Question
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 :
A.
$$\frac{3}{4}$$
B.
$$\frac{1}{2}$$
C.
$$\frac{1}{4}$$
D.
none of these
Answer :
$$\frac{1}{4}$$
Solution :
$$n\left( S \right) = $$ the area of the circle of radius $$r.$$
$$n\left( E \right) = $$ the area of the circle of radius $$\frac{r}{2}.$$
$$\therefore $$ the required probability $$ = \frac{{n\left( E \right)}}{{n\left( S \right)}} = \frac{{\pi .{{\left( {\frac{r}{2}} \right)}^2}}}{{\pi {r^2}}} = \frac{1}{4}$$
$$n\left( S \right) = $$ the area of the circle of radius $$r.$$
$$n\left( E \right) = $$ the area of the circle of radius $$\frac{r}{2}.$$
$$\therefore $$ the required probability $$ = \frac{{n\left( E \right)}}{{n\left( S \right)}} = \frac{{\pi .{{\left( {\frac{r}{2}} \right)}^2}}}{{\pi {r^2}}} = \frac{1}{4}$$