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Qualifiers to the World Cup Russia 2018: An introduction to the theory of probability

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7 Octubre, 2017

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September 11^th, 2017

Supplementary Information (1^st part of the article)

There are 243 configurations in which the standings can be arranged at the end of the competition. In matchday 17 there are 9 possible configurations:

A) Chile vs Ecuador (WL) – Argentina vs Peru (WL)
B) Chile vs Ecuador (DD) – Argentina vs Peru (WL)
C) Chile vs Ecuador (LW) – Argentina vs Peru (WL)
D) Chile vs Ecuador (WL) – Argentina vs Peru (DD)
E) Chile vs Ecuador (DD) – Argentina vs Peru (DD)
F) Chile vs Ecuador (LW) – Argentina vs Peru (DD)
G) Chile vs Ecuador (WL) – Argentina vs Peru (LW)
H) Chile vs Ecuador (DD) – Argentina vs Peru (LW)
I) Chile vs Ecuador (LW) – Argentina vs Peru (LW)

(DD) means a draw. (WL) means that the home team wins and the visitor loses. (LW) means that the home team loses and the visitor wins. Each of these 9 sub-groups is labeled with letters from A to I. In each sub-group matchday 18 provides 27 possible configurations. Below we show all of them.

In Tables A.1, B.1, C.1, D.1, E.1, F.1, G.1, H.1, and I.1, we show the standings. We have grouped 9 standing tables for each sub-group. For each one, Chile has 3 values depending on the match against Brazil (giving a total of 27 possible configurations for each sub-group). The first place qualifies directly to the World Cup, the second place advances to the inter-confederation play-offs.
In Tables A.2, B.2, C.2, D.2, E.2, F.2, G.2, H.2, and I.2, we show the qualifying probability of each team after knowing the result of the match Brazil vs Chile. Note that by “qualifying team” we also imply the team who advances to the inter-confederation play-offs. In parentheses we show the qualifying probability when the team has the best goal difference.
In Tables A.3, B.3, C.3, D.3, E.3, F.3, G.3, H.3, and I.3, we show the probability, P(X), of reaching the last qualifying position with X points. e.g., P (27) indicates the probability of reaching the inter-confederations play-offs (2nd place here) with 27 points.

A) Chile vs Ecuador (WL) – Argentina vs Peru (WL)

Table A.1
= 30 = 29 (27) (26) = 24 = 20	= 29 (27) (26) = 28 = 24 = 21	= 29 (27) (26) = 27 = 24 = 24
= 30 = 27 (26) (29) = 25 = 20	= 28 = 27 (26) (29) = 25 = 21	= 27 (26) (29) = 27 = 25 = 24
=30 = 27 = 26 (27) (29) = 20	= 28 = 27 = 26 (27) (29) = 21	= 27 = 27 = 26 (27) (29) = 24

Table A.2
Qualifying probability	If Brazil beats Chile	If Brazil ties Chile	If Chile beats Brazil
	9/9	8/9 (9/9)	8/9 (9/9)
	3/9	3/9 (6/9)	0
	0	0	0
	6/9	6/9 (9/9)	9/9

Table A.3
Probability, P(X), of reaching the last qualifying position with X points
P(29) = 3/27
P(28) = 3/27
P(27) = 15/27
P(26) = 6/27

B) Chile vs Ecuador (DD) – Argentina vs Peru (WL)

Table B.1
= 30 = 27 (25) (24) = 24 = 21	= 28 = 27 (25) (24) = 24 = 22	= 27 (25) (24) = 27 = 24 = 24
= 30 = 25 (27) (24) = 25 = 21	= 28 = 25 (27) (24) = 25 = 22	= 27 = 25 (27) (24) = 25 = 24
= 30 = 27 = 24 (25) (27) = 21	= 28 = 27 = 24 (25) (27) = 22	= 27 = 27 = 24 (25) (27) = 24

Table B.2
Qualifying probability	If Brazil beats Chile	If Brazil ties Chile	If Chile beats Brazil
	9/9	9/9	8/9 (9/9)
	6/9 (9/9)	3/9 (6/9)	0 (3/9)
	0	0	0
	0 (3/9)	3/9 (6/9)	6/9 (9/9)

Table B.3
Probability, P(X), of reaching the last qualifying position with X points
P(27) = 15/27
P(25) = 9/27
P(24) = 3/27

C) Chile vs Ecuador (LW) – Argentina vs Peru (WL)

Table C.1
= 30 = 26 (23)(24) = 24 = 23	= 28 = 26(23)(24) = 24 = 24	= 27 = 26 = 26(23)(24) = 24
= 30 = 25 = 24 (23)(26) = 23	= 28 = 25 = 24(23)(26) = 24	= 27 = 26 = 25 = 24(23)(26)
= 30 = 27 = 23 = 23 (24)(26)	= 28 = 27 = 24 = 23(24)(26)	= 27 = 27 = 26 = 23(24)(26)

Table C.2
Qualifying probability	If Brazil beats Chile	If Brazil ties Chile	If Chile beats Brazil
	9/9	9/9	9/9
	6/9 (7/9)	5/9 (7/9)	3/9
	2/9 (3/9)	2/9 (3/9)	0 (2/9)
	0	0 (2/9)	4/9 (6/9)

Table C.3
Probability, P(X), of reaching the last qualifying position with X points
P(27) = 9/27
P(26) = 10/27
P(25) = 4/27
P(24) = 4/27

D) Chile vs Ecuador (WL) – Argentina vs Peru (DD)

Table D.1
=29 (27)(26) = 28 = 25 = 20	=29 (27)(26) = 26 = 25 = 21	= 29 (27)(26) =25 =25 = 23
=28 = 27 (29)(26) = 26 = 20	=27(29)(26) =26 =26 =21	=27(29)(26) =26 =25 =23
=28 =28 =26 (27)(29) =20	=28 =26(27)(29) =26 =21	=28 =26(27)(29) =25 =23

Table D.2
Qualifying probability	If Brazil beats Chile	If Brazil ties Chile	If Chile beats Brazil
	4/9 (7/9)	4/9 (6/9)	3/9 (6/9)
	4/9 (7/9)	4/9 (6/9)	3/9 (6/9)
	0	0	0
	5/9 (8/9)	8/9	9/9

Table D.3
Probability, P(X), of reaching the last qualifying position with X points
P(28) = 7/27
P(27) = 4/27
P(26) = 13/27
P(25) = 3/27

E) Chile vs Ecuador (DD) – Argentina vs Peru (DD)

Table E.1
=28 =27 (25)(24) =25 =21	=27(25)(24) =26 =25 =22	=27(25)(24) =25 =25 =24
=28 =26 =25 (24)(27) =21	=26 =26 =25(24)(27) =22	=26 =25(24)(27) =25 =24
=28 =28 =24(25)(27) =21	=28 =26 =24(25)(27) =22	=28 =25 =24 =24(25)(27)

Table E.2
Qualifying probability	If Brazil beats Chile	If Brazil ties Chile	If Chile beats Brazil
	9/9	6/9 (9/9)	4/9 (6/9)
	9/9	6/9 (9/9)	4/9 (6/9)
	0	0	0
	0	0 (5/9)	8/9

Table E.3
Probability, P(X), of reaching the last qualifying position with X points
P(28) = 3/27
P(27) = 4/27
P(26) = 9/27
P(25) = 11/27

F) Chile vs Ecuador (LW) – Argentina vs Peru (DD)

Table F.1
=28 =26(24)(23) =25 =23	=26(24)(23) =26 =25 =24	=26 =26(24)(23) =25 =25
=28 =26 =24 (23)(26) =23	=26 =26 =24 =24(23)(26)	=26 =26 =25 =24(23)(26)
=28 =28 =23 =23(24)(26)	=28 =26 =24 =23(24)(26)	=28 =26 =25 =23(24)(26)

Table F.2
Qualifying probability	If Brazil beats Chile	If Brazil ties Chile	If Chile beats Brazil
	6/9 (7/9)	6/9 (7/9)	4/9 (6/9)
	8/9 (9/9)	8/9 (9/9)	3/9 (6/9)
	3/9	3/9	1/9 (3/9)
	0	0	3/9 (8/9)

Table F.3
Probability, P(X), of reaching the last qualifying position with X points
P(28) = 3/27
P(26) = 18/27
P(25) = 6/27

G) Chile vs Ecuador (WL) – Argentina vs Peru (LW)

Table G.1
= 29 (26)(27) =27 =27 =20	= 29(26)(27) =27 =25 =21	=29(26)(27) =27 =24 =23
=28 =27(26)(29) =27 =20	=28 =27(26)(29) =25 =21	=28 =27(26)(29) =24 =23
=30 =27 =26(27)(29) =20	=30 =26(27)(29) =25 =21	=30 =26(27)(29) =23 =24

Table G.2
Qualifying probability	If Brazil beats Chile	If Brazil ties Chile	If Chile beats Brazil
	3/9	0 (3/9)	0 (1/9)
	9/9	8/9 (9/9)	8/9 (9/9)
	0	0	0
	6/9	6/9 (9/9)	9/9

Table G.3
Probability, P(X), of reaching the last qualifying position with X points
P(29) = 3/27
P(28) = 3/27
P(27) =14/27
P(26) = 7/27

H) Chile vs Ecuador (DD) – Argentina vs Peru (LW)

Table H.1
=27 (24)(25) =27 =27 =21	=27(24)(25) =27 =25 =22	=27(24)(25) =27 =24 =24
=28 =27 =25(24)(27) =21	=28 =25(24)(27) =25 =22	=28 =25(24)(27) =24 =24
=30 =27 =24(25)(27) =21	=30 =25 =24(25)(27) =22	=30 =24 =24(25)(27) =24

Table H.2
Qualifying probability	If Brazil beats Chile	If Brazil ties Chile	If Chile beats Brazil
	6/9 (9/9)	3/9 (6/9)	0 (3/9)
	9/9	9/9	8/9 (9/9)
	0 (3/9)	0	0
	0 (3/9)	3/9 (6/9)	6/9 (9/9)

Table H.3
Probability, P(X), of reaching the last qualifying position with X points
P(27) = 15/27
P(25) = 9/27
P(24) = 3/27

I) Chile vs Ecuador (LW) – Argentina vs Peru (LW)

Table I.1
=27 =27 =26 (23)(24) =23	=27 =26(23)(24) =25 =24	=27 =26 =26(23)(24) =24
=28 =27 =24(23)(26) =23	=28 =25 =24 =24(23)(26)	=28 =26 =24(23)(26) =24
=30 =27 =23 =23(26)(24)	=30 =25 =24 =23(26)(24)	=30 =26 =24 =23(26)(24)

Table I.2
Qualifying probability	If Brazil beats Chile	If Brazil ties Chile	If Chile beats Brazil
	6/9	6/9	3/9
	9/9	9/9	9/9
	3/9	3/9	0 (3/9)
	0	0	3/9 (6/9)

Table I.3
Probability, P(X), of reaching the last qualifying position with X points
P(27) = 9/27
P(25) = 6/27
P(26) = 12/27

September 11th, 2017

Supplementary Information (1st part of the article)

September 11^th, 2017

Supplementary Information (1^st part of the article)