2. Given a regular 2017-gon $A_{1} A_{2} \cdots A_{2017}$ inscribed in the unit circle $\odot O$, take any two different vertices $A_{i} 、 A_{j}$. Then the probability that $\overrightarrow{O A_{i}} \cdot \overrightarrow{O A_{j}}>\frac{1}{2}$ is . $\qquad$
2. $\frac{1}{3}$.

Notice that, $\overrightarrow{O A_{i}} \cdot \overrightarrow{O A_{j}}>\frac{1}{2} \Leftrightarrow \angle A_{i} O A_{j}<60^{\circ}$.
For any given point $A_{i}$, the number of ways to choose point $A_{j}$ such that $\angle A_{i} O A_{j}<60^{\circ}$ is $2\left[\frac{2017}{6}\right]=672$ (where [x] denotes the greatest integer not exceeding the real number $x$).
Therefore, the required probability is $\frac{672}{2016}=\frac{1}{3}$.