2023 AMC 10B Problem 21

Each of 2023 balls is randomly placed into one of 3 bins. Which of the following is closest to the probability that each of the bins will contain an odd number of balls?

\textbf{(A)} \dfrac{2}{3} \qquad \textbf{(B)} \dfrac{3}{10} \qquad \textbf{(C)} \dfrac{1}{2} \qquad \textbf{(D)} \dfrac{1}{3} \qquad \textbf{(E)} \dfrac{1}{4}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.