# 10 Boxes

There are 10 boxes containing blue and red balls.

The number of blue balls in the $$n^\text{th}$$ box is given by $$B(n) = 2^n$$.
The number of red balls in the $$n^\text{th}$$ box is given by $$R(n) = 1024 - B(n)$$.

A box is picked at random, and a ball is chosen randomly from that box. If the ball is blue, and the probability that the $$10^\text{th}$$ box was picked can be expressed as $$\frac ab$$, where $$a$$ and $$b$$ are coprime positive integers, find $$a+b$$.

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