2017 AIME II Problem 2

Teams T_{1}, T_{2}, T_{3}, and T_{4} are in the playoffs. In the semifinal matches, T_{1} plays T_{4}, and T_{2} plays T_{3}. The winners of those two matches will play each other in the final match to determine the champion. When T_{i} plays T_{j}, the probability that T_{i} wins is \frac{i}{i+j}, and the outcomes of all the matches are independent. The probability that T_{4} will be the champion is \frac{p}{q}, where p and q are relatively prime positive integers. Find p+q.