You have all by now, no doubt, seen that the odds are strong that Arsenal will draw Real Madrid in the next round. According to the betting firm Coral, there is a 21.7% chance Arsenal will play Real Madrid.
I understand the basics of how this works. Because of the fact that four teams from Spain made it into the next round plus the fact that Real Madrid were in a group with Borussia Dortmund, Real Madrid have three teams that they can’t draw. That means that each of the remaining five teams that they can draw all have a higher probability of being selected.
Now, each of the five teams that could draw Real Madrid have their own limitations, Arsenal, for example, can’t draw two teams. Juventus only have one team that they can’t draw. What this means, practically, is that Juventus have more options that they could draw and as a result, they have a smaller chance of drawing Real Madrid. Napoli have the same “benefit”. Since they also won their group and there are no other Italian teams left for them to face in the Champions League, they also have a slightly reduced rate (compared to the two English teams) of facing Real Madrid in the next round.
The astute observer will immediately question why Leicester and Arsenal, who both have 6 teams they could face, don’t have the exact same odds of drawing Real Madrid. Similarly, Napoli and Juventus have different odds even though they are both Italian sides and both only have one team they can’t draw. This is where the math gets a bit complicated.
What we are looking at is a classic “Rooks Polynomial” problem. For those of you who want a more in-depth look at the math behind this, the Institute has created a video which explains how this all works. It’s about 18 minutes long and worth your time if you want to understand why there are slight variations in the odds. Just to put it in terms that might make sense, let’s look at Napoli and Juventus: The team that Napoli can’t draw is Benfica and they have just one exclusion. The team that Juventus can’t draw is Sevilla and they have three exclusions. If you think of these selections as interdependent and as “sets” then the fact that Benfica have more combinations available to them means that Napoli have a slightly lower chance of drawing Real Madrid than Juventus. The same thing applies to Arsenal and Leicester, though the difference between the two teams is functionally irrelevant for the average viewer. We are saying that Arsenal will draw Real Madrid 52 more times out of 10,000 draws than Leicester.
What has actually happened here, and which is a problem that needs to be solved, is that the Champions League format has become a bit predictable in terms of which teams are going to face each other. Because UEFA doesn’t want two teams from the same country playing each other in the first knockout phase (why not?), because there is a lack of diversity in which countries have made it into the knockout round (basically all of the biggest leagues have sent their best teams; Spain has 4; England and Germany 3 each; and Italy, France, and Portugal 2 each), and because the group stages had so many big teams doubled up with each other, the result is that many of the big clubs (Arsenal, Man City, Real Madrid, Barcelona, Atleti, and Borussia Dortmund) have a large number of teams that they can’t play.
If you’re going to have 10 of the 16 teams in the finals from three countries but they can’t play teams from their own country, you have set up a mathematical equation that virtually assures many of those teams will play each other in the next round. Just look at the numbers: odds are that Arsenal will play Real, Barcelona v. Bayern, and Man City v. Atleti.
And even if Arsenal finished second in their group, the numbers would have been almost identical that instead of Real Madrid, Arsenal faced either Barcelona or Atletico Madrid. And if the title holders, Real, had won their group as well, it would have ensured Arsenal faced one of those three teams. Hence the reason why Wenger was keen to win his group. What he really needed was for Real Madrid to also win their group. Though had that happened, the odds were almost certain that Arsenal would be complaining about a trip to Bayern Munich or Borussia Dortmund.