Question

Consider the figure given:
In the figure, two semi circles are drawn. $O$ is the centre of the larger one. If we put a dot inside the larger semi circle by closing our eye, what is the probability that the point is inside the smaller semi circle.

Answer 1

Let the radius of the smaller semi-circle be $r$.
Thus area of smaller semi-circle $=\frac{\pi r^2}{2}$
Now, area of larger semi-circle $=\frac{\pi {\left(2r\right)}^2}{2}=4\times \frac{\pi r^2}{2}=4$ (Area of smaller semi-circle)
$P$(Point is inside the semi-circle)$=\frac{Areaofsmallersemi-circle}{Areaoflargersemi-circle}=\frac{\frac{\pi r^2}{2}}{\frac{4\pi r^2}{2}}=\frac{1}{4}$
Thus, probability that the point is inside the smaller semi-circle is $\frac{1}{4}$.