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# Probability question about the UEFA Champions League Watch

1. Some of you may know that the draw for the UEFA champions league knock out stages is tomorrow. An interesting probability question has arisen from this (about Arsenal). Basically, Arsenal will only be able to draw one of four teams, however the probability of selecting each team is not the same.

__

The rules for the draw are:

1) A team from the unseeded pot must play a team from the seeded pot (listed below). A team from the unseeded pot is drawn randomly first, and to see which seeded team they play, a team from the seeded pot is randomly selected immediately afterwards (subject to restrictions below). This process continues until all unseeded teams are drawn seeded teams.

2) Teams from the same group (capital letter) cannot play one another.
Spoiler:
Show
For example, if Arsenal are drawn from the unseeded pot first, they cannot play Shakhtar Donetsk from the seeded pot as they are both from group H (as shown in brackets)

3) Teams from the same country cannot play one another.
Spoiler:
Show
For example, if Arsenal are drawn from the unseeded pot first, they cannot play Manchester United, Chelsea or Tottenham Hotspur. This is because they are all from England (as shown in brackets)

Unseeded pot (group, country)

Inter Milan (A, Italy)
Lyon (B, France)
Valencia (C, Spain)
Copenhagen (D, Denmark)
Roma (E, Italy)
Marseille (F, France)
AC Milan (G, Italy)
Arsenal (H, England)

Seeded pot
(group, country)

Tottenham (A, England)
Schalke (B, Germany)
Manchester United (C, England)
Barcelona (D, Spain)
Bayern Munich (E, Germany)
Chelsea (F, England)
Shakhtar Donetsk (H, Ukraine)

Examples for clarification:
Spoiler:
Show
For instance, if Arsenal are the first team to be drawn from the unseeded pot, by 2) they will be unable to play Shakhtar Donetsk and by 3), they will be unable to play Tottenham, Manchester United and Chelsea. The probability of drawing Schalke, Barcelona, Bayern Munich or Real Madrid are all 1/4.

Say Arsenal draw Schalke; the first match is "Arsenal v Schalke".

Now if Valencia are the second team to be drawn from the unseeded pot, by 1), they cannot play Schalke as they have already been selected from the seeded pot, by 2) they cannot select Manchester United because they are both from group C and by 3) they cannot select Barcelona or Real Madrid because they are from Spain. The probability of selecting the remaining 4 teams is 1/4.

___

Question:

What are the respective probabilities of Arsenal playing the four teams they can draw (Schalke, Barcelona, Bayern Munich, Real Madrid)?
2. Erm well, it would be ...
one quarter, one quarter, one quarter, one quarter...

However, say if Real Madrid have 7 different opponents they could be seeded against, you would say it was a probability of one seventh to play arsenal...

But this makes no sense!?!
The probability of Arsenal vs Real is one quarter, but Real vs Arsenal is one seventh? These are identical cases, but different probabilities are associated with the same outcome?

I HAVE CONFUSED MY SELF SO MUCHHHHH
3. If it was 1/4 each, I wouldn't have taken the effort to make a massive OP

But a team from the unseeded pot is always selected first.
5. This is why Stats in Maths A Level is a killer!!!
6. Bump, the draw is in a few hours.

The probability is not 1/4 for each team; e.g. if Valencia are selected before Arsenal in the draw, Valencia are able to select Bayern Munich and Schalke (who Arsenal can play) however they are unable to pick the two other teams Arsenal can play (Real Madrid and Barcelona) because of rule 3); they are Spanish teams. This will have an effect on the probability for Arsenal.

Have a go!
7. (Original post by n1r4v)
Bump, the draw is in a few hours.

The probability is not 1/4 for each team; e.g. if Valencia are selected before Arsenal in the draw, Valencia are able to select Bayern Munich and Schalke (who Arsenal can play) however they are unable to pick the two other teams Arsenal can play (Real Madrid and Barcelona) because of rule 3); they are Spanish teams. This will have an effect on the probability for Arsenal.

Have a go!
Are you sure? I simulated the draw 100000 times, and Arsenal drew Schalke 25128 times, Barca 24920 times, Bayern 25042 times and Real 24910 times...
8. (Original post by sonofdot)
Are you sure? I simulated the draw 100000 times, and Arsenal drew Schalke 25128 times, Barca 24920 times, Bayern 25042 times and Real 24910 times...
Thanks for the simulation! The results are very interesting; I can't put my finger on why the German teams are selected slightly more than the Spanish teams though.

Can you point out the flaw in my reasoning?

The way I saw it... apart from Valencia, if any other unseeded teams were picked before Arsenal, they'd have an equal probability of selecting Schalke, Barca, Bayern and Real and so I think that it would mean that overall, the probability of Arsenal selecting these teams remains equal (with the condition that Valencia is not selected before Arsenal)

However if Valencia was picked before Arsenal in the draw, they would be unable to select Real and Barca (and would only be able to pick Schalke or Bayern if available) and so it's more likely that Barca and Real would "remain" for Arsenal (I find it hard to express this part into words). And so it would be slightly more likely for them to pick a Spanish team.

Can you see the flaw in the wishy-washy reasoning ?

Though I wouldn't be surprised if I was wrong; 2) and 3) seem to cause all sorts of problems.
9. (Original post by n1r4v)
Thanks for the simulation! The results are very interesting; I can't put my finger on why the German teams are selected slightly more than the Spanish teams though.

Can you point out the flaw in my reasoning?

The way I saw it... apart from Valencia, if any other unseeded teams were picked before Arsenal, they'd have an equal probability of selecting Schalke, Barca, Bayern and Real and so I think that it would mean that overall, the probability of Arsenal selecting these teams remains equal (with the condition that Valencia is not selected before Arsenal)

However if Valencia was picked before Arsenal in the draw, they would be unable to select Real and Barca (and would only be able to pick Schalke or Bayern if available) and so it's more likely that Barca and Real would "remain" for Arsenal (I find it hard to express this part into words). And so it would be slightly more likely for them to pick a Spanish team.

Can you see the flaw in the wishy-washy reasoning ?

Though I wouldn't be surprised if I was wrong; 2) and 3) seem to cause all sorts of problems.
stick it in the maths forum, or i will, someone will blatently help work it out
10. (Original post by jermaindefoe)
stick it in the maths forum, or i will, someone will blatently help work it out
It is in the maths forum
11. (Original post by n1r4v)
It is in the maths forum
fml

12. (Original post by n1r4v)
Thanks for the simulation! The results are very interesting; I can't put my finger on why the German teams are selected slightly more than the Spanish teams though.

Can you point out the flaw in my reasoning?

The way I saw it... apart from Valencia, if any other unseeded teams were picked before Arsenal, they'd have an equal probability of selecting Schalke, Barca, Bayern and Real and so I think that it would mean that overall, the probability of Arsenal selecting these teams remains equal (with the condition that Valencia is not selected before Arsenal)

However if Valencia was picked before Arsenal in the draw, they would be unable to select Real and Barca (and would only be able to pick Schalke or Bayern if available) and so it's more likely that Barca and Real would "remain" for Arsenal (I find it hard to express this part into words). And so it would be slightly more likely for them to pick a Spanish team.

Can you see the flaw in the wishy-washy reasoning ?

Though I wouldn't be surprised if I was wrong; 2) and 3) seem to cause all sorts of problems.
My simulation had a massive hole in it, so just ignore that post...

However, I think I've found a flaw in the draw: what happens if the draw goes like this:
Roma - Chelsea
Marseille - Spurs
Inter - Man U
Valencia - Bayern
Lyon - Shakhtar
AC Milan - Schalke
Then the only teams that are left are Copenhagen and Barca, and they were in the same group, so can't be drawn against each other... Would they just do the draw again?
13. If they would just do it again, I've rerun my simulation to account for this (it happened 24000/100000 times!), and the Spanish teams had higher probabilites of around 0.278, compared to the German teams getting 0.222, so I think you're right
14. Well I suppose it's irrelevant now, but if, after the first five had gone like they did, then Marseille had drawn Barca, Arsenal would have had no-one to play...
15. I actually asked it on a different forum a few days ago, nobody responded but I think they'd have no choice but to restart.

But thanks for the simulation . It's actually a lot more pronounced than I expected.
16. (Original post by n1r4v)
I actually asked it on a different forum a few days ago, nobody responded but I think they'd have no choice but to restart.

But thanks for the simulation . It's actually a lot more pronounced than I expected.
I'm pretty sure that the teams that the picked team cannot play are removed from the draw, e.g. when arsenal were drawn, only Barcelona were left in the bowl.

Roma v Shakhtar Donetsk
AC Milan v Tottenham Hotspur
Valencia v Schalke
Inter Milan v Bayern Munich
Arsenal v Barcelona
Marseille v Manchester United
FC Copenhagen v Chelsea

For your original question, Schalke and Bayern could get any of 7 inc. arsenal, whereas Barça and Real could get any one of 6 inc. arsenal.

This suggests the chances of Arsenal getting Barça or Real was higher, but I can't give you figures
17. (Original post by Mr ABC)
I'm pretty sure that the teams that the picked team cannot play are removed from the draw, e.g. when arsenal were drawn, only Barcelona were left in the bowl.

Roma v Shakhtar Donetsk
AC Milan v Tottenham Hotspur
Valencia v Schalke
Inter Milan v Bayern Munich
Arsenal v Barcelona
Marseille v Manchester United
FC Copenhagen v Chelsea

For your original question, Schalke and Bayern could get any of 7 inc. arsenal, whereas Barça and Real could get any one of 6 inc. arsenal.

This suggests the chances of Arsenal getting Barça or Real was higher, but I can't give you figures
Regarding your first point, I'm not sure how that's relevant, but the point was it is possible for there to be no teams left in the bowl for Arsenal, which would mean that the process is forced to restart.

Regarding your second point, because unseeded teams are selected first, it's more of a case of which team Arsenal get, instead of which team (Sch, Bay, Bar, Real) get Arsenal.
18. Arsene wenger a couple weeks ago: We don't care who we face in the next round even barcelona.

Haha
19. (Original post by sonofdot)
Well I suppose it's irrelevant now, but if, after the first five had gone like they did, then Marseille had drawn Barca, Arsenal would have had no-one to play...
(Original post by n1r4v)
Regarding your first point, I'm not sure how that's relevant, but the point was it is possible for there to be no teams left in the bowl for Arsenal, which would mean that the process is forced to restart.
Surely, if after the first five Marseille was drawn, Barça would have been removed from the pot. I'm not certain but it just seems doubtful that they would wipe all the previous draws and start again.

(Original post by n1r4v)
Regarding your second point, because unseeded teams are selected first, it's more of a case of which team Arsenal get, instead of which team (Sch, Bay, Bar, Real) get Arsenal.
Yes, but they are basically the same thing. My point is that when Arsenal is picked, the two german teams are more likely to have already been picked as they have more potential pairings.
20. (Original post by Mr ABC)
Surely, if after the first five Marseille was drawn, Barça would have been removed from the pot. I'm not certain but it just seems doubtful that they would wipe all the previous draws and start again.
I've looked at the rules for the procedure and it didn't mention anything like that, and they certainly wouldn't remove a team pre-emptively . I think the main "rule" for any failure in the procedure is to restart the whole process (not only failures intrinisic to the procedure itself like the one described, but other failures like repeating a team twice or forgetting to place a team inside a ball)

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