Question

A square is inscribed in a circle such that all the four vertices lie on the circumference of the circle. If $P_1$ is the probability that a randomly selected point inside the circle lies within the square and $P_2$ is the probability that the point lies outside the square(inside the circle), then

• A. $P_1=P_2$
• B. $P_2<P_1$
• C. $P_2>P_1$
• D. $P_1^2-P_2^2<1/3$

Answer 1

Answer: (B), (D)

$P_1^2-P_2^2<1/3$
$P_2<P_1$

Let $r$ be the radius of the circle. The length of the diameter of the circle will be equal to the length of the diagonal of the square.
Hence, the area of the square $=A_s=2r^2$.
The area of the circle $=A_c=\pi r^2$.
$P_1=\frac{A_s}{A_c}=\frac{2}{\pi }>\frac{1}{2}$
$P_2=\frac{A_c-A_s}{A_c}=\frac{\pi -2}{\pi }<\frac{1}{2}$
Hence, $P_1>P_2$
${P_1}^2-{P_2}^2={\left(\frac{2}{\pi }\right)}^2-{\left(\frac{\pi -2}{\pi }\right)}^2=\frac{4\pi -{\pi }^2}{{\pi }^2}=\frac{4}{\pi }-1<\frac{1}{3}$