Scientist: How can we make bigger bombs and cause maximal damage?

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