Question

Two semicircles are drawn in a square as shown. If we put a dot in the figure, without looking into it, what is the probability of the dot being in the shaded region?

Answer 1

Area of a square of side a = $a^2$

If a is the side the square, the diameter of the semicircular region = a

Radius = $\frac{a}{2}$

Area of the semicircle = $\frac{\pi r^2}{2}$

$\therefore$ Area of one of the semicircles = $\frac{\frac{\pi a^2}{4}}{2}$ = $\frac{\pi a^2}{8}$

Area of two semicircles = $\frac{2\times \pi a^2}{8}$= $\frac{\pi a^2}{4}$

Area of the shaded region = Area of the square - Area of the semicircles

= $a^2-\frac{\pi a^2}{4}$

= $\frac{4a^2-\pi a^2}{4}$ = $\frac{a^2(4-\pi )}{4}$

Probablity = $\frac{\frac{a^2(4-\pi )}{4}}{a^2}$= $\frac{a^2(4-\pi )}{4a^2}$= $\frac{4-\pi }{4}$