Question

Suppose you drop a die at random on the rectangular region. What is the probability that it will land inside the circle with a diameter of $1m$?

Answer 1

Probability:

Diameter of the circle $=1m$

Radius of the circle $=\frac{1}{2}m$

Area of the circle $={\mathrm{\pi r}}^2=\mathrm{\pi }\times {\left(\frac{1}{2}\right)}^2=\mathrm{\pi }\times \frac{1}{4}=\frac{\mathrm{\pi }}{4}{m}^2$

Area of the rectangle $=3\times 2=6m^2$

The probability that die will land in the circle $\mathbf{\text{=}}\frac{\mathbf{Area}\mathbf{}\mathbf{of}\mathbf{}\mathbf{circle}}{\mathbf{Area}\mathbf{}\mathbf{of}\mathbf{}\mathbf{rectangle}}$

$=\frac{{\displaystyle \frac{\mathrm{\pi }}{4}}}{6}$

$=\frac{\mathrm{\pi }}{4}\times \frac{1}{6}=\frac{\mathrm{\pi }}{24}$

Therefore, the probability that die will land in the circle $=\frac{\mathrm{\pi }}{24}$