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

  Radius of the circle$=r\Rightarrow n\left(S\right)=\pi r^2$  

             $n\left(E\right)=\pi {\left(\frac{r}{2}\right)}^2=\frac{\pi r^2}{4}$

            $P\left(E\right)=\frac{\frac{\pi r^2}{4}}{\pi r^2}=\frac{1}{4}$

The probability that the point is closer to the centre than the boundary of the  circle is $\frac{1}{4}$