First we select edge 1-3 with probability $\frac{1}{9}$ now common neighbor of (1,3) are (4,2)
Probability of choosing 2 as neighbor is so that {1,2,3} can from triangle is $\frac{1}{2}$ so total probabiltiy is $\frac{1}{9}*\frac{1}{2}$
similar for edge 1-2 and edge 2-3 as both have only 2 neighbour common {3,5} and {1,6} respectively
total probability is $\frac{1}{9}*\frac{1}{2}*3$= $\frac{1}{6}$