Question

Points are selected at random inside the circle until a point is closer to the centre than to the circumference. What is the probability that exactly three points are selected?

• A. $\frac{9}{32}$
• B. $\frac{16}{64}$
• C. $\frac{9}{64}$
• D. None of the above

Answer 1

Answer: (C)

$\frac{9}{64}$

Let the radius of the circle be ${}^{\prime }{r}^{\prime }.$ So the radius of points near to the center of circle will be $\frac{r}{2}.$ It is told that exactly $3$ points are selected which are near to the center of the circle.

$\therefore$ The radius for the three points $=\frac{3}{2}.\frac{r}{2}=\frac{3r}{4}$

$\Rightarrow$ Area of circle $=\frac{\pi {\left(\frac{3r}{4}\right)}^2}{4\pi r^2}$

$=\frac{9\pi r^2}{4\pi r^2}$

$=\frac{9}{64}.$

Hence, the answer is $\frac{9}{64}.$