2022 SSMO Speed Round Problems/Problem 7
Problem
At FenZhu High School, th graders have a 60\% of chance of having a dog and
th graders have a 40\% chance of having a dog. Suppose there is a classroom of
th grader and
th graders. If exactly one person owns a dog, then the probability that a
th grader owns the dog is
for relatively prime positive integers
and
Find
Solution
The probability that a th grader has the only dog is
and the probability for an
th grader is
Then,
The probability is thus so the
sum is