Gleb's Shape Theorem

This is a user suggested problem. The TopsOJ staff thanks and gives full credit to Lamp for their contribution.

Gleb is a 2\text{-d} shape with 12 edges. Each edge of Gleb is colored either red, green or blue. The probability of choosing a blue edge is \frac{1}{12}. The probability of choosing a red edge is \frac{1}{6}. If the probability of choosing a green edge is \frac{a}{b}, where a, b \in \mathbb{Z}, \gcd(a, b) = 1 and a, b > 0, find the value of a+b.