Dr. Gamera
02 Jun 2006, 04:03 PM
Dear all,
I think this is actually on-topic for all eight groups.
Here are the possible results of four-team group play, with three points for
a win, one point for a draw, and no points for a loss, with the top two teams
advancing. These assume identical independently distributed 1/3
probabilities for a win, a draw, or a loss in any group game -- obviously not
the most realistic.
Possible combinations of points:
9-6-3-0: 24 combinations
9-6-1-1: 12 combinations
9-4-4-0: 12 combinations
9-4-3-1: 24 combinations
9-4-2-1: 24 combinations
9-3-3-3: 8 combinations
9-2-2-2: 4 combinations
7-7-3-0: 12 combinations
7-7-1-1: 6 combinations
7-6-4-0: 24 combinations
7-6-3-1: 24 combinations
7-6-2-1: 24 combinations
7-5-4-0: 24 combinations
7-5-3-1: 24 combinations
7-5-2-1: 24 combinations
7-4-4-1: 36 combinations
7-4-3-3: 24 combinations
7-4-3-2: 24 combinations
7-4-3-1: 24 combinations
7-4-2-2: 24 combinations
7-3-2-2: 12 combinations
6-6-6-0: 8 combinations
6-6-4-1: 24 combinations
6-6-3-3: 24 combinations
6-5-4-1: 24 combinations
6-5-2-2: 12 combinations
6-4-4-3: 36 combinations
6-4-4-2: 24 combinations
5-5-5-0: 4 combinations
5-5-4-1: 24 combinations
5-5-3-2: 12 combinations
5-5-3-1: 12 combinations
5-5-2-2: 12 combinations
5-4-4-3: 24 combinations
5-4-4-2: 24 combinations
5-4-3-2: 24 combinations
5-3-3-2: 12 combinations
4-4-4-4: 6 combinations
4-4-4-3: 8 combinations
3-3-3-3: 1 combination
In 153 of these 729 scenarios, the winner of the group has the same number of points as the second-place team in the group; in 207 of these 729 scenarios, the second-place team in the group has the same number of points as the third-place team in the group.
Number of scenarios in which a team with a given number of points finishes
in a given place in a group, and percentage of all such scenarios with that
number of points:
. .G1 .G2 .G3 .G4 . . G1% . .G2% . .G3% . . G4% .G1+G2% .G3+G4%
0 . 0 . 0 . 0 108 . 0.00% .0.00% .0.00% 100.00% . 0.00% 100.00%
1 . 0 . 0 .18 306 . 0.00% .0.00% .5.56% .94.44% . 0.00% 100.00%
2 . 0 . 4 136 184 . 0.00% .1.23% 41.98% .56.79% . 1.23% .98.77%
3 . 1 .33 273 125 . 0.23% .7.64% 63.19% .28.94% . 7.87% .92.13%
4 .14 338 290 . 6 . 2.16% 52.16% 44.75% . 0.93% .54.32% .45.68%
5 148 172 . 4 . 0 .45.68% 53.09% .1.23% . 0.00% .98.77% . 1.23%
6 152 164 . 8 . 0 .46.91% 50.62% .2.47% . 0.00% .97.53% . 2.47%
7 306 .18 . 0 . 0 .94.44% .5.56% .0.00% . 0.00% 100.00% . 0.00%
9 108 . 0 . 0 . 0 100.00% .0.00% .0.00% . 0.00% 100.00% . 0.00%
I think this is actually on-topic for all eight groups.
Here are the possible results of four-team group play, with three points for
a win, one point for a draw, and no points for a loss, with the top two teams
advancing. These assume identical independently distributed 1/3
probabilities for a win, a draw, or a loss in any group game -- obviously not
the most realistic.
Possible combinations of points:
9-6-3-0: 24 combinations
9-6-1-1: 12 combinations
9-4-4-0: 12 combinations
9-4-3-1: 24 combinations
9-4-2-1: 24 combinations
9-3-3-3: 8 combinations
9-2-2-2: 4 combinations
7-7-3-0: 12 combinations
7-7-1-1: 6 combinations
7-6-4-0: 24 combinations
7-6-3-1: 24 combinations
7-6-2-1: 24 combinations
7-5-4-0: 24 combinations
7-5-3-1: 24 combinations
7-5-2-1: 24 combinations
7-4-4-1: 36 combinations
7-4-3-3: 24 combinations
7-4-3-2: 24 combinations
7-4-3-1: 24 combinations
7-4-2-2: 24 combinations
7-3-2-2: 12 combinations
6-6-6-0: 8 combinations
6-6-4-1: 24 combinations
6-6-3-3: 24 combinations
6-5-4-1: 24 combinations
6-5-2-2: 12 combinations
6-4-4-3: 36 combinations
6-4-4-2: 24 combinations
5-5-5-0: 4 combinations
5-5-4-1: 24 combinations
5-5-3-2: 12 combinations
5-5-3-1: 12 combinations
5-5-2-2: 12 combinations
5-4-4-3: 24 combinations
5-4-4-2: 24 combinations
5-4-3-2: 24 combinations
5-3-3-2: 12 combinations
4-4-4-4: 6 combinations
4-4-4-3: 8 combinations
3-3-3-3: 1 combination
In 153 of these 729 scenarios, the winner of the group has the same number of points as the second-place team in the group; in 207 of these 729 scenarios, the second-place team in the group has the same number of points as the third-place team in the group.
Number of scenarios in which a team with a given number of points finishes
in a given place in a group, and percentage of all such scenarios with that
number of points:
. .G1 .G2 .G3 .G4 . . G1% . .G2% . .G3% . . G4% .G1+G2% .G3+G4%
0 . 0 . 0 . 0 108 . 0.00% .0.00% .0.00% 100.00% . 0.00% 100.00%
1 . 0 . 0 .18 306 . 0.00% .0.00% .5.56% .94.44% . 0.00% 100.00%
2 . 0 . 4 136 184 . 0.00% .1.23% 41.98% .56.79% . 1.23% .98.77%
3 . 1 .33 273 125 . 0.23% .7.64% 63.19% .28.94% . 7.87% .92.13%
4 .14 338 290 . 6 . 2.16% 52.16% 44.75% . 0.93% .54.32% .45.68%
5 148 172 . 4 . 0 .45.68% 53.09% .1.23% . 0.00% .98.77% . 1.23%
6 152 164 . 8 . 0 .46.91% 50.62% .2.47% . 0.00% .97.53% . 2.47%
7 306 .18 . 0 . 0 .94.44% .5.56% .0.00% . 0.00% 100.00% . 0.00%
9 108 . 0 . 0 . 0 100.00% .0.00% .0.00% . 0.00% 100.00% . 0.00%