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
.