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.
Catch 22.
Thanks for explaining the maths behind the draw.
I agree the competition is becoming more and more predictable especially for the next draw. I think the reason they keep teams from the same country apart is because they want to keep each country interested for as long as possible. I wouldn’t even be surprised if the driving force was the desire to avoid a Barcelona v Real last 16 game at all costs.
Whether we get end up drawing Real I honestly don’t care. I’m of the opinion that we’re in the competition to play the big teams and considering most Arsenal fans probably don’t think we can win it, why not play as many big games as possible and enjoy them?
I don’t know much about formations and tactics, but watching last nights game, I kept thinking that we’d have even more chance of getting a hammering by Dortmund. I know Real play on the counter but Dortmund were one pass away from being through on goal multiple times last night and in another game could have scored another 2 goals. It’s no coincidence they’ve broken the scoring record. I also think we tend to do better against big teams when we know we have to be on top of our game to win.
Funny you say that. I watched the first half pretty carefully but not the second. What stood out for me was how both teams were trying to press each other, but how much more comfortable Real were at dealing with the press. Being at home helps, but it was clear they had superior technicians throughout their back line. I felt Sokratis and Bartra were weak links for Dortmund, and they are flanked by ageing warriors in Schmelzer and Pisczek. Weidenfeller’s athleticism is not what’s keeping him in the side either. By contrast, Real’s Carvajal, Varane, Ramos and Marcelo are a cut above when in possession.
The other big thing was how much Madrid put into their press early on. I kept thinking, if Dortmund can resist this, Real will tire and lose this game in the second half. Madrid were rewarded with a 2-0 lead, but naturally couldn’t sustain that intensity over 90 minutes and Dortmund were good enough to score two more themselves, having had the luxury of bringing on Reus as a super sub. Madrid though still have Bale, Isco, Kroos, Pepe, etc to add to the team they fielded last night.
It was two good teams and a good game. I think we’d have a tough time against either of them but I know which I prefer to face and it’s not close.
And I really do see Sanchez and Ozil taking apart that Real backline. Yeah they do have a lot of firepower up front but the number of chances BVB got seems to suggest a vulnerability at the back that can be exploited with pace and power.
Once you get to the knock-out rounds, anything can happen. Theoretically, you’re among the best sixteen teams in Europe. Just like when you face a journeyman heavyweight boxer, you can still get knocked out by a single punch if you lower your guard at an inopportune moment. It works both ways i.e. for the champ and for the journeyman. Which are we? It doesn’t really matter. We just have to be ready.
And for those of you (like me) for whom the math is still too complex, a quick application of Sod’s Law will easily explain why we will be playing Real Madrid in the next round.
Hahaha
Real Madrid is probably the best team in world football right now. Talk about talent, depth, application and effort, it’s all there right now. It’s hard to fathom how they finished behind BVB in their group, but it was by slim margins and they represent a nightmare draw for any of the group winners. We often talk about having to face the best teams eventually, the need to focus on the domestic title charge; but I’ll argue it’s going to be good for Arsenal to stay in this competition as long as possible. The longer we are in it, the more excitement and buzz is around the club. Winning can become addictive, infectious even, and this squad of players is hungry for success. The longer they rub shoulders with the best clubs in the world, the more invincible they’ll feel. Getting spanked by Madrid in the round of 16 will do nothing positive for our title challenge. I realize I’m talking about an impossible to quantify intangible here but anyone who watches sports can tell you, winning carries an aura with it which can be punctured and lost 10x as quickly as it was built. It’s imperative that we keep winning in the PL and the CL.
Besides that, there are the more tangible benefits: more money in the coffers, a general sense of progress compared to recent seasons which will be important not just for fan sentiment (and the future of our manager) but also when Alexis and Ozil decide whether they want to commit to Arsenal for the rest of their prime years. Yes, we need progress from the CL round of 16. A semi-final appearance at least is not an unreasonable goal, but Real Madrid would provide a colossal and unfair obstacle. Let us hope the football gods continue to smile on us.
While we may have the largest individual chance of drawing Real Madrid, we also have a 78.26% chance of NOT drawing Real Madrid. So I don’t think it matters that much. I’d say Benfica, Leverkusen, Porto and Sevilla are good draw results for us, not to underestimate any of them, and the combined probability of getting one of them is 61.5%, well over half. Certainly finishing top of the group is beneficial.
I don’t mind us playing Madrid especially now that Bale is injured. They are slightly little less dangerous without him and we might do better against an attacking minded team like Madrid than a more organized team like Sevilla or Athletico (obviously can’t draw Athletico but just making a point). Let’s see what happens though. We won the group against all odds. Maybe we will draw Benfica or Porto against the odds as well.
Very interesting these odds derived mathematically. Still they exist in the realms of probabilities. In the final analysis mother luck takes the day. Mathematically, if a coin is flipped 10 times, we should have 5 heads and 5 tails. In reality (over a small sample like 10 times) we could have any combinations including the ultimate of say 10 heads to zero tails. That’s mother luck at work.
I don’t want us drawn against Real or Bayern. Simply for the same reason of odds, this time to give ourselves better odds for progressing to the next round. But there again luck is sometimes presented disguised. So if you ask me, I’d say that whoever we are drawn against is the best. Am sleeping easy.
Stop with this math already . You have given me a headache.
Besides, these percentages are only correct if the UEFA representative drawing those balls is wearing insulated leather gloves 🙂
I’d rather play real than athletics, Bayern, Dortmund or barca. I think out of that lot real are the choice. The ones who will let us play, and be nervous about playing us for that reason.
Arsenal are good enough for it not to matter who we get, it’s which arsenal turns up for the first leg.
Great explanation! But “odds are that Arsenal will play Real Madrid” isn’t quite correct – they’re our most likely result (a plurality of 22%), but odds are we WON’T play them because the majority of results are non-Real-Madrid results (78%).
I just have to say you’re wrong here but oddly don’t have the energy to explain why.