Difference between revisions of "2017 AIME II Problems/Problem 2"
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Revision as of 11:50, 23 March 2017
Problem
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
.
Solution
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
.
See Also
2017 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 |
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