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 $\text{DOES NOT}$ retain the same front and rear views after the flipping operation?

1. $0$
2. $\frac{1}{3}$
3. $\frac{2}{3}$
4. $1$

Given that, the front and rear view of a disc.

We have a three-axis, and we can flip with respect to $1-1,2-2$ and $3-3:$

• When we flipped disk with axis ${\color{Blue}{1-1:\text{Front View} \Leftrightarrow \text{Rear View}}}$
• When we flipped disk with axis ${\color{Magenta}{2-2:\text{Front View} \nLeftrightarrow \text{Rear View}}}$
• When we flipped disk with axis ${\color{Purple}{3-3:\text{Front View} \nLeftrightarrow \text{Rear View}}}$

Now we have,

• The number of favorable cases $= 2$
• The total number of cases $= 3$

The required probability ${\color{Green}{= \dfrac{\text{The number of favorable cases }}{\text{The total number of cases}}}} = \dfrac{2}{3}.$

$\therefore$ The probability that the disc ${\color{Red}{\text{DOES NOT}}}$ retain the same front and rear views after the flipping operation is $\dfrac{2}{3}.$

Correct Answer $:\text{C}$

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