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 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 to see the answer.
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; 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
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 European teams into pairs
Total number of possiblilities with two pairs of English teams = × =
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)
The draw for the semi-final takes place at the same time as the quarter final draw.
Create the actual quarter-final draw using the simulator below
Produced by R.D.Geach March 2008 |