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Three way football tournament probability?
Football teams A,B and C are evenly matched. Initially A and B play a game, and the winner then plays C. This continues, with the winner always playing the waiting team, until one team has won
two games in a row. That team is then declared champion.
What is the probability that A is the champion ?
Llaffer
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This is a special type of tournament. It will go on until some team has won 2 games in a row.
Julius N
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Surely team A stands a better chance ?
5 réponses
- ?Lv 7il y a 7 ansRéponse favorite
I'm getting the following, but I'm not totally sure.
A and B should have equal chance, whatever that chance is.
Focusing on how C can win, C can't win on "trials" 1 or 2, but can win on the 3rd "trial" if
one of the following occurs - ACC or BCC - the total prob. here is 1/4.
Then C can win on trial 6 if the following occurs - ACBACC, BCABCC - total prob of 1/32.
It looks like it continues in a similar manner - C could win on trial 9, 12, ..
And you get an infinite series 1/4 + (1/32) + (1/256) + ...= 1/4 + [(1/32) + (1/32)(1/8) + ...]=
1/4 + [1/32/ [1 - 1/8]] = 1/4 + 1/28 = 2/7.
So there is 5/7 left for A and B, which gives prob of 5/14 for A and B.
- Divide By ZeroLv 7il y a 7 ans
I agree with Leonard.
aa = 1/4
bcaa = 1/16
acbaa = 1/32
bcabcaa = 1/128
acbacbaa = 1/256
1/(2^2) + 1/(2^4) + 1/(2^5) + 1/(2^7) + 1/(2^8) + ...
= sum from 1/4 to 1/(2^n)
minus the sum from 1/8 to 1/(8^n)
= 1/2 - 1/7 = 5/14 = Pr(A wins)
So Pr(B) is also 5/14 and Pr(C) is 2/7.
This can also be done with a transition matrix.
- Julius NLv 7il y a 7 ans
You need to look at the total sample space:
0.25 for A winning a tournament in two games PLUS
0.25 for B winning after two games PLUS
0.25 for C winning after three games PLUS
THEN there is a .25 chance of a fourth game which will be between A and B with .0625 chance of A winning the tournament and .0625 chance of B winning the tournament.
so the P(A winning) = 1/4 + 1/16 + 1/64 + ... = 1/3
It appears that even though A and B play first all three of them have an equal chance of winning the tournament
- llafferLv 7il y a 7 ans
Do you mean two games in a row? or two tournaments in a row?
If it's just winning two games in a row:
If they are evenly matched, there is a 50% chance of winning against team B.
Then there is a 50% chance of winning against team C.
So that's a 25% chance (1/2 * 1/2 = 1/4) of winning two games in a row, which is winning one tournament.
But if you mean two tournaments in a row, Team A gets a bye for winning the first one, so wins against the victor of Team B and C. With another 50% chance of winning that, to win three games in a row, and two tournaments:
1/4 * 1/2 = 1/8
The probability is 12.5%.
