Problem:
Club Truncator is in a soccer league with six other teams. each of which it plays once. In any of its matches, the probabilities that Club Truncator will win. lose, or tie are each . The probability that Club Truncator will finish the season with more wins than losses is , where and are relatively prime positive integers. Find .
Solution:
The probability that the team has more wins than losses is the same as the probability that the team has more losses than wins, and hence , where is the probability that Truncator has the same number of wins as losses. The probability of three wins and three losses is , the probability of two wins and two losses is , the probability of one win and one loss is , and the probability of no wins and no losses is . Therefore , and . Thus .
The problems on this page are the property of the MAA's American Mathematics Competitions