Bayern Munich, PSG’s most likely opponent in the round of 16

The draw for the knockout stages of the Champions League takes place on Monday, November 7, in Nyon (Switzerland), at the headquarters of the Union of European Football Associations (UEFA), from noon. A meeting eagerly awaited by the qualifiers of the first round. And this year, some draw probabilities are quite extreme. Liverpool, for example, have a 37.1% chance of meeting Bayern Munich, while Paris Saint-Germain (PSG) only have a 10.8% chance of meeting Porto. PSG’s most likely opponent is Bayern Munich (19.5%), followed by Napoli (17.1%). The following table gives the draw probabilities for the sixteen qualified teams:

If Liverpool have so many chances of meeting Bayern Munich, it is because these two teams have only four possible opponents each. Conversely, PSG and Porto each have seven potential opponents.

Why such a disparity? This is the consequence of the constraints that UEFA imposes during the draw. Group winners must be paired with group runners-up. In addition, two teams from the same country or which were part of the same group during the group stage cannot be drawn against each other. Thus, Liverpool, second in group A, cannot meet either the English group winners (Tottenham, Chelsea, Manchester City) or Naples, winner of group A.

Constraints during pulling

Let’s study the probability that PSG fall on Bayern Munich. You would think that PSG, who have seven possible opponents, have a one in seven chance of falling on Bayern. This will be the case if PSG are the first group runners-up drawn, but it only has a one in eight chance of happening. And then the probability of PSG-Bayern cannot in any case be equal to both one in seven (seen from Paris) and one in four (seen from Munich)!

One might also think that, to calculate the probability of two teams A and B meeting, it would suffice to list all of the complete eligible draw results (there are 3,876 this year), and then calculate the proportion of eligible draws where A meets B. For example, among the 3,876 possible results of the draw, there are exactly 702 results for which PSG meets Munich (i.e. 18.1%).

It is tempting to conclude that Paris have an 18.1% chance of meeting Bayern. This would be the case if the draw were to randomly draw one ball from 3,876, with each ball representing a complete qualifying draw. But this is not feasible in practice. The draw follows other rules, and the probability of a PSG-Bayern clash is actually not equal to 18.1%.

To calculate this probability, it is necessary to take into account the way in which UEFA proceeds to impose the constraints during the draw. Eight balls, corresponding to the eight runners-up, are placed in an urn and drawn successively. Each time a second group is chosen, a computer provides the list of possible opponents for this team. One of these opponents is then drawn by lot.

Determining the list of possible opponents can be more complicated than it seems. Imagine for example that the first four clashes drawn are Liverpool-Benfica, PSG-Chelsea, Leipzig-Naples and Bruges-Manchester City, and that Milan is the fifth ball drawn from the urn.

Even if Milan seems a priori to be able to play against Tottenham, Tottenham would not be listed as a possible opponent for Milan. Indeed, pairing Milan and Tottenham would lead to a dead end: Bayern could then only meet a German club or Inter, which is prohibited, Bayern and Inter having already faced each other in the group stage.

The impact of Benfica’s last goal

By considering all the possible scenarios of the draw and by using the formula of the total probabilities, one can calculate the exact probabilities by computer. The draw procedure has an impact on the odds: PSG-Bayern are more likely than they should be (19.5% vs. 18.1%), while Liverpool-Bayern are less likely than they should be being (37.1% versus 39.9%). By “should be”, I mean: if the 3,876 eligible draws were equiprobable.

To illustrate the impact of the procedure on the odds, I have also calculated what the odds would be if, instead of emptying the group runners-up pot, UEFA were to empty the group winners pot. The probabilities would indeed be slightly different. For example, the probability of Liverpool-Bayern would drop slightly, to 36.9%.

Finally, I calculated what the odds would have been had Benfica not scored a sixth goal in added time against Maccabi Haifa on Wednesday night. PSG, who would then have won their group, would have had a 20.6% chance of falling against Liverpool. The probabilities would have been different, not only for PSG and Benfica, but also for all the other teams. It’s amazing how much a goal affects the rest of the competition.

Finally, note that if Bayern is the most likely opponent of PSG, PSG is also the least likely opponent of Bayern (along with Bruges), which may surprise but is not contradictory! It should also be noted that the fact that PSG has a 19.5% chance of meeting Bayern also means that it has a much better chance (80.5%) of not falling on them!