## Champions League eight of finals’ draw: what are the odds?

*[This is a guest post by my friend and colleague Bernardo Nipoti from Collegio Carlo Alberto, Juventus Turin.]*

The matches of the group stage of the UEFA Champions league have just finished and next Monday, the 14th of December 2015, in Nyon, there will be a round of draws for deciding the eight matches that will compose the first round of the knockout phase.

As explained on the UEFA website, rules are simple:

- two seeding pots have been formed: one consisting of group winners and the other of runners-up;
- no team can play a club from their group or any side from their own association;
- due to a decision by the UEFA Executive Committee, teams from Russia and Ukraine cannot meet.

The two pots are:

Group winners: Real Madrid (ESP), Wolfsburg (GER), Atlético Madrid (ESP), Manchester City (ENG), Barcelona (ESP, holders), Bayern München (GER), Chelsea (ENG), Zenit (RUS);

Group runners-up: Paris Saint-Germain (FRA), PSV Eindhoven (NED), Benfica (POR), Juventus (ITA), Roma (ITA), Arsenal (ENG), Dynamo Kyiv (UKR), Gent (BEL).

Giving these few constraints, are there some matches that are more likely to be drawn than others? For example, supporters of Barcelona might wonder whether the seven possible teams (PSG, PSV, Benfica, Juventus, Arsenal, Dynamo Kyiv and Gent) are all equally likely to be the next opponent of their favorite team.

Although it surely could have been done analytically, I decided to tackle this problem with the brute force of simulation and, in few seconds, I got the answer that is summarized by next Table.

Rows refer to teams in the group of winners while columns are dedicated to runner-up teams. In each cell we have the probability that a match between the two teams corresponding to row and column, will be drawn. For example, the probability that Barcelona and Juventus will play a re-match of last year final is approximated by 0.133. A zero in the table indicates that the corresponding match cannot be played since it would violate one of the aforementioned constraints (e.g. Zenit and Dynamo Kyiv cannot meet due to rule 3). As an example, I report the bar graphs displaying the distribution for the next opponent of Arsenal (top), Barcelona (center) and Juventus (bottom).

The most likely match turns out to be Zenit-Arsenal with an approximated probability of 0.232: this can be explained by the fact that Zenit has only 6 and Arsenal only 5 possible opponents.

Finally, if I had to bet on the outcome of next Monday’s draw, I would pick this list of eight matches:

Real Madrid – Roma

Wolfsburg – Benfica

Atlético Madrid – Paris Saint-Germain

Manchester City – Dynamo Kyiv

Barcelona – PSV Eindhoven

Bayern München – Gent

Chelsea – Juventus

Zenit – Arsenal

According to the probabilities reported in the above Table, this is the most likely list of eight matches between all the outcomes that I observed in my simulation. Be aware though that the chances of winning the bet are very low since I observed this exact outcome only 110 times out of one million simulated draws!

Bernardo

luciosaid, on 14 December 2015 at 03:18shouldn’t the probabilities of gent playing against real m. and wolfsburg be equal? i get ~13.15185% for both

Bernardosaid, on 14 December 2015 at 14:45Lucio, you’re surely right. Consider though that the values in the table are relative frequencies (as observed in my simulation) and not exact probabilities, this should explain the difference you noticed.

Julyan Arbelsaid, on 14 December 2015 at 15:14So in the end, out of the most probable draw of eight matches, two were drawn, not bad.

Champions League 8 of finals’ attract: what are the odds? | A bunch of datasaid, on 21 April 2016 at 16:42[…] Supply link […]