Suppose you drop a tie at random on the rectangular region shown in the figure. What is the probability that it will land inside the circle with diameter 1 m? "
Given:
Length of the rectangle $=3\ m$
Breadth of the rectangle $=2\ m$
Diameter of the circle $=1\ m$
To do:
We have to find the probability that the tie will land inside the circle.
Solution:
Area of a rectangle of length $l$ and breadth $b$ is $lb$.
This implies,
Area of the rectangle $=3 \times 2=6 \mathrm{~m}^{2}$
Diameter of circle $=1 \mathrm{~m}$
This implies,
Radius $(r)=\frac{1}{2} \mathrm{~m}$
Area of the circle $=\pi r^{2}=\pi \times(\frac{1}{2})^{2}=\frac{1}{4} \pi\ \mathrm{m}^{2}$
Therefore,
Probability that it will land inside the circle $=\frac{Area\ of\ the\ circle}{Area\ of\ the\ rectangle}$
$=\frac{\frac{1}{4} \pi}{6}$
$=\frac{1 \times \pi}{4 \times 6}$
$=\frac{\pi}{24}$
The probability that the tie will land inside the circle is $\frac{\pi}{24}$.
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