John has a box that contains one red ball and one blue ball. Every minute he reaches into the box and randomly takes out a ball from it (without looking). No matter which ball he took out, he puts it back \({together\ with}\) another ball of the same color. Thus, after \(n\) minutes, the box contains a total of \(n+2\) balls.
The probability that after \(2\) hours, the box contains exactly \(77\) blue balls, can be written as \(\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers. Find \(a+b\).
by
Calvin Lin
