Should I have bet on Leicester City?

If you’re not British, or you live under a stone somewhere, then you may not have heard about one of the most extraordinary sporting stories ever. Leicester City, a football (in the British sense) team that last year only just escaped relegation from the top division, has just won the league. At the start of the season you could have bet on this happening at odds of 5000-1. Just 12 people availed themselves of this opportunity.

Ten pounds bet then would have net me 50000 pounds now, so a natural question arises: should I be kicking myself (the appropriate reaction given the sport) for not placing such a bet? In one sense the answer is obviously yes, as I’d have made a lot of money if I had. But I’m not in the habit of placing bets, and had no idea that these odds were being offered anyway, so I’m not too cut up about it.

Nevertheless, it’s still interesting to think about the question hypothetically: if I had been the betting type and had known about these odds, should I have gone for them? Or would regretting not doing so be as silly as regretting not choosing and betting on the particular set of numbers that just happened to win the national lottery last week?

Here’s a possible argument that the 5000-1 odds at the beginning of the season were about right, or at least not too low, and an attempted explanation of why hardly anybody bet on Leicester. If you’ve watched football for any length of time, you know that the league is dominated by the big clubs, with their vast resources to spend on top players and managers. Just occasionally a middle-ranking club has a surprisingly good season and ends up somewhere near the top. But a bottom-ranking club that hasn’t just been lavished with money doesn’t become a top club overnight, and since to win the league you have to do consistently well over an entire season, it just ain’t gonna happen that a club like Leicester will win.

And here are a few criticisms of the above argument.

1. The argument that we know how things work from following the game for years or even decades is convincing if all you want to prove is that it is very unlikely that a team like Leicester will win. But here we want to prove that the odds are not just low, but one-in-five-thousand low. What if the probability of it happening in any given season were 100 to 1? We haven’t had many more than 100 seasons ever, so we might well never have observed what we observed this season.

2. The argument that consistency is required over a whole season is a very strong one if the conclusion to be established is that a mediocre team will almost never win. Indeed, for a mediocre team to beat a very good team some significantly good luck is required. And the chances of that kind of luck happening enough times during a season for the team to win the league are given by the tail of a binomial distribution, so they are tiny.

However, in practice it is not at all true that results of different matches are independent. Once Leicester had won a few matches against far bigger and richer clubs, a simple Bayesian calculation would have shown that it was far more likely that Leicester had somehow become a much better team since last season than that it had won those matches by a series of independent flukes. I think the bookmakers probably made a big mistake by offering odds of 1000-1 three months into the season, at which point Leicester were top. Of course we all expected them to fall off, but were we 99.9% sure of that? Surely not. (I think if I’d known about those odds, I probably would have bet £20 or so. Oh well.)

3. Although it was very unlikely that Leicester would suddenly become far better, there were changes, such as a new manager and some unheralded new players who turned out to be incredibly good. How unlikely is it that a player who has caught someone’s eye will be much better than anybody expected? Pretty unlikely but not impossible, I’d have thought: it’s quite common for players to blossom when they move to a new club.

4. The fact that Leicester had a remarkable escape from relegation at the end of last season, winning seven of their last nine matches, was already fairly strong evidence that something had changed (see point 2 above). Had they accumulated their meagre points total in a more uniform manner, it would have reduced the odds of their winning this season.

The first criticism above is not itself beyond criticism, since we have more data to go on than just the English league. If nothing like the Leicester story had happened in any league anywhere in the world since the beginning of the game, then the evidence would be more convincing. But from what I’ve read in the papers the story isn’t completely unprecedented: that is, pretty big surprises do just occasionally happen. Though against that, the way that money has come into the game has made the big clubs more dominant in recent years, which would seem to reduce Leicester’s chances.

I’m not going to come to any firm conclusion here, but my instinct is that 5,000-1 was a very good bet to take at the beginning of the season, even without hindsight, and that 1000-1 three months later was an amazing chance. I’m ignoring here the well-known question of whether it is sensible to take unlikely bets just because your expected gain is positive. I’m just wondering whether the expected gain was positive. Your back-of-envelope calculations on the subject are welcome …

35 Responses to “Should I have bet on Leicester City?”

  1. Anonymous Says:

    Loved the post. The phrase “not unduly generous” I had to read a few times. In fact, I am still thinking about it. So it was not not too generous?

    • gowers Says:

      I was defining “generous” to mean that the bookmakers were being kind to the punters. But the sentence was not clear, so I’ve reworded it now.

  2. thomasdevlin Says:

    Also Leicester were on the march from the end of the last season with 7 wins, 1 draw and 1 loss in the EPL during Apr – Jun 2015.

    • gowers Says:

      Ah, I meant to mention that. If you don’t mind, I’d like to add that in.

    • Anonymous Says:

      Please do – I really appreciate you giving your time. If Leicester had ‘maintained’ this record of 22 points per 9 games they would have now had 88 instead of only 77 points per 36 games played. (I didn’t bet on them.)

  3. Richard Baron Says:

    One thing we know is that the expected value was not easily perceived to be positive (assuming that it was in fact positive). If it were, then gamblers would have piled in and the odds would have shortened pretty smartly – not because bookies can tell how football matches will turn out, but because they like to keep a book that will overall ensure a profit for them, whoever wins.

    What you really want is the mirror image of that – a program that will monitor all the odds being offered, and work out a combination of bets you can place, perhaps with several different bookies, such that you will win overall, whatever happens (finding the arbitrage opportunity). But again, if that were easily done, lots of people would be doing it already, and the odds would shift accordingly. (The obstacle to doing it is presumably not the computation, unless someone is going to tell me that the problem is outside P, but the poor odds on offer.)

    • B Says:

      There exist such services already, and they find dozens of arbitrage opportunities on a daily basis. The catch is that the betting houses have methods to detect people systematically making +EV bets, and will stop them from making further wagers.

  4. thomasdevlin Says:

    Off the direct point but … I think a rarer event was the 2003 – 04 season where the last team in each of the top 4 English football leagues was also last alphabetically – and overall in these top four divisions the first and last were also first and last alphabetically).

    • Stickler Says:

      That’s a nice piece of trivia, but to me it feels a bit too much like “result-fishing”. Put otherwise: I can imagine that many year’s final league tables contain equally rare patterns, if you look for them after the fact.

  5. André Rodenburg Says:

    “5,000-1 was a very good bed to take at the beginning of the season” – I understand that those who did, didn’t sleep well on it and had money paid out beforehand

    • Richard Baron Says:

      Yes, I saw those reports about people cashing out half way. But if you can take £10,000 half way through the season, or hold out for a possible £50,000, then at that point you are being invited to bet a sum you would really regret losing at the unimpressive odds of 5 to 1. That is a different proposition from risking £10 at amazing odds.

  6. domotorp Says:

    First of all, there is a typo in the last para: “good bed” should be “good bet”. Second, being British, you are probably aware of the incredibly stupid names of your football leagues and should be careful with writing “win the championship”, as “Championship” is the name of the second level of the football pyramid, which, Leicester, have in fact won just two years ago. And here is a good point to mention that Leicester a little earlier stopped being one of the very low budget teams, as they have a foreign billionaire owner who has invested a lot of money into the club (though much less as the top teams).

    And now, to address whether it would have been a good idea to bet £20 on them mid-season if you think that you knew the odds better than others, depends on your utility function, i.e., how much different quantities of money are worth to you. Most bettors would use the Kelly formula, which says that if you think the odds are 100-1 instead of 1000-1, then you should bet about 1% of your bankroll, i.e, the money that you can afford to loose and might occasionally use for betting. So, if you have £2000 to spend on bets (which is realistic), then it would have been a good idea.

    Thanks — I’ve changed “bed” and “championship”.

  7. deaneyang Says:

    I’m not sure what the definition of a “good bet”, but, as you point out, this bet is not so different from buying a lottery ticket. Mathematicians tend to dismiss such bets as being stupid, because the expected payoff is negative. However, “expected payoff” is important mostly when one is making the similar bets many times and investing a substantial amount of money doing so. But that’s not the case here. Instead, one is contemplating spending a rather trivial fraction of one’s wealth for a potentially massive upside, however improbable. The question is whether it’s worth sacrificing what one would have used the money for otherwise (ranging from a lot of beers at the pub to putting it in an investment) for this near-zero probability of winning a lot of money. I doubt I would ever place such a bet, but I’ve come to see why it’s not so irrational after all to buy lottery tickets or bet on bad soccer teams. I suppose I’m just saying in simplistic terms what others explain using utility functions and all that, but I find it easier to think about it this way.

    • Mark Meckes Says:

      “Mathematicians tend to dismiss such bets as being stupid, because the expected payoff is negative.”

      Even more, in my experience mathematicians in general (as opposed to probabilists and statisticians) tend not to think beyond expected value calculations when analyzing uncertain events. Many of them seem hardly to be aware that probability theory has even developed more sophisticated ways of looking at things (although some have heard the word “variance”).

  8. obryant Says:

    The payoff-to-probability calculation is a little off due to taxes. A ten pound bet wins £50000 at 5000-1 odds, but then your government will take 15% (if I understand British taxes correctly), so you only net £42250. If gambling is taxed just like regular income, then you probably lose 40% and only take home £30000.

    • domotorp Says:

      No, gambling is absolutely tax-free in the UK.

    • Richard Baron Says:

      I don’t think it is tax-free, but the tax comes at a later stage. Rather than coming out of the winnings on a given bet (which it used to do), general betting duty is 15% of the bookmaker’s gross profits (bets taken in minus winnings paid out). The effect on odds is not obvious to the punter, but it is still there.

    • domotorp Says:

      Yes, I meant it is tax-free for the bettor. Also, I don’t think it affects the odds much, bookies usually try to set them so that they make money regardless of the outcome. (Which is, probably, a good reason why it’s not bad for them to offer Leicester at 5000-1 even if the chances are different as long as not many people bet on them. Is there some statistic about how much money was put on different teams and at what odds to win the league?)

    • obryant Says:


    • Richard Baron Says:

      On the effect of the general betting duty on odds, there is indeed no reason why it should affect the ratio of the odds on team C to the odds on team D. But there will be some shortening of all of the odds together, because the bookie will not simply want to be sure of some gross profit or other. For a given size of business, the bookie will want a certain level of gross profit after the duty has been paid, in order to have enough to pay the staff, pay the rent on the shop, and get a decent return on the capital invested. If the Government is taking a rake-off, someone must lose. The loss may however fall partly on the punters (through shorter odds), partly on the staff (through lower wages), and partly on the bookie (through a lower return on capital and on the proprietor’s own hours working in the business), depending on elasticities of supply and demand. Taxes are like that. They have a habit of imposing burdens in non-obvious places, as well as in obvious places.

  9. Ten Things To Read Today  – Cathedral Group Capital Says:

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  10. Chris Aldrich Says:

    In a typical lottery, the primary function at action is statistical mechanics. By comparison, in most sports gambling scenarios one is betting against the pool of other betters and it is this pool that determines the odds. This is known as Parimutuel betting and can be compared with fixed-odds betting arrangements in which one is gambling against the house, which is setting the odds and payoff usually at the time of the bet, and those odds can shift through time until betting window closes as the house tries to spread out their risk to not only cover betting, but to make a profit.

    I’m not sure which sportsbook you’re quoting for the 5000-1 odds, but I would suppose that it is parimutuel and the house “sets” the game and takes a percentage off of the top (known as the rake or vigorish) as does the local government in taxes. The odds are then determined by the number of betters and the teams for which they’re betting. If the pool of betters (presumably all having the same information) is “right”, then the odds set are typically “correct” and the proper/fair payouts follow.

    In cases where the betting pool is overly biased (perhaps there are more die hard fans for Man United who are betting with their hearts and not their brains) this can create situations known as “overlays” in which careful gamblers who notice the shift can take advantage of the situation and potentially have better-than-usual margins on their bets. Naturally these margins need to be big enough to overcome not only the inherent risk of the initial gamble, but also cover the loss of the houses’s rake and the subsequent taxes. Serious handicappers will use all of the best available information available in making their gambling choices, but will then only proceed to place bets on which they perceive to be overlays. In the long run this will give them a slight edge (not over the house, but) over their fellow gamblers. They also typically place their bets just before the close of betting so that they have the most updated odds available (and can ensure their bet is still an overlay) while keeping in mind that their bet and other last-minute bets can potentially sway the ultimate odds. (example: If someone comes in at the last minute with several million pounds on Leicester City then the odds shift away from them, potentially creating overlays for other teams.)

    In most cases, longshots with terrible odds like these, are longshots for a reason, but if a careful gambler has (preferably solid information creating) reason to believe that the odds shouldn’t be as bad as they are, then it creates an overlay which makes the bet more valuable in comparison to all the other gamblers.

    For statisticians, think of it as a normal curve whose local x-value along the y-axis is shifting (usually slightly) over time. If there’s a huge bump in the outlying choices just before betting closes, then one’s statistical likelihood of winning is better and it’s a good bet with a better-than-average payoff. In practice, there are so many people gambling (usually intelligently) that these local shifts (sometimes called “noise” in engineering parlance) are fairly small and that in aggregate averaged over time most gambling situations are the mathematical equivalent of a numerical lottery.

    As a similar somewhat related example, in US baseball, sabermetricians are gathering large amounts of data to attempt to find otherwise unseen inequities in forecasted player talent and then leverage that to try to win more baseball games. In the early days of applying these methods, one team could statistically do better than others, however over time, as more teams begin using the same methods, the information gap is overcome and everyone is on equal footing again. The trouble in most sports is that coming up with accurate measurements of performance isn’t easy (or even statistically significant) and thereby adds a large enough layer of noise on top of the process that gambling on them is again relatively equivalent to the lottery.

  11. Bill Says:

    It’s helpful to compare 5000-1 odds with 500-1 odds that Simon Cowell would become the next British prime minister and 1000-1 odds that Hugh Hefner would reveal that he was a virgin (source: NYT).

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  13. Ian Says:

    It is an interesting thing. I assume bookies might offer crazy odds in order to bring in money from suckers who hope something may come true. For instance Elvis is alive or Leicester will win the league. Perhaps they just offered the odds in competition, but to get the punters tiny bets they took a risk. Interestingly in NZ, where I live, the local highly regulated betting market only offered 1000 to one at the beginning of the season for Leicester. I didn’t even think about taking it, too improbable. However they were still offering 500 to one in November when they hit the top of the table. I put 5 on it then, but thought I would lose it because it seemed improbable. I guess it was a strange season, Chelsea imploding, Man City and Man U stuttering and Spurs taking crucial points off them all, while losing to “lesser” sides. It is not quite a lottery and people make bets for strange reasons (I am a spurs supporter always put a bet on them being relegated and a bet on them winning, sort of emotional insurance) and I am not sure bookies odds make much sense at the extremes either

  14. Colin Says:

    I wonder what odds the bookies would have offered if you tried to bet in April 2015, when the team was still bottom of the table, that Leicester would win the Premiership the following year? Presumably more than 5000-1, as at that stage you’re betting on the conjunction of two unlikely events (that Leicester will avoid relegation, and that they’ll win the League next year), and with the team having an extremely bad performance record in the preceding months.

  15. Nuwasasira Abel Says:

    hello bloggers , can anyone help me figure out the similariy between the numbers 85, 17,19,4 and 2?
    any logic?

  16. Andrew Maclaren Says:

    A bookmaker discusses the issues in a quantitative way at

  17. domotorp Says:

    Actually, what are the odds you would offer for P=NP solved in a year? (Here there are several ways to define what must happen in a year, I had in mind that a draft appears on arxiv and it will be eventually accepted as correct.)

    • gowers Says:

      I don’t know, but I can put some bounds on it. I would definitely bet on it if I was offered odds of 5,000 to 1, since I’ve already seen enough solutions of problems that had a not-in-my-lifetime quality (I’m thinking in particular of Fermat’s Last Theorem and the Poincaré conjecture) to feel that the probability is much higher than that. I think I’d still bet at 100 to 1. At 50 to 1 I’d have to think about it, and at 10 to 1 I wouldn’t be interested.

      My argument for 100 to 1 is roughly this: I feel as though it will probably be solved within a century (though I’m far from certain about this). Also, I have the impression that it needs someone to come up with a surprising new idea, after which it may be that it won’t take years of hard technical slog to get the proof to work. If that hunch is correct, then it is not unreasonable to think of the probability distribution on the time of its solution as something like an exponential distribution with a fairly large mean, so that this coming year is not much less likely than some future year merely by virtue of being much sooner.

      Having written that, however, I think 100 is perhaps too small and that maybe 200 to 1 is where I’d be fairly happy to bet, while 100 to 1 is roughly on my boundary between where I’d bet and where I wouldn’t.

    • domotorp Says:

      And of course, there’s the risk of the person offering the odds already having a proof, and instead of instant fame and recognition, he first wants to pull a tenner out of your pocket…

  18. meditationatae Says:

    I think it’s entirely believable that Leicester at 5000-1 odds was a good bet, at the beginning of the season. I studied Pari-mutuel and odds in horse racing; in Pari-mutuel, the allotted prize is divided among all winning tickets or bets equitably. So for example, 12 winning bets on Leicester at 10 pounds each share the allotted prize equitably. Had it been 24 winning bets on Leicester at 10 pounds each, in Parimutuel the share of the prize would have been half as much, or 25,000 ponds instead of 50,000 pounds.

  19. Jonathan Kirby Says:

    There was a nice discussion of the 5000-1 odds of Leicester winning on BBC Radio 4’s More or Less programme recently (9th May, available as a podcast). Lord Finkelstein’s statistical model suggests that 5000-1 was about right, or perhaps even longer odds would have been appropriate. I am sceptical because the probability that the model breaks down when considering an event with such long odds has not been considered. Of course there is no objective way to do that without considering another model which could itself be just as bad. I suppose actuaries would be the people to ask about estimating the probability of rare events.

  20. Ben Says:

    It’s easy to say that in hindsight, but are you going to bet £5 on Middlesborough winning the league next season? The truth is that you may as well throw £5 into a lake and hope £1000 comes out. In hindsight you say it’s a good bet, but it really was impossible and therefore not a bet worth taking.

    • gowers Says:

      My point is not that I should have expected Leicester City to win. It’s merely that even without the benefit of hindsight one should have realized that the probability of their doing so was more than 1 in 5000. Whether that makes it worth betting on them is an interesting question: see Domotorp’s comment above (May 3rd, 2:02pm).

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