2001 AIME II Problems/Problem 11

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Problem

Club Truncator is in a soccer league with six other teams, each of which it plays once. In any of its 6 matches, the probabilities that Club Truncator will win, lose, or tie are each $\frac {1}{3}$. The probability that Club Truncator will finish the season with more wins than losses is $\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Solution

We first find out how many way there are for the Truncators to win and lose the same amount. They can get 0 ties, 3 wins and 3 losses, 2 ties, 2 wins, and 2 losses, 4 ties, 1 win, and 1 loss, or 6 ties. Since there are 6 games, there are 6!/3!3! ways for the first, and 6!/2!2!2!, 6!/4!, and 1 ways for the rest, respectively, out of a total of $3^6$. This gives a probability of 141/729. 1 minus this is the probability they won't win and lose the same number of games, and half that is the probability they'll finish with more wins. This is 98/243, so the answer is 341.

See also

2001 AIME II (ProblemsAnswer KeyResources)
Preceded by
Problem 10
Followed by
Problem 12
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions