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In the figure, points A, B, C and D are the centres of four circles that each have a radius of length one unit. If a point is selected at random from the interior of square ABCD. What is the probability that the point will be chosen from the shaded region?"

Academic Mathematics NCERT Class 10

Given:

In the figure, points A, B, C and D are the centres of four circles that each have a radius of length one unit.

A point is selected at random from the interior of square ABCD.

To do:

We have to find the probability that the point will be chosen from the shaded region.

Solution:

Radius of each circle $=1$ unit

This implies,

Length of the side of the square $\mathrm{ABCD}=1+1=2$ units.

Area of the square $=2^{2}=4$ sq. units.

Area of four quadrants inside the square at $\mathrm{A}, \mathrm{B}, \mathrm{C}$ and $\mathrm{D}=4 \times \frac{1}{4} \pi r^{2}$

$=\pi(1)^{2}$

$=\pi$ sq. units

This implies,

Area of the shaded region $=4-\pi$ sq. units

Therefore,

Probability that the point will be chosen from the shaded region $=\frac{\text { Area of the shaded region } }{\text { Area of the square }}$

$=\frac{4-\pi}{4}$

$=1-\frac{\pi}{4}$

The probability that the point will be chosen from the shaded region is $1-\frac{\pi}{4}$.

Updated on: 10-Oct-2022

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