THE analytics blog Statsbomb recently published a piece which puts Arsenal’s 12 seasons without a Premier League title in a slightly different light. They went through the period game by game and looked at the match odds - after all, bookies’ odds are nothing more than the probability of a certain outcome - for every game, ran 10,000 simulations for each season based on them and figured out the probability of them winning the league in each of those years.

What they found was that the three likeliest outcomes saw Arsenal winning one league title (32 per cent), two league titles (29 per cent) or even three titles (18 per cent) at some point in those 12 years. The chances that they would come up empty-handed were only 14 per cent.

In other words, Arsenal not winning a title is quite an unexpected occurrence and down to little more than statistical variance. Or, as some will put it, bad luck.

Yes, the old cliche talks about “making your own luck”. But this goes beyond that. This study - with all the caveats you would expect - suggests that Arsenal did more than enough, all things being equal, to win one, two or three titles. That they didn’t is down to the randomness of football.

It is entirely possible that the reason Stan Kroenke kept Arsene Wenger around all those years isn’t just that he’s a (relatively) inexpensive boss who ensures Arsenal remain a cash cow, which is what he really cares about.

It’s also possible that Kroenke gets this randomness and probability thing more than most. And that, in a cold-hearted analytical way, he reckons Wenger, with all his foibles, is still his best option for success. Or, at least, with a bit of regression to the mean, enough success to keep the golden-egg-laying-goose alive.

Pep Guardiola, whose Manchester City side travel to take on Arsenal this afternoon, is in some ways a kindred spirit. If you ever want to anger him, call his football a rigid utopia as one German journalist did a few years back. He will fire back and say that he plays the way he does because he believes that, in the long term, it’s the best way to win consistently.

The cynics out there will say that actually the best way to win consistently is to manage Bayern and Barcelona and have some of the best players in the world (including the best player) at your disposal. But that’s a bit of a facile argument. Because if that’s all it took, then guys like Tata Martino and Jurgen Klinsmann would have won plenty of silverware and, in fact, they did not.

What unites Guardiola and Wenger is their unorthodoxy and their counter-intuitiveness. Whether it’s predicating your game on full-backs and then going into the season with Bacary Sagna, Pablo Zabaleta, Aleksandar Kolarov and Gael Clichy or thinking you didn’t need a reserve central defender because you have Kolarov (who had never played there before this season) and Vincent Kompany (who is congenitally injured). Or whether it’s sitting on a giant pile of cash because you don’t believe there’s “value in the transfer market” or showing “loyalty” by endlessly extending contracts to margin players who contribute little and play even less.

The sense is that the formula that got them this far, the unorthodoxy that earned them their initial success, is difficult to let go. Not because they fail to realise there may be a better way, but because they are convinced their way (possibly with minor tweaks) remains the best way for them. Denying that, or going in a wholly different direction, would in some ways invalidate the work they did before. And things like probability and variance only serve to strengthen their argument. In a game of wafer-thin margins, they can point to bad luck.

That is why, when the going gets rough, they go back to what they know best and what has worked in the past.

LAST week Everton midfielder James McCarthy got himself injured while on international duty with the Republic of Ireland when he pulled up with a bad hamstring during the warm-up against Wales.

Cue Ronaldo Koeman and his Ireland counterpart Martin O’Neill pointing fingers. Koeman complained that Ireland had been told that McCarthy, who had missed the previous two Everton games, needed to be “protected” as he wasn’t fully fit.

O’Neill - who has previous with Koeman - fired back. He questioned Everton’s pre-season schedule, pointing out that McCarthy played three games in the space of eight days shortly after returning from his post-Euro 2016 holidays. And he called Koeman “Master Tactician of the blame game”.

Koeman took to Twitter for his reply, reminding O’Neill that McCarthy had a full three-and-a-half weeks off after the Euros. And signing his tweet “master tactician”.

National teams and club sides effectively share internationals and that is why they are quick to blame each other, but Koeman and O’Neill have taken it to another level. Which, you suppose, is entertaining for the neutrals.

What is more interesting here isn’t establishing “blame”; unless you are the doctor who assessed McCarthy and you know exactly what was said between the player and the Irish coaching staff it’s impossible to know that. Rather, it is the spin between these two that is fascinating.

Koeman evidently doesn’t trust O’Neill, yet he sent him a player who he later said “was not fit to play”. And then, when McCarthy was asked by the Irish coaches whether he was fit, “of course” he said yes even though he was in no state. Wouldn’t you assume that, at some point, a professional athlete has to take responsibility for his actions?

O’Neill is no better when he brings up pre-season and events from seven months ago. When he talks about Koeman needing “quiet introspection” he is being patronising and provocative. And citing “three games in eight days” is, at best, wilfully misleading. Two of those games were friendlies. The other was Wayne Rooney’s testimonial. McCarthy played a total of 152 minutes. What bearing that has on the present only O’Neill knows.

Events like this make you wonder why professional footballers don’t simply look out for themselves a little more. McCarthy earns around £50,000 a week. He could easily afford his own personal doctor: somebody to assess him alongside the Everton and Ireland medical teams and give him an honest assessment. Maybe his conclusion would be identical to that of the Everton and Ireland staffs, but at least it would come from somebody without any kind of vested interest and free from all pressure.