Question

Suppose you drop a die 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?

Answer 1

$\Large\frak{\underline{\underline{Question:}}}$

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 1m?

$\Large\frak{\underline{\underline{Answer:}}}$

$\underline{\bigstar\:\textsf{Area \: of \: rectangle:}}$

$\normalsize\dashrightarrow\sf\ Area \: of \: rectangle = length \times\ breadth \\\\ \normalsize\dashrightarrow\sf\ Area = 3 \times\ 2\\\\ \normalsize\dashrightarrow\sf\ Area = 6 \\\\ \normalsize\dashrightarrow\sf\ Area = \bf\ 6m^2\\\\\\\underline{\bigstar\:\textsf{Area \: of \: circle:}} \\\\\normalsize\dashrightarrow\sf\ Area \: of \: circle = \pi r^2 \\\\ \normalsize\dashrightarrow\sf\ Area = \pi \times\ r \times\ r \\\\ \normalsize\dashrightarrow\sf\ Area = \pi \times\ \frac{1}{2} \times\ \frac{1}{2} \\\\ \normalsize\dashrightarrow\sf\ Area = \frac{\pi}{4}\\\\ \normalsize\dashrightarrow\sf\ Area = \bf\frac{\pi}{4} m^2$

$\rule{120}{2}$

☯ $\underline{\textsf{Probability \: of \: Event:}}$

$\star$ Event : Chances of die will land under circle.

$\normalsize\ : \implies\sf\ P(E) = \frac{No. \: of \: possible \: outcomes}{No. \: of \: total \: outcomes}\\\\\\\normalsize\ : \implies\sf\ P(die) = \frac{Area \: of \: circle}{Area \: of \: rectangle}\\\\\\\normalsize\ : \implies\sf\ P(die) = \frac{\frac{\pi}{4}}{6}\\\\\\\normalsize\ : \implies\sf\ P(die) = \frac{\pi}{24}\\\\\\\normalsize\ : \implies{\underline{\boxed{\sf \red{ P(die) = \frac{\pi}{24} }}}}$

Answer 2

$\huge\underline\mathrm{SOLUTION:-}$

AnswEr:

• Probability is $\mathsf { \frac{\pi}{24} }$.

Need To Find:

• We use area to find probability = ?

ExPlanation:

Here:

• Diameter of circle = 1m
• Radius of circle = 1/2 m

Now,

• Area of rectangle = Length × Breadth

Putting the values according to the given formula:

➠ Area of rectangle = 3 × 2

➠ Area of rectangle = 6 m²

• Hence, the area of rectangle is 6 m².

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And,

• Area of circle = πr²

Putting the values according to the given formula:

➠ Area of circle = $\mathsf {\pi\bigg(\frac{1}{2}\bigg)^2}$

➠ Area of circle = $\mathsf {\pi \times \frac{1}{4} m {}^{2} }$

• Hence, the area of circle is $\mathsf {\pi \times \frac{1}{4} m {}^{2} }$.

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Now,

• P (die will land inside circle)

➠ $\mathsf {\frac{Area\:of\: circle}{Total\:Area}}$

➠ $\mathsf { \frac{\pi \times \frac{1}{4} }{6} }$

➠ $\mathsf { \frac{\pi}{6 \times 4}}$

➠ $\mathsf { \frac{\pi}{24} }$

• So, Probability is $\mathsf { \frac{\pi}{24} }$.

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