Can anyone help me calculate the probabilities of some UEFA champions league draws?

Question

So I've always been highly suspicious of some of the draws in the champions league in the past & I'd love to calculate the probabilities but I can't do it myself. Can anyone help me? Specifically can someone help me find the probability of Arsenal playing Bayern four out of five seasons? First two times in the round of 16 (2012/2013 season & 2013/14 season) then in the group in 2015/2016 and again in the round of 16 in 2016/17. So basically four out of five seasons with only 2014/15 not included

Second: Chelsea vs PSG in the quarter final in 2013/14 and then in the round of 16 in 2014/15 and again in the round of 16 in 2015/16 so basically three seasons in a row.

If you could also calculate the probability of both of these happening that would be very impressive, thanks.

If you need help with all the possible scenarios please let me know & I'll try to link you.

Answer 1

I will not answer your question, but I will make some comments. You say that "I've always been highly suspicious of some of the draws in the champions league in the past", so my answer will be centered around this comment.

You ask: What is the probability of Arsenal playing Bayern four out of five seasons?

I don't believe that this is the question you want to ask. I am assuming that you are interested in figuring out, by looking at the data of the last draws, whether we have reason to suspect that the draws have been fixed in any way.

Furthermore I assume that you would be equally suspicious

• If Some pair of teams played eachother at least 4 times out of the 5 last seasons.
• If Some pair of teams played eachother at least 7 times out of the 9 last seasons. By this I mean that it seems somewhat arbitrary to look at the last $5$ years. Why are you not considering the last $3$ years or even $10$ years, and why not include all seasons since the Champions League started? It seems to me that you may be cherrypicking the number $5$.
• If this had happened in any $5$-year period, say if for instance Ajax and Porto played eachother in 2010,2011,2013 and 2014.

Furthermore, there is some trouble with the data:

• The same teams do not participate in the tournament each year.
• Champions League is a knock-out tournament. In other words, we do not know how many games each team is expected to play each season. We could give a weight the teams by the number of games they have played historically, e.g. in recent years Real Madrid have played $7$ other teams each year, while let's say that Bayern has played $6$. A problem with this however is that if the tournement draws are already being fixed, we have no way of knowing how far a team would go in the tournament if the draws were not fixed.
• The tournament format has changed since the Champions League was created.

Further comments

• Teams from different countries have a higher chance at playing eachother, as the round of 16 draws do not allow teams from the same country to play eachother. This means that calculating the probability that a given pair of teams qualified for the round of 16 play eachother in a given year is in itself quite cumbersome.
• Draws in the group stages are seeded.
• The best teams in a fair tournament have a higher chance of playing eachother as they play more games.

There are undoubtedly even more things one should consider.

Conclusion: I think it would be more interesting to answer a question like this:

"Since the start of the current format of the Champions League, is there reason to believe that an unusually high number of pairs of teams have played eachother frequently?"

Here we of course have to specify what we consider "unusual" and "frequent".

From your question, it seems to me that you consider that 1 pairs of teams play eachother in 80% of the tournaments over a period of $5$ years in addition to 1 pair of teams playing eachother 100% of the tournaments over a period of $3$ years to be frequent. Perhaps someone will do the calculations and make a reasonable conclusion.