Question

A point is selected at random from the interior of the circle. The probability that the point is closer to the centre than the boundary of the circle is

• A. $\frac{1}{2}$
• B. $\frac{1}{3}$
• C. $\frac{1}{4}$
• D. None of these

Answer 1

The correct option is C

$\frac{1}{4}$

Explanation for the correct option:

Step 1: Find the area of space where it will be closer to the centre than the

circumference:

Assume the radius of the circle as $r$.

Then the required area will be in a radius less than $\frac{r}{2}$.

Required area$=\pi {\left(\frac{r}{2}\right)}^2$

Step 2: Find the total area of the circle:

Estimate the total area of the circle as sample space.

$A=\pi r^2$

Step 3: Find the probability of the point to lie in the favourable area:

Probability can be estimated by dividing the favourable area by the total area of the circle.

$\begin{array}{rcl}p& =& \frac{\pi {\displaystyle \frac{r^2}{4}}}{\pi r^2}\\ & =& \frac{1}{4}\end{array}$ $\left[\because p=\frac{\text{Favourable area}}{\text{Total Area}}\right]$

Hence, option (C) is the correct answer.