Question

A point $p$ is selected randomly from the interior of the circle, then the probability that is closer to the center of the circle rather than its boundary.

• A. $\frac{2}{3}$
• B. $\frac{1}{4}$
• C. $\frac{3}{4}$
• D. $\frac{1}{3}$

Answer 1

The correct option is B $\frac{1}{4}$

Let $p$ be a point inside the circle of radius $r$. Point $p$ is closer to the centre of the circle rather than its boundary, then $P$ must lie inside a circle whose radius is less than $\frac{r}{2}$
$\therefore$ Probability of the point $p$ which lies within the circle
$=\frac{\pi {\left(\frac{r}{2}\right)}^2}{\pi r^2}=\frac{1}{4}$