# Using the results from (2)-(b) and (2)-(c), derive |S|, that is the number of possible draws of 8 games?

Part A 16 teams are involved in a soccer tournament. We consider two types of tournaments: (1) There is a group stage: the teams are split into four groups of four teams G1, G2, G3 and G4. In each group, the teams play one against each other and are finally ranked according to their results. For (i, r) ∈ {1, . . . , 4} 2 , we denote by Gi(r) the team finishing at the r-th position in group Gi . The first two teams go through the second stage and the last two are eliminated. (a) For a specific group, how many possible final rankings can be observed? (b) For a specific group, how many pairs of teams can go through? (For two teams T1, T2, {T1, T2} and {T2, T1} are the same pair.) The second stage consists of a series of knock–out rounds which end by a final between two teams. More precisely, the quarter–final consists of the four games: Quarter Final (QF) QF#1 QF#2 QF#3 QF#4 Configuration G1(1) vs G2(2) G1(2) vs G2(1) G3(1) vs G4(2) G3(2) vs G4(1) (c) How many possible last eight configurations can be observed? [15 pts] (2) There is no group stage and 8 games (Game #1, …, Game #8) are drawn at random from the initial pool of 16 teams. We are interested in finding the number of possible draws of 8 games. We consider in this Question that the ordering in the game does not matter i.e. for two teams T1 and T2, the game T1 vs T2 is the same as T2 vs T1. The draw takes the form of a series of 7 draws without replacement, whereby two teams are selected from the pool of available teams, drawing in turns Game #1, …, Game #8. (a) The random experiment consists of the random draw of 8 games. Assign a probability space, that is a sample space S, an event space E and a probability measure P to this random experiment. (b) How many possible games can be formed for Game #1? (c) Game #1 has been drawn, how many possible games can be formed for Game #2? (d) Using the results from (2)-(b) and (2)-(c), derive |S|, that is the number of possible draws of 8 games? [hint: think of a decision tree.] (e) The Republic of Ireland and Germany are 2 of the 16 teams involved in the tournament. Using the previous questions, • how many possible draws feature the game Republic of Ireland vs Germany at Game #1? • how many possible draws feature the game Republic of Ireland vs Germany at Game #2?