The figure below shows the front and rear view of a disc, which is shaded with identical patterns. The disc is flipped once with respect to any one of the fixed axes 1-1, 2-2, or 3-3 chosen uniformly at random. What is the probability that the disc DOES NOT retain the same front and rear views after the flipping operation?
Show Hint
When analyzing the symmetry of objects under rotations or reflections, consider how the object behaves with respect to different axes or lines of symmetry.
The disc is flipped with respect to one of the three axes (1-1, 2-2, or 3-3). Each axis can either preserve the symmetry of the disc or not. Let's examine the effect of flipping on the front and rear views.
1. When flipped along the axis 1-1 (vertical axis), the disc retains its identical pattern on both views. The front and rear views remain the same.
2. When flipped along the axis 2-2 (diagonal axis), the disc's front and rear views will not be identical after the flip. The pattern on the front and rear views will change.
3. When flipped along the axis 3-3 (another diagonal axis), the disc again does not retain identical patterns on both views.
Thus, in 2 out of the 3 possible flips (axes 2-2 and 3-3), the disc does not retain its identical pattern. Therefore, the probability that the disc does not retain the same front and rear views is:
\[
\frac{2}{3}.
\]
