Question

A square is inscribed in a circle. If $p_1$ is the probability that a randomly chosen point of the circle lies within the square and $p_2$ is the probability taht the point lies outside the square, then

• A. $p_1=p_2$
• B. $p_1>p_2$ and ${p_1}^2-{p_2}^2<\frac{1}{3}$
• C. $p_1<p_2$
• D. None of these

Answer 1

Answer: (B)

$p_1>p_2$ and ${p_1}^2-{p_2}^2<\frac{1}{3}$

If $a$ is the radius of the circle, the area of the inscribed square $=2a^2$ and $p_1=\frac{2a^2}{\pi a^2}=\frac{2}{\pi },p_2=1-p_1=\frac{\pi -2}{\pi }$
$\pi <4$ and so $\pi -2<2$ which gives $p_1>p_2$.
${p_1}^2-{p_2}^2=(p_1-p_2)(p_1+p_2)=\frac{4-\pi }{\pi }<\frac{1}{3}$ as $3<\pi <4$.