[OC] Ich habe den 8. Spieltag der Champions League 20.000 Mal simuliert. Hier ist die Wahrscheinlichkeitsverteilung der endgültigen Ligaphasen-Wertung.

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Excellent work! The bulges are amazing

**The Visualisation** This is a „Violin Plot“ showing the density of finishing positions for the 36 UCL teams.

* **The „Bulge“:** The wider the shape at a specific rank, the higher the probability the team finishes there.
* **The Line:** The white internal bars show the interquartile range (where the middle 50% of outcomes fall).

**Data Source**

* **Fixtures & Standings:** Official UEFA Matchday 8 Schedule.
* **Team Strength:** [ClubElo.com](http://ClubElo.com) ratings (used to weight the Poisson simulation).

**Key Takeaways**

1. **The Certainty:** Look at **Arsenal** and **Bayern Munich** at the top. Their shapes are thin and compressed. The model is nearly 100% certain of their finishing position (Rank 1 or 2).
2. **The Chaos:** Look at **PSG (Paris Saint-Germain)**. They have a massive „bimodal“ distribution (two distinct bulges). This means they will likely either qualify comfortably in the Top 8 OR drop significantly; there is very little middle ground for them.
3. **The Bubble:** The fight for Rank 24 (the elimination cutoff) creates a massive spread of probability for the **12 teams straddling the line**, where a single goal can swing a team 5+ positions.

**Methodology**

* **Engine:** Python (NumPy/Pandas) running 20,000 Monte Carlo simulations.
* **Scoring:** Strength-weighted Poisson distribution based on current team Elo ratings for remaining fixtures.
* **Tiebreakers:** Full UEFA Article 18 implementation (H2H, Goal Difference, Goals Scored, Away Goals, Wins, Away Wins).

**Tools Used**

* Python (Matplotlib)
* Visuals rendered in a custom „Midnight Analytics“ theme.
* [**Source Code / GitHub**](https://github.com/ACl365/ucl-permutations)

uau! this is not only beautiful but gorgeous data!
do you have any explanation for the multi modality observed on the different teams?