Champions League Draw

Introduction

There is much excitement and anticipation among football fans at the strong possibility that 4 English clubs (Arsenal, Chelsea, Liverpool and Manchester United) will reach the quarter finals of the UEFA Champions League. This will be clarified on Tuesday 11th March when Liverpool meet Inter Milan, in Italy, for the second leg of their match. They have already beaten Inter Milan 2 – 0 in the first leg, played at home in February.

Last year there was a similar situation in that 3 English clubs reached the quarter finals, and were not drawn against each other. All three went through to the semi-finals but only LIVERPOOL reached the final, where they lost to MILAN, 2 – 1.

The English fans this year, backed by the press, want each of these 4 teams to avoid each other in both the quarter finals and the semi-finals. The draw for these matches takes place on Friday 14th March. Details are available on many websites, including http://www.uefa.com/competitions/ucl/fixturesresults/index.html

What is the probability of this happening and, anyway, is this the best outcome for English football?

We will try to answer the questions in what follows.

Activities

Work out the answers to the questions below and fill in the boxes. Click on the Click this button to see if you are correct button to find out whether you have answered correctly. If you are right then will appear and you should move on to the next question. If appears then your answer is wrong. Click on to clear your original answer and have another go. If you can't work out the right answer then click on Click on this button to see the correct answer to see the answer.

Activity 1
When the quarter finals draw takes place, each of the 8 teams will be paired to make 4 matches.
In how many different ways can this be done? The order of each pair does not matter as they play two games (legs), home and away.

This is quite a complex calculation so let's try to break it down into managable steps.

There are several approaches, including what is called the ‘method of exhaustion’, in which you systematically write out (or maybe use a spreadsheet) each possible pairing. We’ll look at a method that doesn’t require much previous knowledge, just logic!

If we label the teams T1, T2, T3, ..... T8, then one possible outcome of the draw is

T1T2 | T3T4 | T5T6 | T7T8

Here T1 plays T2, T3 plays T4, etc. You could write down many more possibilities, but let’s try to be mathematical.

Let us consider in how many ways can we choose the first pair of teams in the draw. Completing the table below will help!

Complete the table by filling in the spaces to generate all the possible combinations for the pairs of teams drawn for the first match when there are 8 teams to choose from.
To fill in an empty space, first click on one of the teams listed on the right hand side of the grid. Click on an empty space to insert that team into the space. You can change a team currently in a space by selecting a different team and clicking on the slot to change the team

V V V V V V V














V
V
 V V V V
V
 V V V V
V V V V
V V V
V V
V

The total number of different pairs for the first match drawn is  

Activity 2
Consider now the choice of the pairings for the second match to be drawn. How many possibilities are there?

Complete the table, as before, by filling in the spaces to generate all the possible combinations for the pairs of teams drawn for the second match when there are now only 6 teams to choose from. (There are two fewer teams as these are now involved in the first match.)
To fill in an empty space, first click on one of the teams listed on the right hand side of the grid. Click on an empty space to insert that team into the space. You can change a team currently in a space by selecting a different team and clicking on the slot to change the team

V V V V V












V
 V V V
V V V
V V
V

The total number of different pairs for the second match drawn is  

Activity 3
By using your previous results, you should now be able to calculate the number of possible combinations for the third match (when there 4 teams left) and for the fourth match (when there are just two teams remaining to choose from).
HINT: use your completed tables to help you see the combinations with 4 teams and with just 2 teams.

The total number of different pairs for the third match drawn is  

The total number of different pairs for the fourth match drawn is  

Activity 4
Using the results we have calculated above, it is now possible to work out the total number of combinations for the 4 matches using the formula below;

No. of pairs for Match 1 × No. of pairs for Match 2 × No. of pairs for Match 3 × No. of pairs for Match 4

Fill in the spaces in the formula below.

× × × =  

Activity 5
Having calculated all the possible pairings for the 4 mathces we have one further subtle issue to consider. Let us consider the two groups of 4 pairs below as two potential ways in which the draw may have occurred;

T1T2 | T3T4 | T5T6 | T7T8
T7T8 | T5T6 | T3T4 | T1T2

You can see that Match 1 of the top selection is different to Match 1 of the bottom one, Match 2 of the top selection is different to Match 2 of the bottom one and so on. The method we have used so far to determine the total number of different pairings for each match would consider these two arrangements as different. However, if we consider the 4 matches as a whole, they are identical as in both instances, T1 plays T2, T3 plays T4, T5 plays T6 and T7 plays T8. As such we need to allow for this kind of repetition within the calculations completed above.

To determine the total number of repetitions for each arrangement, we need to work out in how many ways we can order the 4 matches. To do this, complete the table below.

Click on a space to select a match. Continue to click to change the match to the next one

The total number of arrangements for four matches is  

Hence, using the results we have calculated in the activities so far, the total number of different combinations of 8 teams paired up into 4 matches is given by

No. of pairs for Match 1 × No. of pairs for Match 2 × No. of pairs for Match 3 × No. of pairs for Match 4
No. of arrangements for four matches

Fill in the blanks in the equation below to give the solution to the formula above.

 =

Activity 6
Now we have succssfully calculated how many different combinations of four mathces there are, we can use it to calculate the probabilities we require.

By distinguishing between the English and the European teams there are only 3 possible outcomes for the 4 matches;

  • There are no matches involving English teams playing each other
  • There is one match where two English teams play each other
  • There are two matches where English teams play against each other

    • In order to calculate the probability of each of the three events described above happening, we first need to caluclate in how many ways each event can occur within the draw format.

    First, let us consider the event that no English team plays against another English team. For this to happen, each English team must play against one of the other 4 European sides. We can consider this as fixing the English teams in place and then arranging the four European sides around them. This is the same scenario as in Activity 5 where we had to find how many ways there are of arranging four objects.

    The number of ways in which the event that no English teams play each other can occur is  

    The probability that no English team meets another in the quarter finals is
    		=
    Remember to cancel your answer

    Activity 7
    Now let's look at the scenario where English teams play each other in two matches. In other words each English team plays another English team.

    Complete the tables below to find the number of ways that the event described above can happen.

    Click on a space to select a team. Continue to click to change the team to the next one
    Combining 4 English teams into pairs

    V V
    V V
    V V

    Combining 4 European teams into pairs

    V V
    V V
    V V

    Total number of possiblilities with two pairs of English teams = × =  

    The probability that there are two all English matches in the quarter finals is
    		=
    Remember to cancel your answer

    Activity 8
    We can now combine the previous two probabilities we've calculated to work out the probability that there is only one match in the quarter finals where two English sides play against each other.

    Probabilty that there is one all English match = 1 - (probability that there are no all English matches + probability that there are two all English matches)

    P(1 all English match) = 1 - (
    		+
    		) =
    Remember to cancel your answer

    Extension Activity
    The draw for the semi-final takes place at the same time as the quarter final draw.

    What is the probability of guaranteeing at least one English side reaching the final?
    HINT: Consider which combination of quarter final matches would gaurentee two English sides going onto the semis!

    Quarter-Final Draw Simulator

    Create the actual quarter-final draw using the simulator below

    MATCH 1
      V  

    MATCH 2
      V  

    MATCH 3
      V  

    MATCH 4
      V  

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    Produced by R.D.Geach March 2008