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

Probability \(= \frac{Area\,inside\,the\,circle}{Area\,of\,the\,rectangle}\)

Since the circle has diameter \(1\,m\),
Radius \(= \frac{1}{2} = 0.5\)
\(\begin{aligned} \text{Area of the circle}, A &= π{r}^2 \\ &= π(0.5)^2\\ &= π(0.25) \\ &= \frac{π}{4} {m}^2 \end{aligned}\)

Area of rectangle \(= length\,×\,breadth = 3 × 2 = 6 {m}^2\)

\(\begin{aligned} \text{Probability} &= \frac{Area\,of\,circle}{Area\,of\,rectangle} \\ &= \frac{\frac{π}{4}}{6} \\ &= \frac{π}{24} \end{aligned}\)
Using π ≈ 3.14,
Probability \(= \frac{3.14}{24} ≈ 0.131\)
So the probability that the dye lands inside the circle is 0.131.