**About this project:** After the USA was drawn into the "group of death" in 2014 (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:

# points | % to advance | % advance without tiebreaker |
---|---|---|

0 | 0.00% | 0.00% |

1 | 0.00% | 0.00% |

2 | 1.23% | 0.00% |

3 | 7.87% | 2.78% |

4 | 54.32% | 25.93% |

5 | 98.77% | 96.30% |

6 | 97.53% | 92.59% |

7 | 100.00% | 100.00% |

8 | - | - |

9 | 100.00% | 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!

Someone wrote in asking how likely it was for a group to finish in some interesting ways. Again, this assumes every game has an equal probability of drawing or either team winning:

There are six games in the group stage, and the probability of a draw is 1/3, so the probability of all six games being draws is (1/3)^6 = 1/729, or about 0.14%.

First we have to pick which team will finish first, second, third, and last: there are 4! = 24 ways to do that. Now that we know that, each match has to end with the higher ranked team defeating the lower ranked team, so like the previous scenario, there's a 1/729 chance of that. So the overall chance is 24/729, or about 3.3%.

Hmm, this is trickier. Let's say the teams are A, B, C, and D. First we have to choose which teams are going to draw with each other; this can be either A/B and C/D, A/C and B/D, or A/D, and B/C, so there are 3 choices. Then for the other 4 games, there must be a cycle; let's say we're in the A/B and C/D draw scenario, there are only two possible series of games. Either A beats C beats B beats D beats A, or A beats D beats B beats C beats A. So there are 3*2=6 determinations of how all the group stage games have to work out, and the overall chance is 6/729, or about 0.82%.

**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