Combinatorics
AIME
Basic
2015
In a new school $40$ percent of the students are freshmen, $30$ percent are sophomores, $20$ percent are juniors, and $10$ percent are seniors. All freshmen are required to take Latin, and $80$ percent of the sophomores, $50$ percent of the juniors, and $20$ percent of the seniors elect to take Latin. Find the probability that a randomly chosen Latin student is a sophomore.
Let's assume there are totally $100$ students. Then the total number of students who take Latin will be $$40+30\times 80\% + 20\times 50\% + 10\times 20\% = 76$$
Among which sophomore is $30\times 80\%=24$. Therefore the result is $$\frac{24}{76}=\frac{6}{19}$$