2022 AIME II Problems/Problem 2

Problem

Azar, Carl, Jon, and Sergey are the four players left in a singles tennis tournament. They are randomly assigned opponents in the semifinal matches, and the winners of those matches play each other in the final match to determine the winner of the tournament. When Azar plays Carl, Azar will win the match with probability $\frac23$ . When either Azar or Carl plays either Jon or Sergey, Azar or Carl will win the match with probability $\frac34$ . Assume that outcomes of different matches are independent. The probability that Carl will win the tournament is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p+q$ .

Solution 1

Let $A$ be Azar, $C$ be Carl, $J$ be Jon, and $S$ be Sergey. The $4$ circles represent the $4$ players, and the arrow is from the winner to the loser with the winning probability on top.

This problem can be solved by using $2$ cases.

$\textbf{Case 1:}$ $C$ 's opponent for the semifinals is $A$

The probability $C$ 's opponent is $A$ is $\frac13$ . Therefore the probability $C$ wins the semifinals in this case is $\frac13 \cdot \frac13$ . The other semifinal game is played between $J$ and $S$ , it doesn't matter who wins because $C$ has the same probability of winning either one. The probability of $C$ winning in the finals is $\frac34$ , so the probability of $C$ winning the tournament is $\frac13 \cdot \frac13 \cdot \frac34$

$\textbf{Case 2:}$ $C$ 's opponent for the semifinals is $J$ / $S$

It doesn't matter if $C$ 's opponent is $J$ / $S$ because $C$ has the same probability of winning either one. The probability $C$ 's opponent is $J$ / $S$ is $\frac23$ . Therefore the probability $C$ wins the semifinals in this case is $\frac23 \cdot \frac34$ . The other semifinal game is played between $A$ and $J$ / $S$ . In this case it matter who wins in the other semifinal game because the probability of $C$ winning $A$ or $J$ / $S$ is different.

$\textbf{Case 2.1:}$ $C$ 's opponent for the finals is $A$

For this to happen, $A$ must have won $J$ / $S$ in the semifinals

To be continued......

~isabelchen

Video Solution (Mathematical Dexterity)

https://www.youtube.com/watch?v=C14f91P2pYc

2022 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 1	Followed by Problem 3
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

Page

Toolbox

Search

2022 AIME II Problems/Problem 2

Contents

Problem

Solution 1

Video Solution (Mathematical Dexterity)

See Also