2006 AIME II Problems/Problem 10
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
The results of the five remaining games are independent of the first game, so by symmetry, the probability that scores higher than in these five games is equal to the probability that scores higher than . We let this probability be ; then the probability that and end with the same score in these five games is .
Of these three cases (), the last is the easiest to calculate (see solution 2 for a way to directly calculate the other cases).
There are ways to to have victories, and ways for to have victories. Summing for all values of ,
Thus . The desired probability is the sum of the cases when , so the answer is , and .
You can break this into cases based on how many rounds wins out of the remaining games.
- If wins 0 games, then must win 0 games and the probability of this is .
- If wins 1 games, then must win 1 or less games and the probability of this is .
- If wins 2 games, then must win 2 or less games and the probability of this is .
- If wins 3 games, then must win 3 or less games and the probability of this is .
- If wins 4 games, then must win 4 or less games and the probability of this is .
- If wins 5 games, then must win 5 or less games and the probability of this is .
Summing these 6 cases, we get , which simplifies to , so our answer is .
We can apply the concept of generating functions here.
The generating function for is for the first game where is winning n games. Since lost the first game, the coefficient for is 0. The generating function for the next 5 games is . Thus, the total generating function for number of games he wins is
The generating function for is the same except that it is multiplied by instead of . Thus, the generating function for is
The probability that wins 0 games is . Since the coefficients for all where
sums to 32, the probability that wins more games is .
Thus, the probability that has more wins than is .
After the first game, there are games we care about-- those involving or . There are cases of these games: wins more than , wins more than , or and win the same number of games. Also, there are total outcomes. By symmetry, the first and second cases are equally likely, and the third case occurs times, by a special case of Vandermonde's Identity (see Combinatorial identity). There are therefore possibilities for each of the other two cases.
If has more wins than in its remaining games, then cannot beat overall. However, if has more wins or if and are tied, will beat overall. Therefore, out of the possibilites, ways where wins, so the desired probability is , and .
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