Dot Products in a $2020$-gon

A regular $2020$-gon $A_1A_2\cdots A_{2020}$ is inscribed in the unit circle $O$. Two distinct vertices $A_i, A_j$ are chosen at random.

1. Find the probability that $\overrightarrow{OA_i} \cdot \overrightarrow{OA_j} > \dfrac{1}{2}$, and compare it with $\dfrac{1}{3}$. 2. Find the probability that $\left|\overrightarrow{OA_i} + \overrightarrow{OA_j}\right| < 1$, and compare it with $\dfrac{1}{3}$.