Probability theory in the FIFA World Cup Qualifiers

stats con chris
2022-03-29
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Supplementary information

Here we show the 243 possible ways in which the standings can end. On the penultimate date there are nine possible configurations, which are:

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 that we have a draw. (WL) means that the home team wins and the away team loses. (LW) means that the home team loses and the away team wins. Each of these 9 sub-groups is labeled with letters A through I. In each sub-group there are 27 possible configurations arising from the last date. Below we show each of them.

In Tables A.1, B.1, C.1, D.1, E.1, F.1, G.1, H.1, I.1, we show the standings. They are grouped in 9 tables for each sub-group, where Chile takes 3 values ​​depending on the match against Brazil; therefore, for each sub-group, we have a total of 27 possible ways in which the standings can end. The first place qualifies directly to the World Cup, the second place qualifies to the intercontinental playoffs.
In Tables A.2, B.2, C.2, D.2, E.2, F.2, G.2, H.2, I.2, we show the qualifying probability of each team after knowing the result of the match Brazil vs Chile. Note that the qualifying probability also includes the team that goes to the intercontinental playoffs. In parentheses we show the probability considering that 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, I.3, we show the probability of the minimum score in the last qualifying spot. e.g., P(27) indicates the probability of reaching the intercontinental playoffs (2nd place here) with 27 points.

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

Tabla A.1
1. =30
2. =29(27)(26)
3. =24
4. =20 		1. =29(27)(26)
2. =28
3. =24
4. =21 		1. =29(27)(26)
2. =27
3. =24
4. =24
1. =30
2. =27(26)(29)
3. =25
4. =20 		1. =28
2. =27(26)(29)
3. =25
4. =21 		1. =27(26)(29)
2. =27
3. =25
4. =24
1. =30
2. =27
3. =26(27)(29)
4. =20 		1. =28
2. =27
3. =26(27)(29)
4. =21 		1. =27
2. =27
3. =26(27)(29)
4. =24
Tabla A.2
Qualifying probability Brazil beats Chile Brazil draws with Chile Brazil loses to Chile
9/9 8/9
(9/9)		 8/9
(9/9)
3/9 0 
(3/9)		 0
(1/9)
0 0 0
6/9 6/9
(9/9)		 9/9
Tabla A.3
Probability of the minimum score in the last qualifying spot
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)

Tabla B.1
1. =30
2. =27(25)(24)
3. =24
4. =21 		1. =28
2. =27(25)(24)
3. =24
4. =22 		1. =27(25)(24)
2. =27
3. =24
4. =24
1. =30
2. =25(27)(24)
3. =25
4. =21 		1. =28
2. =25(27)(24)
3. =25
4. =22 		1. =27
2. =25(27)(24)
3. =25
4. =24
1. =30
2. =27
3. =24(25)(27)
4. =21 		1. =28
2. =27
3. =24(25)(27)
4. =22 		1. =27
2. =27
3. =24(25)(27)
4. =24
Tabla B.2
Qualifying probability Brazil beats Chile Brazil draws with Chile Brazil loses to Chile
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)
Tabla B.3
Probability of the minimum score in the last qualifying spot
P(27) = 15/27
P(25) = 9/27
P(24) = 3/27

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

Tabla C.1
1. =30
2. =26(23)(24)
3. =24
4. =23 		1. =28
2. =26(23)(24)
3. =24
4. =24 		1. =27
2. =26
3. =26(23)(24)
4. =24
1. =30
2. =25
3. =24(23)(26)
4. =23 		1. =28
2. =25
3. =24(23)(26)
4. =24 		1. =27
2. =26
3. =25
4. =24(23)(26)
1. =30
2. =27
3. =23
4. =23(24)(26) 		1. =28
2. =27
3. =24
4. =23(24)(26) 		1. =27
2. =27
3. =26
4. =23(24)(26)
Tabla C.2
Qualifying probability Brazil beats Chile Brazil draws with Chile Brazil loses to Chile
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)
Tabla C.3
Probability of the minimum score in the last qualifying spot
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)

Tabla D.1
1. =29(27)(26)
2. =28
3. =25
4. =20 		1. =29(27)(26)
2. =26
3. =25
4. =21 		1. =29(27)(26)
2. =25
3. =25
4. =23
1. =28
2. =27(29)(26)
3. =26
4. =20 		1. =27(29)(26)
2. =26
3. =26
4. =21 		1. =27(29)(26)
2. =26
3. =25
4. =23
1. =28
2. =28
3. =26(27)(29)
4. =20 		1. =28
2. =26(27)(29)
3. =26
4. =21 		1. =28
2. =26(27)(29)
3. =25
4. =23
Tabla D.2
Qualifying probability Brazil beats Chile Brazil draws with Chile Brazil loses to Chile
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
Tabla D.3
Probability of the minimum score in the last qualifying spot
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)

Tabla E.1
1. =28
2. =27(25)(24)
3. =25
4. =21 		1. =27(25)(24)
2. =26
3. =25
4. =22 		1. =27(25)(24)
2. =25
3. =25
4. =24
1. =28
2. =26
3. =25(24)(27)
4. =21 		1. =26
2. =26
3. =25(24)(27)
4. =22 		1. =26
2. =25(24)(27)
3. =25
4. =24
1. =28
2. =28
3. =24(25)(27)
4. =21 		1. =28
2. =26
3. =24(25)(27)
4. =22 		1. =28
2. =25
3. =24
4. =24(25)(27)
Tabla E.2
Qualifying probability Brazil beats Chile Brazil draws with Chile Brazil loses to Chile
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
Tabla E.3
Probability of the minimum score in the last qualifying spot
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)

Tabla F.1
1. =28
2. =26(24)(23)
3. =25
4. =23 		1. =26(24)(23)
2. =26
3. =25
4. =24 		1. =26
2. =26(24)(23)
3. =25
4. =25
1. =28
2. =26
3. =24(23)(26)
4. =23 		1. =26
2. =26
3. =24
4. =24(23)(26) 		1. =26
2. =26
3. =25
4. =24(23)(26)
1. =28
2. =28
3. =23
4. =23(24)(26) 		1. =28
2. =26
3. =24
4. =23(24)(26) 		1. =28
2. =26
3. =25
4. =23(24)(26)
Tabla F.2
Qualifying probability Brazil beats Chile Brazil draws with Chile Brazil loses to Chile
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)
Tabla F.3
Probability of the minimum score in the last qualifying spot
P(28) = 3/27
P(26) = 18/27
P(25) = 6/27

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

Tabla G.1
1. =29(26)(27)
2. =27
3. =27
4. =20 		1. =29(26)(27)
2. =27
3. =25
4. =21 		1. =29(26)(27)
2. =27
3. =24
4. =23
1. =28
2. =27(26)(29)
3. =27
4. =20 		1. =28
2. =27(26)(29)
3. =25
4. =21 		1. =28
2. =27(26)(29)
3. =24
4. =23
1. =30
2. =27
3. =26(27)(29)
4. =20 		1. =30
2. =26(27)(29)
3. =25
4. =21 		1. =30
2. =26(27)(29)
3. =23
4. =24
Tabla G.2
Qualifying probability Brazil beats Chile Brazil draws with Chile Brazil loses to Chile
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
Tabla G.3
Probability of the minimum score in the last qualifying spot
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)

Tabla H.1
1. =27(24)(25)
2. =27
3. =27
4. =21 		1. =27(24)(25)
2. =27
3. =25
4. =22 		1. =27(24)(25)
2. =27
3. =24
4. =24
1. =28
2. =27
3. =25(24)(27)
4. =21 		1. =28
2. =25(24)(27)
3. =25
4. =22 		1. =28
2. =25(24)(27)
3. =24
4. =24
1. =30
2. =27
3. =24(25)(27)
4. =21 		1. =30
2. =25
3. =24(25)(27)
4. =22 		1. =30
2. =24
3. =24(25)(27)
4. =24
Tabla H.2
Qualifying probability Brazil beats Chile Brazil draws with Chile Brazil loses to Chile
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)
Tabla H.3
Probability of the minimum score in the last qualifying spot
P(27) = 15/27
P(25) = 9/27
P(24) = 3/27

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

Tabla I.1
1. =27
2. =27
3. =26(23)(24)
4. =23 		1. =27
2. =26(23)(24)
3. =25
4. =24 		1. =27
2. =26
3. =26(23)(24)
4. =24
1. =28
2. =27
3. =24(23)(26)
4. =23 		1. =28
2. =25
3. =24
4. =24(23)(26) 		1. =28
2. =26
3. =24(23)(26)
4. =24
1. =30
2. =27
3. =23
4. =23(26)(24) 		1. =30
2. =25
3. =24
4. =23(26)(24) 		1. =30
2. =26
3. =24
4. =23(26)(24)
Tabla I.2
Qualifying probability Brazil beats Chile Brazil draws with Chile Brazil loses to Chile
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)
Tabla I.3
Probability of the minimum score in the last qualifying spot
P(27) = 9/27
P(25) = 6/27
P(26) = 12/27

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