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A round-robin tournament is held among 4 players, where every player faces every other player exactly once. Each match is won by one of the two players with equal probability (1/2 each), and no match ends in a draw. The probability that at the end of the tournament there is no player who won all their matches and also no player who lost all their matches equals a/b, where a and b are coprime positive integers. What is |2a - b|?
- 1
- 2
- 3
- 4
Correct answer: 2
Solution
Out of 64 equally likely outcomes, 40 contain at least one undefeated or winless player (by inclusion-exclusion: 32+32-24=40), leaving 24 valid outcomes. P = 24/64 = 3/8, giving a=3, b=8, and |2*3-8|=2.
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