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 \( \dfrac ab\), where \(a\) and \(b\) are coprime positive integers, find \(a+b\).


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