Adarsh decided to attend CEN-105 lecture feeling he will learn something new. It turns out the class is extremely boring. He begins writing a random number starting with "$0.$". After the decimal point, he writes a long string of numbers, formed by randomly choosing any prime between $1$ and $9$ repeatedly (Ex. $0.7223557333 \dots$).

Let the expected number formed be represented as $\frac{p}{q}$, where $p$ and $q$ are coprime integers, find $p+q$.

You can learn about expected value here