Question

A point is selected at random inside a circle. The probability that the point is closer to the center of the circle than to its circumference is

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

Answer 1

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

Let $S$ denote the set of points inside the circle with radius $r$, and let $A$ denote the set of point inside the concentric circle with radius $\frac{1}{2}r$.
Thus, $A$ consists precisely of those points of $S$ which are closer to the center than to its circumference.
Therefore, $p=P\left(A\right)=\frac{\pi {\left(\frac{r}{2}\right)}^2}{\pi r^2}=\frac{1}{4}$