How MIT students used mathematical thinking to win lotto

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 gotta tell a longer story one that actually I would probably take about 40 minutes to tell us to tell it maybe like a third of a very long story that's uh it's in the book and the story has to do with the lottery in the US 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 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 gonna 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 1 in 200 chance that you win $300 back or should I change the currency Bothwell sorry I didn't think to do that ok well the story takes place in United States so if you play this game a thousand times and people who play the lottery do play a thousand times and 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 you're gonna win well of course there's variation in that it's random but if there's a one to 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 $1500 sounds pretty good until you think about the fact that you spent $2,000 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 gonna win on average per play so that's the mathematical term I gotta 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 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 it's not even a possible value for the ticket to have write that ticket it's either worth nothing or it's worth $300 but it is definitely not worth a dollar in 50 Cent's so now if we had it to do over again we would probably call this the average value but is a much more reasonable summation of what we're actually trying to describe a dollar 50 is how much the average ticket is worth and the other hand 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 or something that's worth a dollar fifty and here you have in a nutshell the mathematical case against plain water and now I'll complicate that a little 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 on you don't have to check it for yourself but what I want to point out is then 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 Jackson the real lotteries have like a whole sequence of lower tier prizes some of which like matching three out of the six numbers in the messages 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 hot if you play every day probably every so often you're gonna 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 lots to do with psychology so that's one way that my simplified example is unrealistic to few tiers of prizes the other way my um my simplified model is unrealistic is there was incredibly generous to the flyer to the players no real lottery pays back at dollar fifty for every as the average value of a $2 ticket 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 it's anybody play euromillions that's a forty cent expected value on it to breath well 0.4 of a euro expected value I'm a two euro ticket that's insanity that's why this would never be tolerated in United States I just want to tell you guys even American lottery players would be like that it's a stinker of a game people who like play this everyday we'd like not play your Emily okay 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 about 2004 2005 a whole year went by without anybody winning the jackpot and they could see it 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 because if nobody wins the jackpot the jackpot pool gets bigger that money that is not given out and 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 gonna win they said let's make a new rule if that jackpot goes over 2 million dollars and nobody wins the jackpot that drawing then it's gonna roll down all that money is gonna 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 from I can always tell how matthean audience is by how big a laugh I can get with a table that's always up so this is what the payoff matrix for for the Massachusetts cash windfall lottery looked like on February 7 2005 so for instance have four of six prize there's a one in a hundred chance of winning you remember in a usual of drawing that was about $150 prize on this roll down day in which no one won the jackpot that prize was actually worth almost $2,400 to stop to think about that there's a one an 800 chance of winning and the prize is worth twenty four hundred dollars to a few about eight hundred tickets you were probably gonna win $2,400 right there for the $1,600 you spent buying 800 tickets and that's just the floor 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 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 hails 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 with 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 in length here I'm gonna tell it to you somewhat briefly what happened well what happened is that on February 7th 2005 the state lottery started getting phone calls they got a phone call from a star market in Cambridge Massachusetts for the convenience store saying some college kids just came in and want to buy 5,000 lottery tickets is that okay so there's a rule you know if somebody from 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 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 - 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 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 and 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 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 at the end so they sort of coalesce 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 up and hear about it well where did Massachusetts get the idea for this role down rule they got it from the roll down game in Michigan that had just closed I'm not sure they asked Michigan Wyatt close but jerry selbee knew why closed because he had made about two million dollars off the Michigan game over the previous seven years or so so he was you 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 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 investments game and buying yet more until by the time this meets 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 we're being sold to a member of one of these three groups so how does this story end 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 and once people perceive that the game is not what it seems then people stop playing and then it doesn't work anymore then no more money flows into the system so so in a way that's the end of the story but if you read it from a mathematical point of view the quanta logical end of the story is not really the end right because as my petitions there are some puzzles that remain at least they remain for me when I was like reading this story and I find to read with Matthews eye and trying 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 of the easy one is hard let's start with the easy one the first puzzle is how could you actually get away with this this is a little weird but let me remind you that the state knows who's winning the lottery right because we have to give you the money so it's not a secret the state knows that all the winning members are coming from the same three convenience stores again and again let me remind you something else and 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 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 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 that the new Massachusetts lottery game had a positive expect to tell you 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 you're 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 and knock yourself out because the next thing that happened was when 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 quit the it spawns the question why didn't the state do anything about it if this thing 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 Brandon's product because I should say is the name of James Harvey and you Ron and James Harvey's team it's the name of his group of people um you might have checked that their strategy was really not very random at all it's that 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 that was where the money was coming from so how did you think of what's going on here I want to remind you one more very important thing 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 gonna get dispersed in prices in one form or another so what does that mean that means that for 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 existing so I think it's safe to say that somehow have 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 are playing the lottery on the non roll down days so that's what this I 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 are playing only on the roll down days with Massachusetts getting 8 80 cents every time a ticket is sold maybe a good analogy is like this 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 some 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 talking about in the breath they said how did the kids and on mighty 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 50 cents for $2 ticket it 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 gonna win some of them are gonna lose but if each one is slightly tilted towards the MIT kids then on the whole they're very likely gonna 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 method from a mathematical point of view is to compare it to the following diagram which is exactly the same the kids from MIT or the other high-volume betters are 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 when 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 have license to the 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 point out which is that jerry selbee and dr. john use 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 a machine that picks random members for you to play and that seems like a good idea right because we all know that you can't predict what numbers lottery or you might come up with any numbers as good as other if you're gonna buy 200,000 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 convert their name did not do this they filled out their tickets by hand 200,000 of them why why do this this is a humongous pain and you know the inspector general's report mentions if they did this but didn't say why and it 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 meth additions when were faced with the problem we don't understand the first thing we do is you try to make it simpler we try to replace our problem with a simpler problem hopefully which has the same features and not the same features the original one that we can use it to gain some insight so that's exactly what I'm gonna 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 3 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 members out of seven for the combinatorics fans in the audience the number of these is 35 which is seven choose three and it's called seven choose three because the number of ways of choosing three things out of seven 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 you

Channel: Edzai Conilias Zvobwo

Views: 30,036

Rating: 3.5599999 out of 5

Keywords: lottery, gambling, mit, win, jackpot, play, math, statistics, data, analysis, understanding

Id: -WM3Yozsdok

Length: 43min 0sec (2580 seconds)

Published: Wed Dec 07 2016