Odd dot product of two random 0-1 vectors

Two random vectors of length $10$ are generated: every entry of $v_1$ is an independent $\text{Bernoulli}(\tfrac12)$ variable, and every entry of $v_2$ is an independent $\text{Bernoulli}(\tfrac34)$ variable. The probability that the dot product $v_1\cdot v_2$ is odd can be written as $\frac12-\frac{1}{a^n}$ with $a$ prime. What is $a\cdot n$?

$42$. Each coordinate product is $1$ with probability $q=\tfrac38$, so the parity of a sum of $10$ independent bits is odd with probability $\tfrac12\left(1-(1-2q)^{10}\right)=\tfrac12-\tfrac12\cdot4^{-10}=\tfrac12-2^{-21}$. Hence $a=2$, $n=21$.