# The Absent-Minded Professor

An absent-minded professor left his office with his umbrella, lectured in three different classrooms, and then returned to his office. The probability that the professor will leave his umbrella in any classroom is $$\frac{1}{4}$$, under the condition that he enters the classroom with his umbrella. Given that the professor returned to his office and noticed that he no longer had his umbrella, the probability that the professor left his umbrella in the first classroom can be written as $$\frac{a}{b}$$, where $$a$$ and $$b$$ are coprime positive integers. Find $$a+b$$.

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