# A PSG-Bayern Munich in the round of 16 of the Champions League? Probably, but...

A little more patience for Paris Saint-Germain (PSG)

A little more patience for Paris Saint-Germain (PSG). The capital club will soon know the name of its opponent for the knockout stages of the Champions League. The draw will begin at midday on Monday 18 December at the Union of European Football Associations (UEFA) headquarters in Nyon, Switzerland. At 1 p.m., the posters for the Europa League play-offs, the lower continental level, will be determined by the same process.

Like last year, PSG qualified by finishing second in their group. And as was the case last year, he could well cross paths with Bayern Munich in the next round, if the odds are to be believed (17.30%). Kylian Mbappé and his teammates, on the other hand, are less likely to meet one of the four Spanish teams – FC Barcelona, ​​Real Madrid, Atletico Madrid and Real Sociedad – and the London team Arsenal (13.59%). A little more for Manchester City (14.75%).

This year, the odds are surprisingly even, varying only from 13.21% to 20%, while some matches often exceed 30% or even 40%. The draw is therefore very open. The explanation lies in the fact that the four Spanish teams and the two English teams having validated their ticket for the round of 16 all finished first in their group. The three qualified Italian clubs all finished second.

Each round of 16 pits a group winner against a runner-up from another group and UEFA rules state that two teams from the same country cannot face each other at this stage of the competition. This season, the geographical constraint only concerns German teams: Bayern Munich and Borussia Dortmund having finished at the top of their group; RB Leipzig, second.

## A sequential procedure

But how do we calculate these probabilities? Since PSG has seven possible opponents, one might believe that the Parisians have a one in seven chance of meeting Bayern. But if we follow this reasoning, since Munich only has six possible opponents, the probability of PSG-Bayern would be 1/6. However, said probability cannot be equal to 1/7 and 1/6 at the same time.

It is tempting to believe that the probability of PSG-Bayern is 1,810/10,595 = 17.08%, since this year there are 10,595 tables of possible eighths, and that, among them, 1,810 have the poster between Parisians and Bavarians. In fact, this would be true if the draw consisted of choosing a ball, containing a complete draw of the round of 16, at random from 10,595. But, this is not how the draw is carried out.

Because the drawing procedure is, in fact, sequential. A runner-up from the group is drawn at random, then an algorithm identifies their eligible opponents – in other words, those who guarantee that the draw will not lead to a dead end. A team is therefore drawn at random from these eligible opponents. And we start the process again, until all the runners-up in the group have been drawn.

A combination of graph theory and probability theory makes it possible to calculate exactly the chances of having a PSG-Bayern in the next round. They are therefore 17.30%. Note that while Munich is Paris' most likely opponent, there is still more than an 82% chance that the Parisians will avoid the Munich residents. It is therefore much more likely that PSG will fall against a Spanish team (54.36%) than against Bayern.

The probabilities will change from the first team drawn on Monday. The “live probability calculator”, built with the help of my students at the Ecole des Ponts ParisTech, allows you to both simulate draws and visualize how they evolve during the draw, in a way that sometimes extreme. For example, last year, PSG-Bayern was the eighth and final match drawn, but from the fourth known match, the die was already cast: the probability of PSG-Bayern had increased to... 100%!

## What about French clubs in the Europa League?

This year, it is the Europa League play-off probability table that is the most interesting. The most likely match is Lens-Sporting Portugal (24.33%), due to the fact that the Sang et Or, who cannot meet Olympique de Marseille, Rennes and Toulouse, have only five possible opponents and Sporing , he only has six.

For OM, Rennes and Toulouse, the opponent probabilities vary little: Braga, Benfica and AC Milan being the slightly more likely opponents than the others. Toulouse has, in particular, a 15.25% chance of facing the Lombard club which has the same owner, the American fund Redbird.

The live probability calculator also allows you to simulate Europa League play-off draws and follow the evolution of the probabilities live during the real draw.

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