2023 AMC 12B Problem 19

Counting assignments where all three bins are odd with the parity filter gives $$\scriptsize\frac{1}{8}\sum_{S\subseteq\{1,2,3\}}(-1)^{|S|}(3-2|S|)^n=\frac{3^n-3}{4}$$ for odd $n.$ Dividing by the $3^n$ total assignments, the probability is $\dfrac{3^n-3}{4\cdot 3^n},$ which for $n=2023$ is extremely close to $\tfrac14.$ Thus, the correct answer is E.