https://www.youtube.com/watch?v=nAIDNr4__kY
- https://doc.trackmania.com/play/how-to-play-cotd/
- https://www.trackmania.com/tracks/MsCfHde9bpRbpTu79eQorrjVr12
- https://trackmania.io/#/totd/leaderboard/9689807f-4482-4198-bbc7-afe02b818343/MsCfHde9bpRbpTu79eQorrjVr12
- https://trackmania.io/#/cotd/5064
Likelyhood that the two plays the same day?
For simplicity we'll postulate 100% even though it is not realistic. This should be adjusted for more accurate results.
P(BothSameDay) = 1
100%
Likelyhood that they both qualify into division 1?
Each player individually.
P(Div1) = 64 ÷ number_of_players
P(Div1) = 64 ÷ 3 551
P(Div1) ≃ 0,018
1,8 %
Both players together.
P(BothDiv1Generic) = P(Div1Generic) × P(Div1Generic)
P(BothDiv1Generic) = P(Div1Generic)²
P(BothDiv1) = 0,018 × 0,018
P(BothDiv1) = 0,018²
P(BothDiv1) ≃ 0,000324
0,03 %
Now one could and should ponderate this with each player's level. A higher level player has a better likelyhood to win than a novice. But we're gonna ignore that here ;) This also applies to the rounds!
Likelyhood of each player's survival per rounds.
P(Survival[1-24]) = number_of_survivable_spots ÷ number_of_remaining_players
P(Survival[1-24]) = (number_of_remaining_players - number_of_ko_this_round) ÷ number_of_remaining_players
No KO.
P(Survival1) = 64 ÷ 64
P(Survival1) = 1
Round 1: 100 %
4 KOs per round.
P(Survival2) = (64 - 4) ÷ 64
P(Survival2) = 60 ÷ 64
P(Survival2) = 15 ÷ 16
P(Survival2) ≃ 0,9375
Round 2: 93,75 %
P(Survival3) = (60 - 4) ÷ 60
P(Survival3) = 56 ÷ 60
P(Survival3) = 14 ÷ 15
P(Survival3) ≃ 0,9333
Round 3: 93,33 %
P(Survival4) = (56 - 4) ÷ 56
P(Survival4) = 52 ÷ 56
P(Survival4) = 13 ÷ 14
P(Survival4) ≃ 0,9286
Round 4: 92,86 %
P(Survival5) = (52 - 4) ÷ 52
P(Survival5) = 48 ÷ 52
P(Survival5) = 12 ÷ 13
P(Survival5) ≃ 0,9231
Round 5: 92,31 %
P(Survival6) = (48 - 4) ÷ 48
P(Survival6) = 44 ÷ 48
P(Survival6) = 11 ÷ 12
P(Survival6) ≃ 0,9167
Round 6: 91,67 %
P(Survival7) = (44 - 4) ÷ 44
P(Survival7) = 40 ÷ 44
P(Survival7) = 10 ÷ 11
P(Survival7) ≃ 0,9091
Round 7: 90,91 %
P(Survival8) = (40 - 4) ÷ 40
P(Survival8) = 36 ÷ 40
P(Survival9) = 9 ÷ 10
P(Survival8) = 0,9
Round 8: 90 %
P(Survival9) = (36 - 4) ÷ 36
P(Survival9) = 32 ÷ 36
P(Survival9) = 8 ÷ 9
P(Survival9) ≃ 0,8889
Round 9: 88,89 %
P(Survival10) = (32 - 4) ÷ 32
P(Survival10) = 28 ÷ 32
P(Survival10) = 7 ÷ 8
P(Survival10) = 0,875
Round 10: 87,5 %
P(Survival11) = (28 - 4) ÷ 28
P(Survival11) = 24 ÷ 28
P(Survival11) = 6 ÷ 7
P(Survival11) ≃ 0,8571
Round 11: 85,71 %
P(Survival12) = (24 - 4) ÷ 24
P(Survival12) = 20 ÷ 24
P(Survival12) = 5 ÷ 6
P(Survival12) ≃ 0,8333
Round 12: 83,33 %
P(Survival13) = (20 - 4) ÷ 20
P(Survival13) = 16 ÷ 20
P(Survival13) = 4 ÷ 5
P(Survival13) = 0,8
Round 13: 80 %
2 KOs per round.
P(Survival14) = (16 - 2) ÷ 16
P(Survival14) = 14 ÷ 16
P(Survival14) = 7 ÷ 8
P(Survival14) = 0,875
Round 14: 87,5 %
P(Survival15) = (14 - 2) ÷ 14
P(Survival15) = 12 ÷ 14
P(Survival15) = 6 ÷ 7
P(Survival15) ≃ 0,8571
Round 15: 85,71 %
P(Survival16) = (12 - 2) ÷ 12
P(Survival16) = 10 ÷ 12
P(Survival16) = 5 ÷ 6
P(Survival16) ≃ 0,8333
Round 16: 83,33 %
P(Survival17) = (10 - 2) ÷ 10
P(Survival17) = 8 ÷ 10
P(Survival17) = 4 ÷ 5
P(Survival17) = 0,8
Round 17: 80 %
1 KO per round.
P(Survival18) = (8 - 1) ÷ 8
P(Survival18) = 7 ÷ 8
P(Survival18) = 0,875
Round 18: 87,5 %
P(Survival19) = (7 - 1) ÷ 7
P(Survival19) = 6 ÷ 7
P(Survival19) ≃ 0,8571
Round 19: 85,71 %
P(Survival20) = (6 - 1) ÷ 6
P(Survival20) = 5 ÷ 6
P(Survival20) ≃ 0,8333
Round 20: 83,33 %
P(Survival21) = (5 - 1) ÷ 5
P(Survival21) = 4 ÷ 5
P(Survival21) = 0,8
Round 21: 80 %
P(Survival22) = (4 - 1) ÷ 4
P(Survival22) = 3 ÷ 4
P(Survival22) = 0,75
Round 22: 75 %
P(Survival23) = (3 - 1) ÷ 3
P(Survival23) = 2 ÷ 3
P(Survival23) ≃ 0,6667
Round 23: 66,67 %
P(Survival24) = (2 - 1) ÷ 2
P(Survival24) = 1 ÷ 2
P(Survival24) = 0,5
Round 24: 50 %
Chances of going to finale COTD once qualified.
P(SurvivalToFinale) = P(Survival1)
× P(Survival2)
× P(Survival3)
...
× P(Survival23)
P(SurvivalToFinale) = 1
× 15 ÷ 16
× 14 ÷ 15
× 13 ÷ 14
× 12 ÷ 13
× 11 ÷ 12
× 10 ÷ 11
× 9 ÷ 10
× 8 ÷ 9
× 7 ÷ 8
× 6 ÷ 7
× 5 ÷ 6
× 4 ÷ 5
× 7 ÷ 8
× 6 ÷ 7
× 5 ÷ 6
× 4 ÷ 5
× 7 ÷ 8
× 6 ÷ 7
× 5 ÷ 6
× 4 ÷ 5
× 3 ÷ 4
× 2 ÷ 3
× 1 ÷ 2
P(SurvivalToFinale) = 4 ÷ 16
× 4 ÷ 8
× 1 ÷ 8
P(SurvivalToFinale) = 1 ÷ 64
P(SurvivalToFinale) ≃ 0,01563
1,56 %
Chances of winning including qualifier.
P(WinCOTD) = P(Div1) × P(SurvivalToFinale) × P(Survival24)
P(WinCOTD) = 64 ÷ 3 551
× 1 ÷ 64
× 1 ÷ 2
P(WinCOTD) = 1 ÷ 3 551
× 1 ÷ 2
P(WinCOTD) = 1 ÷ 7 102
P(WinCOTD) = 0,0001409
0,014 %
Likelyhood that both players survives per rounds.
P(SurvivalBoth[1-23]) = P(Survival[1-23]) × P(Survival[1-23])
P(SurvivalBoth[1-23]) = P(Survival[1-23])²
Round 24 is a trap, hence it's ignored. See below for the explanation.
P(SurvivalBoth1) = 1²
P(SurvivalBoth1) = 1
Round 1: 100 %
P(SurvivalBoth2) = (60 ÷ 64)²
P(SurvivalBoth2) ≃ 0,8789
Round 2: 87,89 %
P(SurvivalBoth3) = (56 ÷ 60)²
P(SurvivalBoth3) ≃ 0,8711
Round 3: 87,11 %
P(SurvivalBoth4) = (52 ÷ 56)²
P(SurvivalBoth4) ≃ 0,8622
Round 4: 86,22 %
P(SurvivalBoth5) = (48 ÷ 52)²
P(SurvivalBoth5) ≃ 0,8521
Round 5: 85,21 %
P(SurvivalBoth6) = (44 ÷ 48)²
P(SurvivalBoth6) ≃ 0,8403
Round 6: 84,03 %
P(SurvivalBoth7) = (40 ÷ 44)²
P(SurvivalBoth7) ≃ 0,8264
Round 7: 82,64 %
P(SurvivalBoth8) = (36 ÷ 40)²
P(SurvivalBoth8) ≃ 0,81
Round 8: 81 %
P(SurvivalBoth9) = (32 ÷ 36)²
P(SurvivalBoth9) ≃ 0,7901
Round 9: 79,01 %
P(SurvivalBoth10) = (28 ÷ 32)²
P(SurvivalBoth10) ≃ 0,7656
Round 10: 76,56 %
P(SurvivalBoth11) = (24 ÷ 28)²
P(SurvivalBoth11) ≃ 0,7347
Round 11: 73,47 %
P(SurvivalBoth12) = (20 ÷ 24)²
P(SurvivalBoth12) ≃ 0,6944
Round 12: 69,44 %
P(SurvivalBoth13) = (16 ÷ 20)²
P(SurvivalBoth13) = 0,64
Round 13: 64 %
P(SurvivalBoth14) = (14 ÷ 16)²
P(SurvivalBoth14) ≃ 0,7656
Round 14: 76,56 %
P(SurvivalBoth15) = (12 ÷ 14)²
P(SurvivalBoth15) ≃ 0,7347
Round 15: 73,47 %
P(SurvivalBoth16) = (10 ÷ 12)²
P(SurvivalBoth16) ≃ 0,6944
Round 16: 69,44 %
P(SurvivalBoth17) = (8 ÷ 10)²
P(SurvivalBoth17) = 0,64
Round 17: 64 %
P(SurvivalBoth18) = (7 ÷ 8)²
P(SurvivalBoth18) ≃ 0,7656
Round 18: 76,56 %
P(SurvivalBoth19) = (6 ÷ 7)²
P(SurvivalBoth19) ≃ 0,7347
Round 19: 73,47 %
P(SurvivalBoth20) = (5 ÷ 6)²
P(SurvivalBoth20) ≃ 0,6944
Round 20: 69,44 %
P(SurvivalBoth21) = (4 ÷ 5)²
P(SurvivalBoth21) = 0,64
Round 21: 64 %
P(SurvivalBoth22) = (3 ÷ 4)²
P(SurvivalBoth22) ≃ 0,5625
Round 22: 56,25 %
P(SurvivalBoth23) = (2 ÷ 3)²
P(SurvivalBoth23) ≃ 0,4444
Round 23: 44,44 %
Since only one player can win, there is not a single chance of both winning! It makes no sense to compute this one ;)
P(SurvivalBoth24) = 0
Round 24: 0 %
Global chances of both player surviving until the finale once qualified.
P(SurvivalBothToFinale) = P(SurvivalBoth1)
× P(SurvivalBoth2)
× P(SurvivalBoth3)
...
× P(SurvivalBoth23)
P(SurvivalBothToFinale) = 1
× (60 ÷ 64)²
× (56 ÷ 60)²
× (52 ÷ 56)²
× (48 ÷ 52)²
× (44 ÷ 48)²
× (40 ÷ 44)²
× (36 ÷ 40)²
× (32 ÷ 36)²
× (28 ÷ 32)²
× (24 ÷ 28)²
× (20 ÷ 24)²
× (16 ÷ 20)²
× (14 ÷ 16)²
× (12 ÷ 14)²
× (10 ÷ 12)²
× (8 ÷ 10)²
× (7 ÷ 8)²
× (6 ÷ 7)²
× (5 ÷ 6)²
× (4 ÷ 5)²
× (3 ÷ 4)²
× (2 ÷ 3)²
P(SurvivalBothToFinale) = 60² ÷ 64²
× 56² ÷ 60²
× 52² ÷ 56²
× 48² ÷ 52²
× 44² ÷ 48²
× 40² ÷ 44²
× 36² ÷ 40²
× 32² ÷ 36²
× 28² ÷ 32²
× 24² ÷ 28²
× 20² ÷ 24²
× 16² ÷ 20²
× 14² ÷ 16²
× 12² ÷ 14²
× 10² ÷ 12²
× 8² ÷ 10²
× 7² ÷ 8²
× 6² ÷ 7²
× 5² ÷ 6²
× 4² ÷ 5²
× 3² ÷ 4²
× 2² ÷ 3²
P(SurvivalBothToFinale) = 2² ÷ 64²
P(SurvivalBothToFinale) = 4 ÷ 4 096
P(SurvivalBothToFinale) = 1 ÷ 1 024
P(SurvivalBothToFinale) ≃ 0,001
1 in 1024
About 1 ‰ or 0,1 %
Chances of being in a COTD cup together.
P(BothFinale) = P(BothSameDay) × P(BothDiv1) × P(SurvivalBothToFinale)
P(BothFinale) = 1
× (64 ÷ 3 551)²
× 2² ÷ 64²
P(BothFinale) = 64² ÷ 3 551²
× 2² ÷ 64²
P(BothFinale) = 2² ÷ 3 551²
P(BothFinale) = 4 ÷ 12 609 601
P(BothFinale) ≃ 1 ÷ 3 152 400
P(BothFinale) ≃ 0,3172 × 10⁻⁶
P(BothFinale) ≃ 0,0000003172
1 in 3 152 400
About 0,00003172 %