In a library, $60$ people are reading fiction books and $45$ people are reading non-fiction books. Some people are reading both fiction and non-fiction. If one of the people reading a non-fiction book is selected at random, the probability that this person is also reading a fiction book is $\frac{1}{3}$. If instead, someone reading a fiction book is selected at random, what is the probability that this person is also reading a non-fiction book?