All That Glitters Is Gold

There are three boxes:
1. a box containing 2 gold coins,
2. a box containing 2 silver coins,
3. a box containing 1 gold coin and 1 silver coin.

After choosing a box at random and picking out one coin, you withdrew a gold coin. If the chances that the other coin in the box is gold can be represented as \(\frac {a}{b}\) where \(a\) and \(b\) are co-prime, find the value of \(a + b\).


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