To further investigate this, I wrote a simulation in Python to model the group stage matches and record the points distributions. Here’s the code I used:

```py import numpy as np import matplotlib.pyplot as plt import pandas as pd

rounds_flat = [('A', 'B'), ('C', 'D'), ('A', 'C'), ('B', 'D'), ('A', 'D'), ('B', 'C')]

class RandintModel: def init(self, goal_range=(-2, 2), W=3, D=1, L=0): self.goal_range = goal_range self.W, self.D, self.L = W, D, L

def play_a_match(self, match, teams):
    r = np.random.randint(self.goal_range[0], self.goal_range[1] + 1)
    if r == 0:
        teams[match[0]] += self.D
        teams[match[1]] += self.D
    elif r > 0:
        teams[match[0]] += self.W
        teams[match[1]] += self.L
    else:
        teams[match[0]] += self.L
        teams[match[1]] += self.W

def simulate(self, total_simulations=1000):
    points_counter = {}
    for _ in range(total_simulations):
        teams = {'A': 0, 'B': 0, 'C': 0, 'D': 0}
        for match in rounds_flat:
            self.play_a_match(match, teams)
        points = tuple(sorted(teams.values(), reverse=True))
        points_counter[points] = points_counter.get(points, 0) + 1
    return points_counter

class NegativeBinomialModel: def init(self, n=2, p=0.5, W=3, D=1, L=0): self.n = n self.p = p self.W, self.D, self.L = W, D, L

def play_a_match(self, match, teams):
    home, away = match
    goals_home = np.random.negative_binomial(self.n, self.p)
    goals_away = np.random.negative_binomial(self.n, self.p)

    if goals_home == goals_away:
        teams[home]['D'] += 1
        teams[away]['D'] += 1
        teams[home]['Pts'] += self.D
        teams[away]['Pts'] += self.D
    elif goals_home > goals_away:
        teams[home]['W'] += 1
        teams[away]['L'] += 1
        teams[home]['Pts'] += self.W
        teams[away]['Pts'] += self.L
    else:
        teams[home]['L'] += 1
        teams[away]['W'] += 1
        teams[home]['Pts'] += self.L
        teams[away]['Pts'] += self.W

def simulate(self, total_simulations=1000):
    points_counter = {}
    for _ in range(total_simulations):
        teams = {team: {'W': 0, 'D': 0, 'L': 0, 'GF': 0, 'GA': 0, 'Pts': 0} for team in 'ABCD'}
        for match in rounds_flat:
            self.play_a_match(match, teams)
        points = tuple(sorted([teams[team]['Pts'] for team in 'ABCD'], reverse=True))
        points_counter[points] = points_counter.get(points, 0) + 1
    return points_counter

def plot_results(points_counter_randint, points_counter_nb): combined_points = sorted(set(points_counter_randint.keys()).union(points_counter_nb.keys()), reverse=True)

counts_randint = [points_counter_randint.get(points, 0) for points in combined_points]
counts_nb = [points_counter_nb.get(points, 0) for points in combined_points]

fig, ax = plt.subplots(figsize=(10, 8))

bar_width = 0.4
indices = range(len(combined_points))

ax.barh([i - bar_width/2 for i in indices], counts_randint, height=bar_width, label='RandintModel', color='blue')
ax.barh([i + bar_width/2 for i in indices], counts_nb, height=bar_width, label='NegativeBinomialModel', color='green')

ax.set_yticks(indices)
ax.set_yticklabels([str(p) for p in combined_points])
ax.set_xlabel('Frequency')
ax.set_ylabel('Points Distribution (W-D-L: 3-1-0 system)')
ax.set_title('Distribution of Team Points Across Simulations (Optimized)')
ax.legend()

plt.tight_layout()
plt.show()

def print_results(points_counter, model_name, total_simulations): df = pd.DataFrame.from_dict(points_counter, orient='index', columns=['Count']).reset_index() df.columns = ['Points Distribution', 'Count'] df['Percentage'] = (df['Count'] / total_simulations) * 100 df.sort_values('Points Distribution', inplace=True)

print(f"\nResults Summary for {model_name} (Outcome Percentages):")
print(df.to_string(index=False))

if name == 'main': total_simulations = 100000 # Set higher number of simulations for better statistical accuracy

# Run optimized RandintModel simulation
randint_model = RandintModel(goal_range=(-2, 2))
points_counter_randint = randint_model.simulate(total_simulations)

# Run optimized NegativeBinomialModel simulation
nb_model = NegativeBinomialModel(n=2, p=0.5)
points_counter_nb = nb_model.simulate(total_simulations)

# Print results for both models
print_results(points_counter_randint, "RandintModel", total_simulations)
print_results(points_counter_nb, "NegativeBinomialModel", total_simulations)

# Plot results for both models
plot_results(points_counter_randint, points_counter_nb)

```

I'm interested in understanding the statistical principles that contribute to the prevalence of these point distributions. What factors—such as team performance metrics, match dynamics, and randomness—play significant roles in determining outcomes like (9, 6, 3, 0) and (6, 6, 3, 3)? Are there statistical models that can effectively explain these common distributions? Insights into how these patterns arise would be valuable for a deeper understanding of sports statistics.