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
![\[A = \left(30 \cdot \left(\frac{2}{5}\right)^{29} \cdot \frac{3}{5}\right) \cdot \left(\frac{3}{5}\right)^{10}\]](http://latex.artofproblemsolving.com/5/9/1/5917c8e032694ce29feace325ba864266eb9db63.png)
and the probability for an th grader is
![\[B = \left(\frac{2}{5}\right)^{30} \cdot \left(10 \cdot \left(\frac{3}{5}\right)^{9} \cdot \frac{2}{5}\right).\]](http://latex.artofproblemsolving.com/d/c/6/dc66ab61d058c251382699bf70ee221c882c4481.png)
Then,
![\[\frac{A}{B} = \frac{30 \cdot 2^{29} \cdot 3 \cdot 3^{10}}{2^{30} \cdot 10 \cdot 3^9 \cdot 2} = \frac{30 \cdot 3 \cdot 3}{2 \cdot 10 \cdot 2} = \frac{27}{4}.\]](http://latex.artofproblemsolving.com/0/4/a/04a7a382dca2315a2ce81982e89b386fc049cb77.png)
The probability is thus 
so the
sum is 
