2006 AIME A Problems/Problem 10
Problem
Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 points. The total points are accumilated to decide the ranks of the teams. In the first game of the tournament, team
beats team
The probability that team
finishes with more points than team
is
where
and
are relatively prime positive integers. Find
Solution
You can break this into cases based on how many rounds A wins out of the remaining 5 games.
If A wins 0 games, then B must win 0 games and the probability of this is .
If A wins 1 games, then B must win 1 or less games and the probability of this is .
If A wins 2 games, then B must win 2 or less games and the probability of this is .
If A wins 3 games, then B must win 3 or less games and the probability of this is .
If A wins 4 games, then B must win 4 or less games and the probability of this is .
If A wins 5 games, then B must win 5 or less games and the probability of this is .
Summing these 6 cases, we get , which simplifies to
, so out answer is
.