Probability
Classical Probability
Grade 12
Question:
<p>\(2^n\) players of equal strength are playing a knock out tournament. If they are paired at randomly in all rounds, find the probability that out of two particular players \(S_1\) and \(S_2\), exactly one will reach in semi-final \((n \in N,\ n \geq 2)\).</p>
<p>\(\dfrac{8(2^n - 4)}{2^n(2^n - 1)}\)</p>
<p>\(\dfrac{8(2^n - 4)}{2^n(2^n - 1)(2^n - 2)}\)</p>
<p>\(\dfrac{4(2^n - 4)}{2^n(2^n - 1)}\)</p>
<p>\(\dfrac{8(2^n - 2)}{2^n(2^n - 1)}\)</p>
Step-by-Step Solution
Key Concept: For exactly one of S₁, S₂ to reach the semifinal, they must NOT meet before the semifinal (so both reach quarterfinals), then exactly one must win their semifinal match. Calculate the probability they avoid each other until semifinals, then one loses the semifinal.
<p><strong>Step 1:</strong> For exactly one of S₁, S₂ to reach the semifinal, both must reach the semifinal stage, then exactly one must lose in the semifinal.</p><p><strong>Step 2:</strong> The tournament has 2ⁿ players. The semifinal means 2 players remain after (n-1) rounds. For S₁ and S₂ to both reach the semifinal without meeting earlier, they must be in different halves of the bracket at each stage up to the semifinals.</p><p><strong>Step 3:</strong> Probability that S₁ and S₂ are placed in different halves of the 2ⁿ bracket = 2ⁿ⁻¹/(2ⁿ-1) (once S₁ is placed, there are 2ⁿ⁻¹ spots out of 2ⁿ-1 remaining spots in the opposite half).</p><p><strong>Step 4:</strong> Given they're in different halves and both reach their respective semifinal brackets, the probability that exactly one wins their semifinal = 1/2 (since they have equal strength, P(S₁ wins semifinal) = P(S₂ wins semifinal) = 1/2; exactly one wins = P(S₁ wins and S₂ loses) + P(S₂ wins and S₁ loses) = 1/4 + 1/4 = 1/2).</p><p><strong>Step 5:</strong> Probability both reach semifinal given they're in different halves = 1/2 × 1/2 = 1/4 (each must win their bracket independently with probability 1/2).</p><p><strong>Step 6:</strong> Total probability = [2ⁿ⁻¹/(2ⁿ-1)] × (1/4) × (1/2) = 2ⁿ⁻³/(2ⁿ-1)</p><p>∴ Answer: A</p>
Correct Answer: A