maa. . .

so in the book how not to be wrong I talked about a lot of mathematical notions which on the one hand are not so complicated they’re simple to explain but they’re very broad in their application and deep in their meaning and one of those things what I want to talk about right now is the idea of expected value and this is the fundamental way that mathematicians talk about the value of something whose true value is unknown or subject to uncertainty subject to some process of chance so a typical example of such a thing a lottery ticket a lottery ticket is something that you buy and you don’t actually know how much it’s worth because you don’t know whether it’s a winner or a loser so for instance you could imagine a lottery ticket where there’s a one in a hundred chance that the lottery ticket comes with a fifty dollar prize but a 99 and 100 chance that the lottery ticket comes with no prize at all because the numbers are pulled out of the cage and they don’t match what’s on your ticket so what’s the expected value of that lottery ticket well here’s one of these many places where mathematical terminology kind of diverges from the usual use of the English language because the value you expect that ticket to have is zero that’s by far the most likely outcome that the numbers pulled out of the cage will not match your ticket but that’s actually not what we mean in mathematics by expected value it might have been better if we were deciding on the notation from scratch to call it average value because what it really measures is if you had a lot of those tickets let’s say you bought one every day for a year how much would they be worth on average so if you bought a hundred of those tickets or if you bought a thousand let’s say probably ten of them would be winners right because it’s a 1/100 chance those ten winners would give you five hundred dollars out of your thousand plays so they would be worth on average fifty cents each and that’s what we mean by expected value the expected value of the ticket is 50 cents this is why it’s kind of a funny term because 50 cents is not even a possible value that it might have it might be worth noting or it might be 50 bucks nonetheless are we talking about its expected value 50 cents is what we mean and this is a very useful notion for determining how much you should rationally be willing to pay for the ticket because if that ticket cost you two dollars to buy and you buy one every day on average you’re going to be spending $2 in gaining 50 cents every one you do and that is not a good bet and that in fact is the way most real life lotteries are structured in a sense they kind of have to be that way because if the average person was making money playing the lottery then the state which ones the lottery would be on the average losing money and then there would not really be much point in the state doing it so I write about this in the book and when I was researching this figuring out what to say I came across a rather interesting story was a story from the Boston Globe from 2012 and it was about a group of students at MIT who had found a way to win money reliably in the Massachusetts state lottery they went about three and a half million dollars this is very strange because you’re not supposed to be able to win the lottery even if you go to MIT because there’s no real strategy in the lottery right you buy a ticket as a ticket how on earth could they have been winning so this seems like a thread to pull so I went and learned about this it turned out that the Massachusetts lottery had changed their rules people were getting discouraged people hadn’t been winning people were had stopped playing so they changed the rules to make it seem like a better deal for the player and when they did that they did a little bit too good of a job they actually made the game really be a good bet for the player only on certain days not on all days um but the MIT kids and it turned out actually not just them but other groups of people had figured this out and what ended up happening is that these lured these groups of betters including the MIT students we’re buying hundreds of thousands of tickets at a time for each of these drawings and they were actually comprising something like eighty to ninety percent of all the tickets that were bought so the whole lottery had become kind of a private deal between the state of Massachusetts and these small groups of betters so this was an interesting story in several dimensions the first thing I wondered when I was reporting on this and studying it was how did the state not figure out about this I mean like the state knows who wins they knows which tickets are paying out so they could easily have seen that the same convenience store in Cambridge was reporting tens of thousands of winning tickets every time that’s sort of not a normal pattern than you might see by chance and when I dug into that what I eventually figured out was that it’s very simple to explain how the state didn’t know it’s that the state actually did know which leads to another question if the state did know why did they not care that these guys were taking all this money and the answer is that from the point of view of the state the lottery is like a tax the state gets 80 cents for every lottery ticket that’s sold and the state doesn’t care who wins doesn’t matter the state takes its 80 cents the rest of the money goes back out in prizes and the state doesn’t care who it goes to so paradoxically enough the fact that those guys were buying so many tickets was actually good for the state they weren’t taking money away from the state they were taking money away from the other players that was one puzzle the second puzzle ended up taking me in a lot of surprising mathematical directions which was this the other groups of players which were one group of a family from Michigan and then a dr.John lottery Club which is a bunch of biochemists from southern parts of Boston they used what’s called the Quick Pick machine it’s a little computer that kind of just picks random numbers for you and will tell you however many tickets you want the MIT cos did not do this they chose their numbers and filled out their tickets by hand 200,000 tickets that’s a lot of work and you’ve got to ask yourself why they did that using the mechanism of expected value every lottery ticket has the same expected value so it shouldn’t matter which tickets you choose why not just choose the random ones um and this I found quite strange so I spent a long time thinking about it and what I eventually came to understand must have been the case I did find these guys but they wouldn’t tell me so then I got obsessed and I had to figured out myself um what I came to understand is that if the question is what’s the expected value what’s your expected winnings on average then it doesn’t matter which tickets you have but if you ask about risk if you ask about how much variance in there is there and how much you might win then which tickets you choose really does matter and in fact there’s a way to choose the tickets that actually guarantees you that you cannot lose so it’s like not gambling at all in some sense it’s literally guaranteed payoffs and that’s quite important because after all if you’re buying 200,000 tickets if you’re sort of borrowing large sums of money from all your friends to go pay play the lottery you better be pretty sure you’re gonna win you probably don’t want to make that play lose hundreds of thousands of dollars of your friends money and then say well mathematically speaking on average will eventually come out of the head that’s not such a great play for a bunch of college students to do so in the book this takes us through the realm of finite projective geometry of perspective in painting and eventually to error correcting codes in the development of information theory all by way of talking about a beautiful mathematical object called a combinatorial design which you can use to select large groups of lottery tickets which are oh so cleverly arranged so that you are guaranteed to have so many winning combinations that you can’t actually lose money as I said I was never able to get these guys to tell me what their strategy was so I don’t know if the combinatorial design I found is what they actually used but if it isn’t I think it’s what they should have used. .

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