Question

There are four players $A_1{,A}_2,A_3$ and $A_4$ in pool $A$ and four players $B_1{,B}_2,B_3$ and $B_4$ in pool $B$. They are arbitrarily paired in their respective pools to play against each other and one winner is decided in each pair to play in semifinal. It is known that whenever $A_1$ plays with $A_2,A_1$ wins the game and whenever $B_1$ plays with $B_2,B_1$ wins.

Find the probability that $A_1$ and $A_2$ reach in the semifinal.

• A. $\frac{1}{3}$
• B. $\frac{1}{6}$
• C. $\frac{11}{72}$
• D. $\frac{1}{72}$

Answer 1

The correct option is B $\frac{1}{6}$

Group stage
Way $1:a_1{a}_2$ and $a_3{a}_4$
Way $2:a_1{a}_3$ and $a_2{a}_4$
Way $3:a_1{a}_4$ and $a_2{a}_3$
Possibilities for semifinals
Way $1:a_1{a}_3$ or $a_1{a}_4$
Way $2:a_1{a}_3$ or $a_1{a}_4$ or $a_2{a}_3$ or $a_3{a}_4$
Hence $P\left(a_1{a}_2\right)=\frac{1}{3}(0.25+0.25)=\frac{1}{6}$