Question

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 the value of $\frac{q-13}{p}$.

Answer 1

$ P(E) =P\left(P_4\right.$ and $P_2$ and $\left.P_4\right)+\left(P_4\right.$ and $P_3$ and $\left.P_4\right)$
$ =\frac{4}{5} \times \frac{2}{5} \times \frac{4}{6}+\frac{4}{5} \times \frac{3}{5} \times \frac{4}{7} $
$ =\frac{32}{150}+\frac{48}{175}=\frac{256}{525}=\frac{ p }{ q } $
$\therefore \frac{ q -13}{ p }=\frac{512}{256}=2 $