2017 AIME II Problems/Problem 2
The teams , , , and are in the playoffs. In the semifinal matches, plays , and plays . The winners of those two matches will play each other in the final match to determine the champion. When plays , the probability that wins is , and the outcomes of all the matches are independent. The probability that will be the champion is , where and are relatively prime positive integers. Find .
There are two scenarios in which wins. The first scenario is where beats , beats , and beats , and the second scenario is where beats , beats , and beats . Consider the first scenario. The probability beats is , the probability beats is , and the probability beats is . Therefore the first scenario happens with probability . Consider the second scenario. The probability beats is , the probability beats is , and the probability beats is . Therefore the second scenario happens with probability . By summing these two probabilities, the probability that wins is . Because this expression is equal to , the answer is .
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