Before you, Brilli the Ant places 2013 urns labelled 1 through 2013. In urn \(i\), there are \( i-1 \) white balls and \( 2013 - i \) black balls. If you randomly choose an urn, and then randomly choose 13 balls without replacement, the probability that all balls are the same color is \( \frac{a}{b} \), where \(a\) and \(b\) are positive coprime integers. What is the value of \(a + b \)?
by
Calvin Lin
