Three way football tournament probability?

Question

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 ?

Leonard · Accepted Answer

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 Zero · Answer

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 N · Answer

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

llaffer · Answer

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%.

? · Answer

Yes! Amen to 3, 8, 10, 13 and 19!

Three way football tournament probability?

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