Problem:
Azar, Carl, Jon, and Sergey are the four players left in a singles tennis tournament. They are randomly assigned opponents in the semifinal matches, and the winners of those matches play each other in the final match to determine the winner of the tournament. When Azar plays Carl, Azar will win the match with probability . When either Azar or Carl plays either Jon or Sergey, Azar or Carl will win the match with probability . Assume that outcomes of different matches are independent. The probability that Carl will win the tournament is , where and are relatively prime positive integers. Find .
Solution:
There are two cases, depending on whether Azar and Carl meet in the semifinals. If they do, which occurs with probability , Carl will win the tournament if and only if he beats Azar and goes on to beat the winner of the other semifinal match, which occurs with probability . If they do not, which occurs with probability , Carl must beat Jon or Sergey in the semifinal match, which occurs with probability , and go on to win the final match. If Azar wins her semifinal match, which occurs with probability , Carl must beat Azar, which occurs with probability . If Azar loses her semifinal match, which occurs with probability , Carl must beat Azar's opponent, which occurs with probability . Thus Carl will win the tournament with probability
The requested sum is .
The problems on this page are the property of the MAA's American Mathematics Competitions