Question

In a knockout tournament 16 equally skilled players namely $P_1,P_2,------P_{16}$ are participating. In each round players are divided in pairs at random and winner from each pair moves to the next round. If $P_2$ reaches the semifinal, then the probability that $P_1$ will win the tournament is

• A. $\frac{3}{64}$
• B. $\frac{1}{16}$
• C. $\frac{1}{20}$
• D. $\frac{1}{15}$

Answer 1

The correct option is C

$\frac{1}{20}$

Let $E_1=P_1$ win the tournament, $E_2=P_2$ reaches the semifinal since all players are equally skilled and there are 4 persons in the semifinal $P(E_2=\frac{{}^{15}{C}_3}{{}^{16}{C}_4})=\frac{4}{16}=\frac{1}{4}{E}_1\cap E_2=P_1\&P_2$ both are in semifinal and $P_1$ wins in semifinal and final $(E_1\cap E_2)=\frac{16{-}^2{C}_2}{{}^{16}{C}_4}.\frac{1}{2}.\frac{1}{2}=\frac{3}{16.15}=\frac{1}{80}$ $P\left(\frac{E_1}{E_2}\right)=\frac{P(E_1\cap E_2)}{P\left(E_2\right)}=\frac{\frac{1}{80}}{\frac{1}{4}}=\frac{1}{20}$