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 point and the loser gets points. The total points are accumulated 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:
Each team has to play six games in all. Team and team each has more games to play, and they do not play against each other, for a total of possible outcomes. For team to finish with more points, it has to win at least as many games as team does. The number of outcomes in which the two teams win the same number of games is
Of the remaining outcomes, wins more than in half of them. Thus the requested probability is , and .
Team and team each has more games to play. For , the th game for team and the th game for team will change the difference between the scores of team and team by , or , and a change of is twice as likely as each of the other changes. Hence consider the coefficients of the generating function , and find the sum of all the coefficients of terms of the form , where . Note that
Thus the sum is
so the requested probability is , and .
The problems on this page are the property of the MAA's American Mathematics Competitions