Question

Thirty-two players ranked $1$ to $32$ are playing in a knockout tournament. Assume that in every match between any to players, the better-ranked player wins, the probability that ranked $1$ and ranked $2$ players are winner and runner up respectively

• A. $\frac{16}{31}$
• B. $\frac{1}{2}$
• C. $\frac{17}{31}$
• D. $\frac{18}{31}$

Answer 1

The correct option is A $\frac{16}{31}$

1) Consider first round. Player with rank 1 can play with other 31 players. But, player with rank 1 should not play with player of rank 2 or else player 2 will get knocked out and it is not allowed.

Thus, total possible outcomes are 31 and favorable outcomes are 30

Thus, probability of first round is $\frac{30}{31}$

2) Consider second round. 16 players will get knocked out in first round so 16 players will be remaining.

Player with rank 1 can play with other 15 players but he will not play with player of rank 2 as player 2 is supposed to appear in final round.

Thus, total possible outcomes are 15 and favorable outcomes are 14

Thus, probability of second round is $\frac{14}{15}$

3) Consider third round. 8 players will be remaining in this round.

Total possible outcomes are 7 and favorable outcomes are 6.

Thus, probability of third round is $\frac{6}{7}$

4) Consider fourth round. 4 players will be remaining in this round.

Total possible outcomes are 3 and favorable outcomes are 2

Thus, probability of fourth round is $\frac{2}{3}$

5) In last and final round, only two players with rank 1 and 2 will play with each other. Thus, player with rank 1 will be winner and other will be runner up as per the requirement.

Thus, probability of final round is 1.

Thus, total probability is,

$\frac{30}{31}\times \frac{14}{15}\times \frac{6}{7}\times \frac{2}{3}\times 1$

$=\frac{16}{31}$