10 Boxes

There are 10 boxes containing blue and red balls.

The number of blue balls in the nthn^\text{th} box is given by B(n)=2nB(n) = 2^n.
The number of red balls in the nthn^\text{th} box is given by R(n)=1024B(n)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 10th10^\text{th} box was picked can be expressed as ab \frac ab, where aa and bb are coprime positive integers, find a+ba+b.


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