Here are two more Bags filled with Balls

Suppose that we have two bags each containing black and white balls.
One bag contains three times as many white balls as black balls.
The other bag contains three times as many black balls as white balls.

Suppose we choose one of these bags at random. For this bag we select five balls at random, replacing each ball after it has been selected.

The result is that we find 4 white balls and one black.

If the probability that we were using the bag with mainly white balls is given by \(\frac ab\), where \(a\) and \(b\) are coprime positive integers, find \(a+b\).

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