AMC 12 2023b Problem 19

Each of $2023$ balls is placed in 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?

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