World Cup: Advancing out of the Group Stage

About this project: After the USA was drawn into the "Group of death" (with Germany, Portgual, and Ghana), I wondered how many points it would take to get out of the group stage.

The way the group stage works is: four teams, each plays each other one time, wins are worth 3 points, draws are worth 1 point. The top two teams advance.

To simplify things I just assumed every game had an equal probability of drawing or either team winning. For tiebreakers, I assumed a uniform distribution of who gets to go on. So, without further adieu:

00.00%0.00%
10.00%0.00%
21.23%0.00%
37.87%2.78%
454.32%25.93%
598.77%96.30%
697.53%92.59%
7100.00%100.00%
8--
9100.00%100.00%

Observations:

• There's a huge jump between 3 and 4 points and 4 and 5 points. 4 points is kind of average (one win, one draw, and one loss), so it makes sense that you have around a 50-50 chance of advancing.
• There's also a weird tiny dip between 5 and 6 points, probably because if you have 5 points you know there are two draws, which lowers the total amount of available points to other teams in the group.
• Also, even if you only get two points you're not mathematically eliminated!

Source files:

• worldCupSimulator.py - the Python script that generated these. Code is very hacky because I wrote it quickly. It just enumerates all 3^6 possibilities and tallies them up.
• rplotscript.txt - script to generate the graph