Mock AIME 2 2010 Problems/Problem 4

Anderson is bored in physics class. His favorite numbers are $1, 7$, and $33$. He writes $0.$, and randomly appends one of his favorite numbers to the end of the decimal he has already written. Since physics class is infinitely long, Anderson writes an infinitely long decimal number. (An example of such a number is $0.1337173377133733 \ldots$) If the expected value of the number Anderson wrote down is of the form $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, find $a + b$.

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