Question

Sixteen players S₁, S₂,..., S₁₆ play in a tournament. They are divided into eight pairs at random from each pair a winner is decided on the basis of a game played between the two players of the pair. Assume that all the players are of equal strength. . 

(a) Find the probability that the players S₁ is among the eight winners. 

(b) Find the probability that exactly one of the two players S₁ and S₂ is among the eight winners.

Answer 1

(a) Probability of S₁ to be among the eight winners = (Probability of S₁ being a pair ) x (Probability of S₁ winning in the group).

= 1 x 1/2 = 1/2 ('.'S₁ is definitely in a group)

(b) If S₁ and  S₂ are in the same pair, then exactly one wins.

If S₁ and S₂ are in two pairs separately, then exactly one of S₁ and S₂ will be among the eight winners. If S₁ wins and S₂ loses or S₁ loses and S₂ wins. 

Now, the probability of S₁, S₂ being in the same pair and one wins = (Probability of S₁, S₂ being the same pair)