Teams T₁, T₂, T₃ and T₄ are in the playoffs. In the semifinal matches T₁ plays T₄ and T₂ plays T₃. 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 $\frac{\mathrm{i}}{\mathrm{i}+\mathrm{j}}$ and the outcomes of all the matches are independent. The probability that T₄ will be the champion is $\frac{\mathrm{p}}{\mathrm{q}},$ where p and q are relatively prime positive integers.
Find the value of $\frac{\mathrm{q}-13}{\mathrm{p}}.$