Question

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

Answer 1

There is a problem to find the probability that land inside the circle with diameter $1m$
Probability is a type of ratio where we compare how many times an outcome can occur compared to all possible outcomes.
${\text{Probability = }}\dfrac{{\left( {{\text{The number of wanted outcomes}}} \right)}}{{\left( {{\text{The number of possible outcomes}}} \right)}}$

Complete step-by-step answer:
The given figure is:

It is given that a die is dropped at random on the rectangular region shown in fig.
We need to determine the probability that it will land inside the circle with diameter $1m$.
Here we use the area to find the probability.
Now we have the diameter of the circle is $1m$.
Thus we get the radius of the circle is $\dfrac{1}{2}m$ .
The area of the circle is
$\Rightarrow \pi {r^2}$
Substituting the values in given,
$\Rightarrow \dfrac{{22}}{7} \times \dfrac{1}{2} \times \dfrac{1}{2}$
$\Rightarrow \dfrac{{11}}{{14}}{m^2}$
Again it is given that,
The length of the rectangle is $3m$.
The breadth of the rectangle is $2m$.
Thus the area of the rectangle is =$3 \times 2 = 6{m^2}$
Probability (P) that the die will land inside the circle
$\Rightarrow \dfrac{{\left( {{\text{The number of wanted outcomes}}} \right)}}{{\left( {{\text{The number of possible outcomes}}} \right)}}$
Probability formula for the problem can be written as,
$\Rightarrow \dfrac{{{\text{The area of the circle}}}}{{{\text{the area of the rectangle}}}}$
$\Rightarrow \dfrac{{\dfrac{{11}}{{14}}}}{{\dfrac{6}{1}}}$
Simplifying we get,
$\Rightarrow \dfrac{{11}}{{14}} \times \dfrac{1}{6}$
$\Rightarrow \dfrac{{11}}{{84}}$
Hence we get, the probability that the die will land inside the circle with diameter $1m$ is$\dfrac{{11}}{{84}}$.

Note: A rectangle is a 2D shape in geometry, having four sides and four corners. Its two sides meet at right angles. Thus, a rectangle has four angles, each measuring $90^\circ$. The opposite sides of a rectangle have the same lengths and are parallel.
${\text{Area of the rectangle}}$ = $Length \times breadth$
A circle is a shape consisting of all points in a plane that are a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant.
Area of the circle is $\pi {r^2}$.