World Cup Direct Qualifying

In the World Cup, the 32 participating teams have been divided into 8 groups. A team plays only 1 league match with each of the other 3 teams in its group. A Win gives the team 3 points, a Draw gives the team 1 point; a Loss brings no p oint. No league match is abandoned.

From each league, 2 teams will qualify to play in the next round, based on the p oints in their group. A team is said to qualify "directly", if it has the highest or the second highes t points in the league. If there is a tie for the highest points between 2 teams, both will qualify "dir ectly". In all other cases , other complex factors like goals scored for, goals scored a gainst, etc. are considered, so that qualification is not "direct".

Suppose the initial Points Table looks like:

 Team  No. of Games Played     Won     Draw     Loss        Points
   A      -                    -        -        -           -
   B      -                    -        -        -           -
   C      -                    -        -        -           -
   D      -                    -        -        -           - 
What is the probability that
  1. Team A will qualify "directly"?
  2. Both Team A and Team B will qualify "directly"?
[I asked a few clarifying questions of Sudipta and learned we should assume that each team is of equal skill, so the outcome of any one game is equally likely to be a win, draw, or loss for a particular team. Sudipta feels you can just analyze the results on all the possible Point Tables. I feel you have to start at the game level for a pure analysis. The analysis here is for JUST the 6 games among 4 teams and finding the obvious top 1 or 2 among those 4 teams. - KD]

Source: Reader Sudipta Das

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