<p>Sixteen players \(S_1, S_2, S_3, \ldots, S_{16}\) 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.<br>(a) Find the probability that the player \(S_1\) is among the eight winners.<br>(b) Find the probability that exactly one of the two players \(S_1\) and \(S_2\) is among the eight winners.</p>
Step-by-Step Solution
Key Concept: Since all players have equal strength, S₁ has a 50% chance to win their paired match regardless of pairing structure. For part (b), use conditional probability: P(exactly one wins) = P(S₁ wins, S₂ loses) + P(S₁ loses, S₂ wins), considering whether they're paired together.
<p><strong>Part (a):</strong></p><p><strong>Step 1:</strong> By symmetry, since all 16 players are of equal strength, each player has an equal probability of being among the 8 winners.</p><p><strong>Step 2:</strong> Total players = 16, winners = 8. By symmetry, P(S₁ is a winner) = 8/16 = <strong>1/2</strong></p><p><strong>Alternatively:</strong> In S₁'s pairing, S₁ plays against one opponent with equal strength. P(S₁ wins) = 1/2 regardless of who they face.</p><p><br/><strong>Part (b):</strong></p><p><strong>Step 1:</strong> Consider two cases based on whether S₁ and S₂ are paired together.</p><p><strong>Step 2:</strong> P(S₁ and S₂ paired together) = 1/15 [Once S₁ is selected, there are 15 possible partners for S₁, and S₂ is one of them]</p><p><strong>Step 3:</strong> If S₁ and S₂ are paired: Exactly one wins = they play each other, so P(exactly one wins | paired) = 1</p><p><strong>Step 4:</strong> If S₁ and S₂ are NOT paired: P(not paired) = 14/15. They play different opponents. P(exactly one wins | not paired) = P(S₁ wins and S₂ loses) + P(S₁ loses and S₂ wins) = (1/2)(1/2) + (1/2)(1/2) = 1/2</p><p><strong>Step 5:</strong> By total probability: P(exactly one of S₁, S₂ wins) = P(paired)·P(exactly one | paired) + P(not paired)·P(exactly one | not paired)</p><p>= (1/15)(1) + (14/15)(1/2) = 1/15 + 7/15 = <strong>8/15</strong></p>
Correct Answer: (a) 1/2, (b) 8/15