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World Cup soccer match draws. Every 4 years the world鈥檚 32 best national soccer teams compete for the World Cup. Run by FIFA (F茅d茅ration Internationale de Football Association), national teams are placed into eight groups of four teams, with the group winners advancing to play for the World Cup. Chance(Spring 2007) investigated the fairness of the 2006 World Cup draw. Each of the top 8 seeded teams (teams ranked 1鈥�8, called pot 1) were placed into one of the eight groups (named Group A, B, C, D, E, F, G, and H). The remaining 24 teams were assigned to 3 pots of 8 teams each to achieve the best possible geographic distribution between the groups. The teams in pot 2 were assigned to groups as follows: the first team drawn was placed into Group A, the second team drawn was placed in to Group B, etc. Teams in pots 3 and 4 were assigned to the groups in similar fashion. Because teams in pots 2鈥�4 are not necessarily placed there based on their world ranking, this typically leads to a 鈥済roup of death,鈥� i.e., a group involving at least two highly seeded teams where only one can advance.

1. In 2006, Germany (as the host country) was assigned as the top seed in Group A. What is the probability that Paraguay (with the highest ranking in pot 2) was assigned to Group A?
2. Many soccer experts viewed the South American teams (Ecuador and Paraguay) as the most dangerous teams in pot 2. What is the probability one of the South American teams was assigned to Group A?
3. In 2006, Group B was considered the 鈥済roup of death,鈥� with England (world rank 2), Paraguay (highest rank in pot 2), Sweden (2nd highest rank in pot 3), and Trinidad and Tobago. What is the probability that Group B included the team with the highest rank in pot 2 and the team with one of the top two ranks in pot 3?
4. In drawing teams from pot 2, there was a notable exception in 2006. If a South American team (either Ecuador or Paraguay) was drawn into a group with another South American team, it was automatically moved to the next group. This rule impacted Group C (Argentina as the top seed) and Group F (Brazil as the top seed), because they already had South American teams, and groups that followed these groups in the draw. Now Group D included the eventual champion Italy as its top seed. What is the probability that Group D was not assigned one of the dangerous South American teams in pot 2?

Step by step solution

01

Important formula

The formula for probability is$\mathbf{P}\mathbf{=}\frac{\mathrm{Favourable}\mathrm{out}\mathrm{comes}}{\mathrm{Total}\mathrm{out}\mathrm{comes}}$

02

(a) Find the probability of Paraguay being allocated to group A

Let,

${\mathbf{A}}_1$=The event (Paraguay is allocated to group A)

${\mathbf{A}}_2$=The event (Ecuador is allocated to group A)

${\mathbf{B}}_1$=The event (Paraguay is allocated to group B)

${\mathbf{B}}_2$=The event (Sweden or a top team in pot 3 is allocated to group B)

Now the probability is

$\begin{array}{l}\mathbf{P}\mathbf{(}{\mathbf{A}}_1\mathbf{)}\mathbf{=}\frac{1}{8}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{125}\end{array}$.

So, the probability of Paraguay for group A is 0.125

03

(b) Determine the probability one of the South American teams was assigned to Group A

$\begin{array}{l}\mathbf{P}\mathbf{(}{\mathbf{A}}_1鈭�{\mathbf{A}}_2\mathbf{)}\mathbf{=}\mathbf{P}\mathbf{(}{\mathbf{A}}_1\mathbf{)}\mathbf{+}\mathbf{P}\mathbf{(}{\mathbf{A}}_2\mathbf{)}鈭�\mathbf{P}\mathbf{(}{\mathbf{A}}_1鈭�{\mathbf{A}}_2\mathbf{)}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{1}{8}\mathbf{+}\frac{1}{8}鈭�\mathbf{0}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{25}\end{array}$

Hence, the probability of Ecuador for group A is 0.25.

04

(c) Find the probability that Group B included the team with the highest rank in pot 2 and the team with one of the top two ranks in pot 3

$\begin{array}{c}\mathbf{P}\mathbf{(}{\mathbf{B}}_1鈭�{\mathbf{B}}_2\mathbf{)}\mathbf{=}\mathbf{P}\mathbf{(}{\mathbf{B}}_1\mathbf{\left)}\mathbf{.}\mathbf{P}\mathbf{(}{\mathbf{B}}_2\mathbf{\right)}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{1}{8}\mathbf{.}\frac{2}{8}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{0312}\end{array}$

Accordingly, the probability is 0.0312.

05

(d) Evaluate the probability that Group D was not assigned one of the dangerous South American teams in pot 2

The total no. of ways to fill the group (A, B, C, D). Group c can be filled in 6 ways, and group D can be filled in 7 ways. Group A can be filled in 6 ways, and Group B can be filled in 5 ways. Then possible ways are $\mathbf{6}\mathbf{脳}\mathbf{7}\mathbf{脳}\mathbf{6}\mathbf{脳}\mathbf{5}\mathbf{=}\mathbf{1260}$.

Group c can be filled in 6 ways; group D can be filled in 2 ways. Group A can be filled in 6 ways, and Group B can be filled in 5 ways. Then possible ways are $\mathbf{6}\mathbf{脳}\mathbf{2}\mathbf{脳}\mathbf{6}\mathbf{脳}\mathbf{5}\mathbf{=}\mathbf{360}$.

The probability that group C will not feature Paraguay or Ecuador whereas group D will have both is$\frac{360}{1260}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{286}$

The probability that one of the deadly South American teams from pot 2 was not allocated to the group is$\mathbf{1}鈭�\mathbf{0}\mathbf{.}\mathbf{286}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{714}$.

Therefore, the probability is 0.714.

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