Ex.15.1 Q20 Probability Solution - NCERT Maths Class 10

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Question

Suppose you drop a die at random on the rectangular region shown in figure. What is the probability that it will land inside the circle with diameter \(1\rm{m}\)?

 

Text Solution

 

What is known?

A die is dropped at random on the rectangular region as shown in figure.

Length of rectangular region \(=3\,\rm{m}\)

Breadth of rectangular region \(=2\,\rm{m}\)

Diameter of the circle \(=1 \,\rm{m}\)

∴ Radius of the circle \(=0.5\,\rm{m}\)

What is unknown?

The probability that the die dropped at random will land inside the circle with diameter \(1\,\rm{m}\)?

Steps:

Area of rectangular region

\[\begin{align}\text{}& =L\,\times B \\ {} & =3\,\times 2 \\{} & =6{{\text{m}}^{2}} \\\end{align}\]

Diameter of circular region \(=1 \rm{m}\) \(\) 

Radius of circular region \(=\frac{1}{2}\text{m}\) 

\[\begin{align}\text{Area of circular region}&= \pi r^{2} \\ & =\pi \,\times {{\left( \frac{1}{2} \right)}^{2}} \\ &=\frac{\pi }{4}\end{align}\]

Probability that it will land inside the circle

\[\begin{align} \text{Probability that it will land inside the circle} & =\frac{\text{ Number of possible outcomes }}{\text{ No of favourable outcomes }} \\    &  =\quad \frac{\text{Area of circular region}}{\text{Area of rectangular region}}  \\    & =\quad \frac{\frac{\pi }{4}}{6}  \\   &  =\frac{\pi }{24}  \\
\end{align}\]

The probability that it will land inside the circle is \(\begin{align}\frac{\pi}{24}\end{align}\)