Mathematician and author Adam Kucharski talks about his new book *The Perfect Bet: How Science and Math Are Taking the Luck Out of Gambling* (Basic Books, 2016).

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Mathematician and author Adam Kucharski talks about his new book *The Perfect Bet: How Science and Math Are Taking the Luck Out of Gambling* (Basic Books, 2016).

** Steve Mirsky: **Welcome to

Adam, what is the perfect bet?

** Adam Kucharski: **In simplest terms, I would probably say the perfect bet is a scientific bet, really looking throughout history at how people have taken on these games and really looked for loopholes and ways to beat the house. There's obviously been a lot of systems or magic formulas and superstitions going around, but really the most successful strategies seem to have a solid basing in science, really tackling those gambling questions as if it's a scientific question and using experiments and analyses accordingly.

** Mirsky: **So the perfect bet is trying to take the gambling out of gambling.

** Kucharski: **In a way, yes, exactly. It's trying to separate this element of luck from games which traditionally are entirely viewed as games of chance, so things like roulette or card games which are really seen as complete gambling, and finding ways to bring an element of skill into that.

** Mirsky: **You talk in the book about the DiCristina case, which caught my eye because our editor-in-chief has married a DiCristina, who I don't think is any relation to the DiCristina who you talk about. But what's fascinating about the DiCristina case, first of all, why don't you talk about what the DiCristina case is.

** Kucharski: **Sure. So this was back in I think it was 2011 when all this was coming up, and it was a real crackdown on the poker industry in the United States. There had been a huge expansion in playing poker. In New York, there was one case brought against someone who had been running a poker room, and this was seen as a form of gambling and so the case went to court. But, unlike many games where in the law they're specified as gambling, poker isn't, so it was really up for debate as to what element of gambling was involved. Under federal law, gambling is defined as something which is predominantly a game of chance, so you had this fascinating legal case which essentially relied on determining whether poker is predominantly a game of chance or predominantly a game of skill, and you had economists and people with a lot of knowledge of these kind of games of _____ games of chance arguing over whether poker is indeed something that's predominantly by chance. In the end, the judge ruled that there was enough element of skill in the game for it not to be defined as gambling. This was the first time that a judge had ever ruled on whether poker was a game of chance or not.

** Mirsky: **Although there's a short story by Mark Twain written over, well more than 100 years ago because he's been dead for at least 100 years, that is the exact same scenario. A guy running a poker game is brought in for violating the law because it's a game of a chance, and he says, "No, it's not a game of chance. It's a game of skill," and he demonstrates to the judge by, if I remember it right, the judge tries to play poker against cardsharps and never wins, and so it's clearly a game of skill.

** Kucharski: **It's fascinating. I think these kind of situations have cropped up a lot, and the judge in the ruling of that poker case mentioned this fascinating case back in the 1920s where someone had stolen an airplane, and they were convicted for grand theft auto. But, at the time, grand theft auto did define vehicles as vehicles running on land, and obviously that didn't cover aircraft. At the time, the ruling made the point of we shouldn't just assume something is the same just because at face value it kind of looks like that. I think it's the same with poker, that just because it resembles other games that are games of chance, we shouldn't just automatically assume it's gambling. I think there's a lot of debate potentially to be had there about what actually constitutes gambling and what is predominantly skill.

** Mirsky: **Let's talk about the many ways that people have tried to make what appears to the layperson as gambling something that's not gambling. You have many examples in the book. Computer programs, obviously card counting in Black Jack, lots of ways to take the chance out of games of chance.

** Kucharski: **Exactly, and I think card counting, as you mentioned, is one of the really good examples. Black Jack is this game which on the face of it is random. You've got cards being dealt by the dealer and your objective is to get to 21 and not going over, and then beating the dealer. Back in the '50s, some soldiers actually out at the Aberdeen training ground were looking at these games, and they realized that because the dealer has one card face up, if you account for what the dealer has when you're making your decisions, you can improve your chances at winning. Unfortunately for them, their perfect Black Jack strategy still resulted in a loss over time, so it wasn't beating the house, it was just making the house have a very low edge. But Edward Thorp, who is a mathematician, subsequently found that if you take into account which cards have been dealt before, then actually that could give you an edge because of course card deals aren't random. The cards that appear in a particular round depend on what's come before because the same card can't appear twice in succession. With that system, he showed that that can be a very effective way of gambling.

** Mirsky: **Basically, the more information you have, the better chance you have to put the odds closer to your favor.

** Kucharski: **Exactly, and a lot of these games are about information. It's about taking advantage of data and information that are there which other people perhaps haven't spotted as useful as they could be.

** Mirsky: **The card counting situation, for example, I think the book the example you use is if the dealer's card is a six versus a ten, that completely changes your strategy as the other player.

** Kucharski: **Exactly, and the dealer has to follow fixed rules in Black Jack. They have to draw up to a certain point and continue, and so if the dealer has a six, they're more likely to have to draw more cards, which will send them over 21 and go bust, but if the dealer potentially starts with quite a strong situation, you don't want to take so many risks against them because there's a stronger chance they'll come out on top. So it's really about adjusting your strategy based on what the other person is doing and what information you're seeing.

** Mirsky: **So the casinos figured out that some people were really good at this, and they started to institute multi-deck Black Jack games, where there might be five or six decks in play at the same time. I mean this is a famous scene in the movie

** Kucharski: **I mean perhaps there are some people out there with that level of memory, but I think for the average person, that's incredibly difficult to card count, and of course the more decks there are, the less useful that information you had about past things. But an interesting footnote to this story was a lot of the gamblers realized that they could change their strategy, so if you're dealing with six decks, it makes it much harder to card count, but it also makes it much harder to shuffle those decks properly. At the time, the casinos were doing these riffle shuffles, where you separate it in two and then riffle them together, and they were only doing one riffle shuffle. If you think of a deck of cards that's in perfect order, if you riffle shuffle it once, all you're going to do is offset the cards by either one or two slots, so actually there's a huge amount of information still there. Many of these card counting groups instead used hidden computers to record the order of the deck, and then when this one riffle shuffle happened, well, each card that's coming out they knew could only be one of two options. So, inadvertently, the casinos had given them even more of an advantage by changing the game that was meant to stop them.

** Mirsky: **Yeah, it didn't come out, I mean I remember when the papers came out maybe 25 years ago about how many shuffles you actually need to truly randomize a deck.

** Kucharski: **Exactly, and I think it's about six shuffles for a riffle shuffle, and actually that's the really fascinating part of _____, all this how much shuffling and how much kind of disorder they need to introduce before something is not predictable.

** Mirsky: **So these computers that the Black Jack players were using to track the deck, they're wearable, tiny instruments. Talk about how these guys would smuggle them into the casinos and try to use them.

** Kucharski: **In many cases, they would be wearable, as you mentioned, hidden computers, and they were used for Black Jack and used for games like roulette, as well. In some cases, they were hidden in shoes, so you would be kind of tapping away at your toes to record the information and have the computer kind of crunching out what was likely to come up, and this required a huge amount of ingenuity. Actually, the first ever wearable computer was designed for Black Jack back in the '50s, so historically, these kind of things are really important in terms of technical development. Then, likewise, a lot of the earlier _____ they were amateurs doing this, and a lot of the technical problems, like I think a couple of times the gamblers got electric shocks when these things didn't quite work. But I think just as a sort of testament to the amount of innovation and just intellectual curiosity behind these things, these guys were building computers and doing things that had never been attempted just for the sake of beating a game. They weren't making a huge amount of money at it, but it was just the challenge of beating these kind of puzzles.

** Mirsky: **Right, and these people tended not to be, in fact I don't think any of them that you mentioned were professional gamblers first. They were scientists first who thought of the casino as an excellent laboratory.

** Kucharski: **Exactly, and throughout history, and one of the really interesting stories in the book was Karl Pearson, so he's a very prominent statistician, and really a lot of the concepts we use about sort of testing hypotheses come from his work. A lot of his early studies used casino games as his laboratory. He saw Monte Carlo as just a source of random data to develop his statistical methods on. Actually his first, the concept of a P value, so what's the probability of observing an event as extreme as the one you've actually observed. It's a good way of seeing whether something is actually plausible in reality or whether the data you're getting suggests that there's something non random going on, and that was actually developed around a lot of these games of supposed chance in casinos.

** Mirsky: **We often hear of some study that used the Monte Carlo method.

** Kucharski: **Exactly, the Monte Carlo method. Another example coming from card games, so this was during the Manhattan Project, so Los Alamos, there was a mathematician called Stanislaw Ulam who essentially didn't like the kind of long analytical equation-solving that's involved in a lot of math, and he realized during one card game it was easier just to lay out the cards and see what happens. If you've got this random process, rather than trying to calculate all the probabilities and do all the equations, just lay out some cards and see what happens, and that computational technique he developed had subsequently made it into computer graphics, into all sorts of simulations and forecasting. It's this technique of rather than trying to solve the mathematics, just generate some random simulations and see what happens, and that's actually a very powerful tool in modern technology.

** Mirsky: **We see this a lot now in, for example, sports analytics, that you don't theorize about what's going on, you just collect huge datasets and let the data tell you what's actually going on.

** Kucharski: **Exactly, and there's a lot more data obviously. I think the U.S. sports in particular has always been very good at having a lot of data and statistics. I think in the U.K. in particular, so with kind of soccer predictions, that really lagged behind just because the data wasn't there. I think in the U.S., people are almost a bit spoiled with the amount of statistics they had. Then of course it becomes very important in how you interpret that data, so you have these syndicates. You have, for instance, horseracing, huge amounts of data on racing, but how do you actually interpret that and convert that into something which predicts a winner?

** Mirsky: **The problems become much more difficult when it's not just two participants. I mean in poker, it's much easier if you just have two players against each other to figure out the odds. If you have a horserace and it's one horse versus another horse, but once there are eight horses, or five players at the table, things get really complex.

** Kucharski: **Exactly, it becomes a lot messier from a statistical point of view, and poker, as you mentioned, is a heads-up poker, which is this two-player game, there's been a lot of attention on because it's fairly neat from a mathematical point of view. You've got what you're doing, what the other person is doing, you don't have to worry about the other situations. Talking to all these teams who work on a lot of the theory of poker, so these strategic techniques, they point out that situations where you've got two people, if one guy wins, the other person loses, but as soon as you have more people, you can get sort of coalitions and people inadvertently ganging up on a particular person.

** Mirsky: **You quote early in the book, I think it's Nick the Greek, the famous gambler, and somebody, I forget who, asked him, "How do you win so consistently?" and it's because he said, "I'm not playing against the house, I'm playing the other players at the table and I know who these people are and I know how they behave." So for him, it was like a psychology experiment.

** Kucharski: **Completely, and that was actually Dick Feynman was the physicist. I think he liked Vegas for the girls and the food but didn't like the gambling so much. But he was couldn't understand how you had a guy who made his money as a professional gambler, and, as you mentioned, Nick really understood the odds so well that he could actually take people on in side bets. People had these superstitions and were trying to place these kind of big money on things which weren't optimal, and that was really the case in lots of gambling. Subsequently, people have done well, and just in a lot of other areas of life, people who understand situations and understand the odds a lot better generally can perform much better in those kind of situations.

** Mirsky: **You mentioned before the historical aspect, and the very beginning of your book talks about the fact that nowadays we think of probability theory as ways to analyze these games. But, historically, probability theory came from gambling, because gambling has been around for thousands of years, probability theory has not been. So, when mathematics started to become really sophisticated in, well, the Middle Ages and then into the Renaissance and the Enlightenment, you really had some people like Pascal starting to analyze what was going on in gambling, and that's where we get our modern probability theory.

** Kucharski: **I found this really surprising, actually, _____ in the book, because I think there's a lot of quite well known examples of people using math and science to beat the house with card counts and this sort of thing, but really, the relationships work both ways. It's not just gamblers using science. In many cases, science has benefited enormously from people studying the house. Back in _____, actually, probability theory was developed to study these games. I mean can you imagine having bets where it's not actually clear what a fair game is. You know, if I was to come up with two sixes, that's just good luck. There's no way of measuring the possible outcomes and what could happen. _____ this theory from probability statistics, and actually more recently things like game theory and _____ theory originated with studying games of chance. I think science actually has benefited a lot from people's curiosity about gambling.

** Mirsky: **Yeah, you have a specific example, I think it was Pascal, and the question was is it more likely for me to roll a six with one die in four chances, or to roll two sixes with two dice in 24 chances, which if you just look at the arithmetic of it, it looks like the same bet but it's not –

** Kucharski: **Exactly.

** Mirsky: ** —from a ballistic point of view.

** Kucharski: **Exactly, and that had been around a while, those kind of bets and those kind of questions, and so it was Fermat and Pascal who developed _____ theory, and one of the crucial things was this concept of an expected value. You know, if you play a game repeatedly, what do you expect to win on average? Until you have that kind of theory in place, it's very hard to actually compare two bets directly and work out which one is more preferable.

** Mirsky: **Right, so you're really in the dark until you have all the data necessary to know what the true expected value is.

** Kucharski: **Exactly, yeah, and until you can do those calculations, and if you can imagine from these kind of _____ gamblers' point of view, as soon as you have those kind of methods, you've just got a huge advantage over anyone else you're betting against.

** Mirsky: **Absolutely. In fact, now that we're talking about expected value, let's talk about some of the lottery cases that you discuss in the book, because for certain lotteries, there is an expected value for your ticket that suddenly makes it actually reasonable to play the lottery. We actually had this just recently, and I forget what the Powerball was worth, $1.4 or $1.5 billion, and once it got to a certain level, your $2.00 ticket actually had an inherent value over $2.00, not by much, $2.02 maybe. It was still a sucker's bet, but from a computer mind point of view, now it's reasonable to at least buy one ticket because you have potential value greater than – I mean, obviously, the big potential value is $1.5 billion, but your actual value of that ticket before the numbers are drawn is more than $2.00. You talk about these particular lotteries. They tend to be fairly low in value compared to the Powerball, they're only a couple million, but because of a quirk in the rules, some people figured out that there comes a time when it's really almost a sure thing to bet in a certain way.

** Kucharski: **Exactly, and that's expected value, when you have lotteries that tip into this situation, where on average the ticket is worth more than actually the amount you're paying to buy it. But really, it's not just the expected value you have to consider, so for a lot of these teams, that's the calculation that motivates them to carry it out but it's the logistics of buying up the ticket. So a good example was, this is over 20 years ago now, in _____ _____ they worked out that it would cost about £1 million to buy up all the tickets. So, by definition, you would have the winning ticket if you bought everything, which was feasible if you had enough people and enough bankroll to carry that out and actually just buy up all the options. Even though the lottery tried to stop them, they actually managed to get most of it in the end and net the jackpot.

But I think things like the Powerball, that issue of bankroll and almost money management becomes more important, because whether the expected value might be positive, so you might, on average, if you play the game lots of times, expect to make money, the chance of you going bankrupt, if you have to buy millions and millions of tickets to make the win, for most people that's not a reasonable option. I think, historically, that's been quite an interesting debate from a kind of mathematical point of view because the mathematicians would value stuff on expected value, whereas, for instance, an economist might look more at utility, so this concept of how much is something worth to you. A good example is the insurance industry. If you buy insurance on something, on average, you're losing money, but I think most people would rather kind of have a predictable loss than suddenly take a big hit, and it's the same with the lottery. That very small chance of a big win for most people isn't worth the almost certain chance of losing the price of their ticket.

** Mirsky: **Right, but these consortiums have been put together in various lotteries at various places around the world where a large number of people all get together and figure out a system for buying up all the tickets.

** Kucharski: **They do, and a really good example was a few years ago in the Massachusetts state lottery. This actually started with a college project. Someone was just comparing lotteries and seeing which would be profitable, and actually noticed that one of them would be profitable, and there was this loophole where if you bought enough tickets in certain weeks, you could almost certainly guarantee that you're going to come away with money. I think what they called the statistical sweet spot was about 300,000 tickets.

** Mirsky: **The key is that the value of the win is worth more than the price of all the tickets.

** Kucharski: **Exactly, yes. You've got the value of the win, and of course for lotteries, often you scoop lower-tier prizes. It's not just the jackpot; you get the lower levels, as well. So that makes the math a bit more difficult, but really, they worked out that this loophole was there and engineered all these ticket purchases. In one great case, actually, this profitable situation was meant to happen when the jackpot reached $2 million, because that's when the lottery would redistribute the prizes. In one week, they actually worked out how to nudge the prize to activate a couple of weeks early, so everyone else is expecting this roll-down to happen I think at the end of the month, and this syndicate bought about 700,000 tickets, so not just exploiting a loophole but actually nudging the system into something that would profit them and no one else, so fantastic example I think of just ingenuity.

** Mirsky: **It's a fairly common story with these consortiums that if something goes wrong and they're not able to buy up 100 percent of the tickets, they might get 80 percent of the tickets, and so when the drawing comes they're on tenterhooks because they have invested hundreds of thousands of dollars, and although the odds are now hugely in their favor, they still have a one in five, one in four chance of losing.

** Kucharski: **That's a concern. I mean obviously if you've invested that much money, and the example of the Irish lottery a few years ago, I think they had about 80 percent of the tickets, and that does give you a huge amount of nerves when the lottery comes. One of the things I found remarkable talking to people who run a lot of these very successful syndicates is they _____ _____ celebrate their wins, because if you think, if there's that much uncertainty when the event comes, you're still gambling, really. There's still a lot of luck involved. So, if you've got something which is a solid system, you shouldn't really be celebrating or commiserating too much because you should have some control over what's happening. So I think in those cases where you have people who are nervously awaiting their draw, that suggests that they didn't quite get there in terms of perfecting their system.

** Mirsky: **You talk in the book about the fact that poker is really hard to play with a computer, there are challenges in poker that are different from other games that computers do really well in. What are those challenges, and where are the poker bots, and will they be taking over eventually all the online gaming sites?

** Kucharski: **Poker I think is a really good example to look at when you're investigating artificial intelligence. So, historically, you've seen games like checkers a few years ago, which was the optimal strategy was found. Obviously, chess, there was a lot of focus with Deep Blue and these kind of chess computers.

** Mirsky: **And now Go.

** Kucharski: **Now with Go, exactly. One of the reasons that poker is very different is what's known as the incomplete information game. If you think of something like chess, everything you see is front of you, all the information you need, which means that from a computational point of view, there's not really any element of chance or unknown information. So, in theory, if someone came up with a perfect strategy, they could just implement that and they'd get the same result every time. But poker is, in many ways, a more realistic game because in reality, in many situations, we do have hidden information. We don't know what the other person is thinking, what actions that they could take, and that's why I think from a computer point of view, it's a more challenging and more interesting problem, potentially, because you have to deal with this risk, you have to make decisions anticipating what the other person is doing. You can't just rely on probability. You have to, to some extent, account for what they're doing, you know, why are they placing low bets, why are they placing high bets, how can I adapt my strategy to work out what the other person is up to?

** Mirsky: **Online gambling is hugely popular, but is the future of it just going to be people having their bots gamble, or are actual human beings going to be playing these games? Certainly, human beings will be playing, but will they ever win?

** Kucharski: **That's a really good question, and it's been incredibly fast, actually, how these bots have improved over recent years, even since I started writing the book. I think some of the teams were a bit cagey about how good they were, and now clearing up these games, so even when you've got multiple players and very high stakes, arguably better than most humans. One of the things that was really interesting is actually a lot of the teams and researchers behind these bots, by their own admission, aren't very good poker players. I think that's a really interesting element of artificial intelligence, that these bots aren't good because you've got a human to tell them how to play; you've essentially created something which can learn on itself. Even if you go back to Alan Turing's kind of famous imitation game of can a computer convince someone that it's a human, in some ways these bots, I think, are edging away at those kind of issues because you've got these bots which can play so much better than their creators.

If you read interviews about a lot of these people who've played the bots, they talk about them like humans because the bots trick them, they deceive them, they're regressive, and these are just computers that have learnt by playing against themselves billions of times. There's no human element in there, and yet they've worked out how to bluff, they've worked out how to deceive people. I think it's really kind of fascinating insight into the learning process, and in some cases, things like bluffing that we think are very human traits actually are just the mathematically optimal thing to do.

** Mirsky: **That's really interesting. Well, we all think of our computers as human no matter how sophisticated the computer is, because we yell at our computers, we smack them on the side of the head and say, "What are you doing? Why did you do that?" Why did you write the book?

** Kucharski: **Really, throughout my career, so I trained as a mathematician and I'm now working kind of _____ _____ public health. But there's always been an interest in these games and gambling. I think anyone with a kind of mathematical mindset is always interested in looking for how games can be beaten and understanding risks. I mean I think during my Ph D , when a lot of these funds which – so you got recruiters kind of e-mailing you from banks and from all these other kind of familiar candidates, but it was people who had the betting equivalent of hedge funds, people who were making money predicting sports. It was something that I was so unaware of was a viable industry and seeing how that had grown, and actually the more I started digging into the history, realizing that mathematicians from Galileo to Turing had been interested in gambling.

They weren't professional gamblers and that wasn't their career choice, but for them it was just a way to play around with ideas. I think that really explained to me why I had this fascination with these kinds of problems, because all the people I've talked to researching the book, although many of them don't actually work as professional gamblers, they've got that scientific mindset and they've got that curiosity about games. I think that was just a fascinating thing to explore, and many of the stories in the book I think kind of illustrate how close science and gambling have been historically. Actually today, there's many of the interesting developments, as well, which are coming out of this relationship between the two industries.

** Mirsky: **Yeah, you have a quote in the book, I forget who it's by, but it says something like once you have enough information, your gambling site is no longer gambling, you're the manager of a hedge fund.

** Kucharski: **Exactly, so I think that was a quote I think from one of the columnists. It was about these funds, they're essentially hedge funds that are offering sports betting as an asset class. I mean they're very much calling it an alternative asset class. Their argument is if you've got a guy who knows a lot about commodities and can outplay the markets, how is that different to a guy who knows a huge amount about sports and can outplay a betting market. Whereas, arguably, in betting you've got a lot of people who have very little expertise playing, and if you've got an edge, you can make money. I think it's much smaller in terms of scale, and not all of these funds, some have been quite successful, some have had more limited success, but I think it's an interesting debate to have over actually what constitutes an investment, and where are we comfortable putting our money in these kind of activities that involve risk.

** Mirsky: **Yeah, I don't know if you've been following here in the states, there have been legal cases about some betting sites, about whether they should be legal or not because the proprietors say it's a game of skill. New York state, for example, is going after I think Draft Kings, and I forget what the other site is called, but saying, "No, this is gambling." Statistics came out that a tiny percentage of the players are winning most of the money because they're not necessarily the guy who knows the most about sports; they're mathematicians and physicists that have Ph D 's. They've created algorithms that track all the data, especially in a sport like baseball, where it's a lot of discreet events, so the statistics become very meaningful for trying to predict the future. These guys are winning almost all the money, and so this is a perfect example. It's not gambling for them. For a lot of them now it's a full-time job.

** Kucharski: **I think the emergence of fantasy sports, so some of these W eb sites you mentioned, is a really interesting development in the U.S., because the argument of course is it's skill because you're predicting which players or which teams are going to do well, and you're essentially placing money on that prediction being correct. But of course sports betting, where you make a prediction about which teams are going to go well and place money on that, is still defined as betting in a lot of places, and I think it's really showing that almost the divide between skill and luck isn't very neat. We can't separate things into, "This is gambling and this is not gambling." A lot of these games where, as you said, if you've got _____ betters making most of the money by using their advantage in terms of how they can analyze the data, and certainly that happens in finance all the time, it happens in investments all the time, I mean if you're calling fantasy betting a game of skill, or fantasy sports a game of skill, then that I think will have implications for how we view other aspects of sports betting, as well.

** Mirsky: **So you're now getting a master's in public health. Are you going to be doing massive database assemblages and crunching and tracking epidemic pathways, things like that?

** Kucharski: **At the moment, actually, I'm working, so I've got a Ph D already in that.

** Mirsky: **Oh, sorry. I thought you said you were working on it.

** Kucharski: **That's fine. Yeah, so I mean I actually work in epidemic forecasting, so we did a lot of work during the Ebola outbreak, and more recently on some of the Zika outbreaks, as well. Although it seems like a very different industry, a lot of the methods we use Monte Carlo simulations all the time dealing with hidden information, I think one of the classic questions we have to ask almost every day is, "Is this a pattern or is this something that's chance?" and that's fundamentally the question which gamblers and scientists have been asking for centuries. That's why I think a lot of these questions about risk are really interesting, not because I want to become a professional gambler, but in life and I think in a lot of these situations where you need to manage risk, you need to understand what you're looking at, and you need to be able to identify whether you're looking at something which is a potential pattern and a problem.

** Mirsky: **We talked about expected value in gambling. In epidemics, you have the expected number of infections that you're going to create from one already-infected person, so it's very, very similar.

** Kucharski: **In many ways, yeah, and you also have the issue of these unlikely events. The most likely outcome perhaps might be quite _____ _____, you know, with a particular bet you might have the situation where, on average, you might make a bit, but there's a very small chance of something extreme happening. In the case of an epidemic it's, again, a different situation but the same analogy, whereas in the majority of cases it might not be a problem, but it's that extreme event you want to know about and you want to be able to measure it properly. That's where a lot of these techniques around probability and statistics become very important.

** Mirsky: **For public health, you really want to be able to game the system.

** Kucharski: **I think it's a situation where there obviously is a lot of risk involved, and you want to make sure that, yeah, you make the right decision.

** Mirsky: **The prescient Mark Twain short story I mentioned is called

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