can South Africa qualify? a study of mathematical possibility

[gusl]

The first stage of the World Cup Finals consists of a number of groups of 4 who all play each other. The top two teams of each group qualify for the next round. A win is worth 3 pts, a draw 1pt.

9 pts, i.e. 3-0-0: 1st place guaranteed
7 pts, i.e. 2-1-0: 1st place not guaranteed but likely; qualification guaranteed
6 pts, i.e. 2-0-1: anything can happen, but qualification is very likely (not guaranteed: in case the 3 top teams all lose 1 game and win 2, one of them has to stay out. example)
5 pts, i.e. 1-2-0: anything can happen (in case the top 3 teams all draw with each other, 1 has to stay out)
4 pts, i.e. 1-1-1: anything can happen, e.g. all teams could end up with 4 points. example
3 pts, i.e. 1-0-2, 0-3-0: 1st place impossible, qualification is unlikely but possible (in the 0-3-0 situation, 1 non-draw between other teams makes first place impossible; 2 non-draws between other teams makes qualification impossible) The 1-0-2 situation is symmetrical to the 2-0-1 above.
2 pts, i.e. 0-2-1: 1st place impossible, qualification is very unlikely but possible (in case the 3 bottom teams all draw with each other, 1 must still rise)
1 pt or less: qualification is impossible

(Back when they had 24 teams, 4 out of 6 3rd-places would make it to the playoffs)


Here are the Group A results so far:

South Africa 1 x 1 Mexico
France 0 x 0 Uruguay
South Africa 0 x 3 Uruguay
France x Mexico
South Africa x France
Uruguay x Mexico

Question: can South Africa qualify, given that it has 1 draw and 1 loss?

If they draw with France, they end up with 2pts, and in this case they cannot qualify this way (since no team is going to have 3-0-0).

So let's assume that South Africa beats France. Would they qualify then? The answer is a definite MAYBE, depending on the performance of other teams.


Possible scenarios:

France beats Mexico, Uruguay beats Mexico: MAYBE
UR 7
SA 4
FR 4
MX 1

France and Mexico draw, Uruguay beats Mexico: YES
UR 7
SA 4
FR 2
MX 2

Mexico beats France, Uruguay beats Mexico: MAYBE
UR 7
SA 4
MX 4
FR 1

France beats Mexico, Uruguay and Mexico Draw: MAYBE
UR 5
SA 4
FR 4
MX 2

France and Mexico draw, Uruguay and Mexico Draw: YES
UR 5
SA 4
MX 3
FR 2

Mexico beats France, Uruguay and Mexico Draw: NO
UR 5
MX 5
SA 4
FR 1

France beats Mexico, Mexico beats Uruguay: MAYBE FOR EVERYBODY, South Africa could even come out first!
UR 4
SA 4
MX 4
FR 4

France and Mexico draw, Mexico beats Uruguay: MAYBE
MX 5
UR 4
SA 4
FR 2

Mexico beats France, Mexico beats Uruguay: MAYBE
MX 7
UR 4
SA 4
FR 1


France x Mexico will collapse within the next 24h. South Africa is hoping for a draw. If France wins against Mexico, South Africa will have to beat a fierce France by enough goals (fierce being the opposite of "indifferent"). If Mexico wins, Uruguay and Mexico will be happy to draw with each other in the final match... which would certainly start conspiracy theories about fixed matches.

Tie-breaking rules

Comments

[rweba.livejournal.com]

Excellent! I was wondering about this.

I wonder if World Cup would be more exciting if it was a single elimination tournament like "March Madness"?

Might lead to a lot of penalty shoot outs, but maybe that would be better than what we have now ....

[gustavolacerda.livejournal.com]

I wouldn't want that... give everyone their 3 matches!

[rweba.livejournal.com]

Thats a good point, especially since they traveled so far...

[easwaran.livejournal.com]

Doesn't it become that after they advance out of the brackets? I think this way every team gets to play at least three games - it would be a shame for everyone to fly half way around the world and then half of them to just quit after their first game.

[rweba.livejournal.com]

True, I suppose it also allows a team to recover from one bad game.

[rweba.livejournal.com]

Mexico beat France, will be interesting Uruguay and Mexico both play for the draw.

One reason not to play for the draw is to avoid the winner of group B which is probably Argentina.

At least both SA and France will be going for the win.

[gustavolacerda.livejournal.com]

right... if they draw, Uruguay wins the group. The winner probably gets to play South Korea, which should be much easier than Argentina.

So I can see Mexico pushing the offense. And if they win, there is hope for South Africa, but they have to overcome the Goal Difference.

With meager wins, SA has -2, Uruguay has +2... SA would need to put 5 x 0 on France to match +2 and qualify on GF (which is the 2nd level of tie-breaking)