$13+2$ Marksl See the fiaure below. We will estimate the value of $$ usina this fiaure.

$($a$)$ Simulate sampling $1000$ data points $($RANDOM DATA$)$ and the equation taught in the class to estimate the area of the circle.

$1.$ We define the system, it$$s necessary conditions

$2.$ Take random data points

$3.$ Process

$4.$ Observe the outcome and Calculate the result

So$,$ the Probability of Hit

$=($no$.$ of darts inside the circle$/$ no$.$ of total darts thrown$)=($Area of Circle$/$ Area of Rectangle$)$

$=$ pi r$2/(2$r $*2$r$)[$for fig left $($high prob$)]$

$=$ pi $/4$

So$,$ Pi $=4*($no of hits$/$ no$.$ total darts$)$

Area:

Again let$$s consider the Prob. of hit where we don$$t know the Area of the Circle.

the Probability of Hit

$=($no$.$ of darts inside the circle$/$ no$.$ of total darts thrown$)=($Area of Circle$/$ Area of Rectangle$)$

$=$ Area of Circle $/($L $*$ W$)$

$=$ Area of Circle $/($L $*$ W$)$

So$,$ Area of Circle $=($L $*$ W$)*($no of hits$/$ no$.$ total darts$)$

$($b$)$ Make a scatter plot with your sampled data points. Mark points inside the circle red and outside the circle green.