Probability
Classical Probability
Grade 12
Question:
<p>Thirty-two players ranked 1 to 32 are playing in a knockout tournament. Assume that in every match between any two players, the better-ranked player wins, the probability that ranked 1 and ranked 2 players are winner and runner up, respectively, is</p>
<p>16/31</p>
<p>1/2</p>
<p>17/31</p>
<p>none of these</p>
Step-by-Step Solution
Key Concept: Player 1 must win the tournament and player 2 must reach the final without ever meeting player 1 before the final. This requires player 2 to be placed in the opposite half of the draw from player 1.
<p><strong>Step 1:</strong> Understand the tournament structure. A 32-player knockout tournament has two halves of 16 players each. Player 1 will win their half regardless of bracket arrangement. Player 2 must be in the opposite half to avoid meeting player 1 before the final.</p><p><strong>Step 2:</strong> Calculate probability that player 2 is placed in the opposite half. Once player 1's position is fixed, there are 31 remaining positions for player 2. Exactly 16 of these positions are in the opposite half of the bracket.</p><p><strong>Step 3:</strong> Probability = (Number of favorable positions for player 2)/(Total remaining positions) = 16/31</p><p><strong>Step 4:</strong> Given this favorable bracket arrangement, player 2 will definitely reach the final (by virtue of being ranked 2, they beat all others in their half), and player 1 will definitely win (they beat all in their half and player 2 in the final).</p><p>∴ Answer: A (probability = 16/31)</p>
Correct Answer: A