2022 AIME II Problems/Problem 2
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 . When either Azar or Carl plays either Jon or Sergey, Azar or Carl will win the match with probability . Assume that outcomes of different matches are independent. The probability that Carl will win the tournament is , where and are relatively prime positive integers. Find .
Let be Azar, be Carl, be Jon, and be Sergey. The circles represent the players, and the arrow is from the winner to the loser with the winning probability as the label.
This problem can be solved by using cases.
's opponent for the semifinal is
The probability 's opponent is is . Therefore the probability wins the semifinal in this case is . The other semifinal game is played between and , it doesn't matter who wins because has the same probability of winning either one. The probability of winning in the final is , so the probability of winning the tournament in case 1 is
's opponent for the semifinal is or
It doesn't matter if 's opponent is or because has the same probability of winning either one. The probability 's opponent is or is . Therefore the probability wins the semifinal in this case is . The other semifinal game is played between and or . In this case it matters who wins in the other semifinal game because the probability of winning and or is different.
's opponent for the final is
For this to happen, must have won or in the semifinal, the probability is . Therefore, the probability that won in the final is .
's opponent for the final is or
For this to happen, or must have won in the semifinal, the probability is . Therefore, the probability that won or in the final is .
In Case 2 the probability of winning the tournament is
Adding case 1 and case 2 together we get so the answer is .
Video Solution (Mathematical Dexterity)
Video Solution by Power of Logic
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