World Cup: Advancing out of the Group Stage

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

graph of probabilty of advancing versus points


Some other group probabilities

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:

All four teams getting 3 draws

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%.

One team getting 9 points, one team getting 6 points, one team getting 3 points, one team getting 0 points

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%.

All teams getting one win, one draw, and one loss

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%.

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