Oxford Sparks

Emily Elias: The most that I’ve ever won on a lottery ticket is, like, maybe £2, and that’s a big deal for me. If you want to hit that jackpot, well, get in line. In this episode of the Oxford Sparks Big Questions Podcast, we are asking, can you predict the lottery? Hello, I’m Emily Elias, and this is the show where we seek out the brightest minds at the University of Oxford and we ask them the big questions…and for this one, we have found a researcher who wants to help make chaos theory make sense when it comes to the lottery.

Dr James Munro: Hello, I’m James Munro, I’m the admissions coordinator for Maths at Oxford, and I also teach a little bit of maths at New College.

Emily: And today you have decided to grace us with your brilliance to talk about the lottery.

James: That’s right. I’m looking forward to talking about, uh, randomness and chaos and selecting things at random.

Emily: Before we talk about all the randomness and things, can we just maybe start with a little primer? For those who are not initiated, how does the lottery work in the UK?

James: Great question. So you choose six numbers from 59, and then they draw numbers from a machine, and if those numbers match yours, you win a prize…except they’re changing it this year. At the time of recording, they’re changing it. So there’ll be two rounds – there’ll be round one where they get the machine and they draw some numbers, and then round two, where they get, I think, an identical machine and they draw six numbers from that machine.

So they’re saying you get two chances to win – once with the first machine and then once with the second machine. The rules don’t say what happens if you win both of them. If your numbers come out with the first machine and then they switch on the second machine and the exact same numbers come out again and you win twice! I think you just get the one jackpot.

Emily: So when does the new lottery rules, when do those come into effect?

James: They come in effect from the 7th of June, 2026. They’ve been switched to the new lotto. A once in a generation change using these two rounds. Round one and round two, two chances to win.

Emily: Okay. You say two chances to win like a marketing person! Does that mean I would be more likely to win the lottery? What are the odds of me winning in the old version versus the new, updated, two machine version.

James: So you are right, I am saying the marketing thing at the moment. Of course, it’s more complicated than that. Before, the chances that you chose the right six numbers that would come out of the machine were about one in 45 million, and with two machines, well, your chances have doubled.

So now it’s pretty much one in 22 million. The complication is that the definition of winning has also changed a little bit as well, and that being twice as likely to win the lottery has been somewhat balanced by the fact that what you win under the new lottery rules is going to be slightly worse.

Emily: So I’m not going to go home with the big satisfying jackpot necessarily.

James: Well, the rules from the jackpot have always been a bit interesting. You’re required to share it with other people who picked the same numbers as you, other people who have won the jackpot that week. And with both machines running – round one and round two – the jackpot gets shared across both of those rounds. So you might win in round one and be happy, but somebody else might then win in round two and you, you have to share the jackpot with them.

Emily: So the odds are slightly better that I’ll win something, but the jackpot remains one jackpot. Is there a strategy that I should go into this with picking numbers? Is it the best thing to pick 1, 2, 3, 4, 5, 6? Or should I be opting for something else?

James: Well, I suppose, if you tell everyone you’re picking 1, 2, 3, 4, 5, 6, and if I say those are great numbers, then well, if everyone picks those numbers, that will be part of the sharing of the jackpot that will happen when those numbers come up: 1, 2, 3, 4, 5, 6. Lots of people will share. So it’s going to be, I think, more important that you choose numbers, which ideally no one else has chosen. They might be special to you, but my general advice is not to pick numbers that are based on a particular date, because other people might have other things going on for that date, and not to pick numbers that are just generally popular.

Emily: But I mean, you’re a mathematician. You’re a man of science, you’re a man of numbers. There’s got to be some sort of theory behind this that explains what is happening when these machines whirl all these balls around and spit numbers out, to make any form of sense of the world, surely?

James: Yeah, so it’s science we’ve understood for hundreds of years, right? Isaac Newson knows about how things falling under gravity and bouncing off the sides of things. Isaac Newson knows how that machine works, even though he’s never seen one. The physics is quite easy to write down, but then the behaviour of the machine, the fact that small changes in how the machine is set up and small changes in where the balls start, or changes while it’s running – a slight wobble or a gust of air – they have big consequences, those small changes. That’s an example of something called ‘chaos theory’, where very small changes can have massive effects later on. Where here, the massive effect might be you matching two numbers and winning a lottery ticket versus matching all six and winning a slice of the jackpot.

Emily: And I suppose if you have two different machines, then two different things could happen to slightly affect them. Like, there wouldn’t be a scenario where – since you have two machines, next to each other – they would both spit out the exact same numbers?

James: Right, brilliant. This is an effect called the butterfly effect, where you imagine a really tiny change, perhaps it’s caused by a little butterfly, that goes on to have larger and larger consequences. The butterfly flaps its wings, and then the weather is different somewhere completely different. Here they’re going to set up, I think, two identical machines next to each other and put the same numbers in each of them. But I’m not expecting to see the same outcome from those two machines, because tiny differences in how they set this up will of course be amplified until you get different outcomes, almost always.

Emily: And what are these tiny differences that we’re actually, sort of like, sweating over? Stuff, like the temperature of the room, or if one of the balls is slightly twisted in a different way in one machine than the other? That sort of thing? Is that what we’re talking about?

James: Yeah, sure. The machines involve stirring, so there’s a motor in there. There’s things that are moving in slightly different rates. There are things about how they set the machine up, like you’re saying – maybe there’s slight angles and positions involved. All of those are things that you could imagine measuring and probably they don’t even give people access to the machine to go in and measure everything before they start. But even if you did, even if you had really good information about the setup of the machine at the start, your predictions would break down at some point because of chaos theory.

Emily: But like, okay, so there’s no historical precedence of like a lottery having the exact same numbers twice in one go… It’s usually each time the numbers are different?

James: So I want to say it’s never happened… that the numbers coming out of the machine are different every time because of chaos theory. But this one time in Bulgaria, they ran the lottery and twice in a row it gave the same numbers. Now, some mathematicians online have argued that this is an example of the birthday paradox, that if you run lots of lotteries, then eventually two of them will have the same outcome. There are only so many outcomes and you get coincidences faster than you might expect. However, I’m not sure that that analysis really grapples with the fact that it was two consecutive lotteries in the same country. They were back-to-back lotteries. So that sparked allegations of fraud, and that was investigated at the time in Bulgaria.

Emily: Okay. But what if I decided on a completely different tactic and I just went out there and I bought every single combination of numbers that exists, that these six numbers could possibly be. What if I bought all the lottery tickets? Well, obviously I would win the lottery, but would it be worth it, the expense?

James: You would double win the lottery under the new rules. You would win round one and round two. Yeah. Even if different numbers come out because you’ve bought every ticket, all 45 million possible combinations of six numbers chosen from 59. The problem with the lottery is that the jackpot’s never really large enough to justify that. Buying all of the tickets costs you, I think, about £90 million, which is quite a lot of money. The jackpot this week, at time of recording, is about £7 million. So the payoff is not really there for it to be worth doing.

And that’s not even thinking about all the admin of going out and buying all of the 45 million tickets!

Emily: Is there a world where it has happened, though? That people have gone in and bought all the lottery tickets – the jackpot was just worth that amount of money that the calculation balanced out, that buying all the tickets made sense?

James: Yeah, so, a group of people did this in 2023 to the Texas lottery. It’s a consortium approach where people pulled resources. Controversially, they had access to machines to help them print millions of lottery tickets so they could get close to having every single ticket. I think they won $95 million on that from the Texas lottery, and it’s currently being litigated.

How and why they had access to the machines for printing all the tickets. But that’s a case recently where the expected value was positive and somebody swooped in and claimed the jackpot. It’s being litigated – partly internally in Texas and partly the person who won the following week, who would’ve won a big rollover jackpot if it hadn’t been claimed the previous week. That person is litigating as well.

Emily: No, I could see that being a very complicated world. But if you haven’t put people off about trying to go for their big Lotto win, keeping all of the stuff that we know about chaos theory in mind, is there anything you could try to do to win the lottery and predict these numbers?

Or is that just a fool’s errand?

James: Well, so predicting the numbers is really difficult. If you want a version of this strategy where you guarantee some sort of a win, then there’s a strategy where you purchase 27 tickets. Some mathematicians at the University of Manchester have worked this out. You purchase just 27 tickets and you can guarantee yourself a win at the lowest level where you match two numbers.

They did some pretty good combinatorics to work this out, but with just 27 tickets  spread across the different numbers that might come out of the machine, you can guarantee that you match a pair, and guarantee that you are, in the lowest sense, a lottery winner. The prize, I think that you win currently for matching two is that you win another lottery ticket.

But maybe that one will win you the jackpot!

Emily: When was the last time you bought a ticket?

James: I’ve actually never bought a lottery ticket! I am waiting for the time that it can go so high that buying one lottery ticket will guarantee that I get some positive return on expectation. In some sense, the only way to win is not to play, unless it’s the case that you enjoy playing and that putting money into the lottery and having that chance of winning a million pounds or winning more lottery tickets or being in subsets, part of something that’s happening across the nation, maybe that means something to you. Or maybe you like the way that some of the money from the lottery that doesn’t go back to you goes to charitable causes. The lottery in the UK is supporting lots of arts and social causes, as well as returning some money to some small number of jackpot winners, and those things might be important to you.

Emily: Good luck to you on all your future lottery endeavours, if you decide to take the plunge.

James: Mm, yes. Well, I’ve got to work out the tech for buying all the tickets. Get ready for my version of the heist movie, right?

Emily: This podcast was brought to you by Oxford Sparks from the University of Oxford, with music by John Lyons, and a special thanks to Dr James Munro. Tell us what you think about this podcast. We are on the internet @Oxford Sparks, or you can go to our website, oxfordsparks.ox.ac.uk. I’m Emily Elias. Bye for now.