Question

The figure shows a circle with centre O and radius 8 cm. $\angle$ BOD =${72}^{\circ }$. The radius of the smaller circle is 4 cm. A point is selected randomly inside the larger circle, find the probability that the point lies neither inside the smaller circle nor inside the sector.

• A. $\frac{13}{20}$
• B. $\frac{3}{5}$
• C. $\frac{11}{20}$
• D. $\frac{1}{2}$

Answer 1

The correct option is C $\frac{11}{20}$

Area of larger circle = $\pi$$\times$$8^2$= 64$\pi$
Area of sector = $\frac{72}{360}$$\times$area of larger circle
= $\frac{1}{5}$$\times$64$\pi$
= $\frac{64\pi }{5}$
Area of smaller circle = $\pi$$\times$$4^2$ = 16$\pi$
Probability of event $E_1$,
P(point lies inside the sector)=$\frac{1}{5}$
Probability of event $E_2$,
P(point lies inside the smaller circle)=$\frac{1}{4}$

Let $E_3$ be the event of the point lying neither inside the smaller circle nor inside the sector.

Probability of an event $E_3=1-E_1-E_2$ (Sum of complementary events is $1$).

P( the point lying neither inside the smaller circle nor inside the sector.) $=1-\frac{1}{5}-\frac{1}{4}=\frac{11}{20}$