2012 AMC 12A Problem 15

A $3\times3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated $90^\circ$ clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability that the grid is now entirely black?

$\textbf{(A)}\ \dfrac{49}{512}\qquad\textbf{(B)}\ \dfrac{7}{64}\qquad\textbf{(C)}\ \dfrac{121}{1024}\qquad\textbf{(D)}\ \dfrac{81}{512}\qquad\textbf{(E)}\ \dfrac{9}{32}$

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