Problem:
Teams , and are in the playoffs. In the semifinal matches, plays , and plays . The winners of those two matches will play each other in the final match to determine the champion. When plays , the probability that wins is , and the outcomes of all the matches are independent. The probability that will be the champion is , where and are relatively prime positive integers. Find .
Solution:
Team will be the champion if and only if it wins its semifinal match (which it will do with probability ) and then beats whoever wins the other semifinal match. Considering the two possible outcomes of the other semifinal match gives the probability
The requested sum is .
The problems on this page are the property of the MAA's American Mathematics Competitions