Much has been written in the days since Leicester City stunned the football world by winning the Premier League, with commentators, bloggers, and modellers now attempting post hoc rationalizations to explain how Leicester’s against-all-odds climb to the top happened, why it happened, and why it was always more likely to have happened than everyone previously expected.

In this respect, Leicester City’s Premier League title represents an example of what philosopher and statistician Nassim Nicholas Taleb calls a “rare phenomenon,” a rare event that is hard to predict, comes as a surprise, but once observed, is explained in hindsight by a failure to account for relevant data that was available all along. Reason is turned on its head, with outcomes being used to explain causes.

Was Leicester’s win predictable ?

What were Leicester’s odds of winning the 2015/16 Premier League at the start of the season, and did the bookmakers underrate their chances? Of course, being a rare phenomenon, attempting to look back and answer these questions runs the risk of hindsight bias. Nevertheless, let’s give it a try.

One possible way is to use the match betting odds market as a guide to how Leicester City was expected to perform. Taking Pinnacle’s Premier League match betting odds for the 2014/15 season and removing their margin, we can run a Monte Carlo simulation to determine the expected points total for all 20 teams and from this, their expected finishing positions.

Although admittedly retrospective, the model is inherently Bayesian, reflecting changes of market opinion in the relative quality of teams, defined by betting odds, as the season progressed. Some argue that a betting market is not necessarily accurate, since bettors suffer from behavioral biases, information cascades, and groupthink. But such market inefficiencies are typically small, insufficient for the majority of bettors to profit from anything other than chance. Moreover, given that a betting market really reflects many hundreds or thousands of models, is there good reason to doubt it being less wise than the mathematical analyses of a few so-called experts? The more diverse the set of opinions, the wiser the collective judgment is likely to be.

The table below shows the results of a 100,000-run Monte Carlo simulation. Leicester City was crowned champion of the 2014/15 season in just one of those runs.

The past is the key to the future

Let’s assume that Leicester City would have been expected to perform more or less the same in 2015/16 as our model retrospectively expected them to perform in 2014/15. Despite winning 7 of their remaining 9 games last season to escape relegation, by August 2015 there was little other reason to expect any major change in expected performance.

Former manager Nigel Pearson, who had arguably engineered their Premier League survival, had left, Esteban Cambiasso, the club’s player of the year had departed, while new manager Claudio Ranieri had never won a league title, having recently been sacked by the Greek national team after losing to the Faroe Islands.

Most pundits and commentators were predicting relegation or a narrow escape. Even a supercomputer predicted just 45 points and a 13th-place finish for Leicester in October, when the team was already in 5th place.

Arguably then, in August 2015, 100,000/1 (1001.00 in decimal) represented the fair odds for Leicester City winning the 2015/16 Premier League. Conceivably, bookmakers actually shortened their odds to 5000/1 (5001.00 in decimal) due to the Favorite–Longshot Bias.

Assessing Leicester’s performance for 2016/17

So what should Leicester City’s odds be for retaining the Premier League title in 2016/17? Another 100,000-run Monte Carlo simulation using this season’s match odds sees the team with an expected points total of 53 and an expected finishing position of 9th. Assuming, as before, that this season’s market opinion of the team will be broadly similar for the 2016/17 season, this translates into fair odds for becoming champions of roughly 625/1 (626.00 in decimal), considerably longer than the currently published odds ranging between 25/1 (26.00 in decimal) and 40/1 (41.00 in decimal).

Some have argued that it’s inappropriate to consider the whole 2015/16 season. Leicester obviously improved during the season, and since the betting market allegedly takes time to respond to a change in a team’s ability, wouldn’t it be better to consider the market’s view of Leicester in the second half of the season, once they were already leading the Premier League? Repeating the simulation now assuming the market had viewed Leicester in the first half as it did in the second sees them top just 1 in 111 times, an expected 8th place and 57 points, just 4 more than for the original simulation.

Skill or luck ? The final verdict

Finally, it’s worth noting that Leicester City’s expected points total for 2015/16 is just 10 and 12 more respectively than their expected (43) and actual (41) points total for the 2014/15 season. This modest change in the betting market’s opinion about the team is a reflection of their genuine improvement in ability.

By implication, the additional 27 points Leicester City accumulated has been achieved purely by chance. Given that the standard deviation in a team’s modeled expected points is typically about 7, their Premier League title truly represents a rare phenomenon.

Last season, the biggest difference between expected and actual points total was just 9 (Everton). If this modeling has any validity, it would seem we are still waiting for Leicester City’s performance to regress to their mean. They may be better than they were, but most of what has happened this season is arguably the result of good fortune.