Binary Dot

Let $v_1$ and $v_2$ be two vectors of length $10$. The $10$ elements of $v_1$ are IID $\text{Bernoulli}\left(\dfrac{1}{2}\right)$ random variables. The $10$ elements of $v_2$ are IID $\text{Bernoulli}\left(\dfrac{3}{4}\right)$ random variables. Find the probability that $v_1 \cdot v_2$ is odd, where the $\cdot$ is dot product of two vectors. The probability is in the form $\dfrac{1}{2} - \dfrac{1}{a^n}$, where $a$ is prime. Find $an$.