Source: 2012 AMC10A Problem #20 and AMC12A Problem #15
February 8, 2012
rotationprobabilitygeometrygeometric transformationAMCAMC 12AMC 12 A
Problem Statement
A 3×3 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 the rotated 90∘ 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?<spanclass=′latex−bold′>(A)</span>51249<spanclass=′latex−bold′>(B)</span>647<spanclass=′latex−bold′>(C)</span>1024121<spanclass=′latex−bold′>(D)</span>51281<spanclass=′latex−bold′>(E)</span>329