Problem 17 · 2022 Math Kangaroo
Hard
Counting & Probability
careful-counting
In a tournament with 8 participants the players are randomly paired up in four teams for the first round and the winner of each encounter then proceeds to the second round. There are two games in the second round and the two winners then play the final. Anita and Martina are the two best players and will win against all others; in case they have to play against each other, Anita will win. How big is the chance that Martina will get to the final?
Show answer
Answer: E — \(\tfrac{4}{7}\)
Show hints
Hint 1 of 2
Martina reaches the final unless she is drawn against Anita in round 1 or in the semifinal.
Still stuck? Show hint 2 →
Hint 2 of 2
Multiply the chance of dodging Anita each round.
Show solution
Approach: avoid the stronger player each round
- Round 1: Martina's opponent is one of 7; chance it is not Anita is 6/7.
- Semifinal: among the four winners she meets one of 3; chance it is not Anita is 2/3.
- P(reaches final) = (6/7)(2/3) = 12/21 = 4/7.
Mark:
· log in to save