Probability
Probability
Allen Star Batch
Grade 12
Question:
Eight players $P_1, P_2, \ldots P_8$ play a knock out tournament. It is known that whenever the players $P_i$ and $P_j$ play, the player $P_i$ will win if $i < j$. Assuming that the players are paired at random in each round, then the probability that the player $P_4$ reaches the final is ______.
Step-by-Step Solution
Key Concept: Divide total pairings by $4!$ to account for the ordering of pairs, then count favorable outcomes where exactly one pair of top seeds faces each other.
Eight players are paired in four pairs in $\frac{1}{4!}\binom{8}{2}\binom{6}{2}\binom{4}{2}\binom{2}{2} = 105$ ways. For at least two players to reach the second round, exactly two players must play against each other in the first round from the top four seeds, with the remaining player playing one of the bottom four. This occurs in $\binom{4}{2} \times 4\binom{3}{1} \times \binom{2}{1} = 36$ ways. The required probability is $\frac{4}{35}$.
Correct Answer: 0.1142