Sum of Two points

Let the two numbers be $x , y$ on two co-ordinate axes.
$0 \le x,y \le 1$
The area of the entire region $A = 1 \cdot 1 = 1$ .
$x^{2} + y^{2} \le 1 , 0 \le x,y$ denotes the region inside a quarter circle of radius 1.
Area of this region $A_{1} = \dfrac{\pi}{4}$
Probability $= \dfrac{A_{1}}{A} = \dfrac{\pi}{4}$

Let $x$ and $y$ denote the random variables associated with the given points. Since equal intervals are assumed to have equal probabilities, the density functions of $x$ and $y$ are given respectively by:

$f_{1}$ $(x)$ = $1\ when\ 0\le x\le 1$ ; otherwise, it is $0$

$f_{2}$ $(y)$ = $1\ when\ 0\le y\le 1$ ; otherwise, it is $0$

Since $x$ and $y$ are independent, the joint density function is given by:

$f(x,y)= f_{1}(x)f_{2}(y)$ = $1\ when\ 0\le x\le 1$ and $\ 0\le y\le 1$ ; otherwise, it is $0$

It follows that the required probability is given by: $P(x^{2}+y^{2}\le 1)=$ $\large\iint_Rdxdy$

where $\large R$ is th region defined by $x^{2}+y^{2}\le 1$ for non-negative $x$ and $y$ , which is a quarter of a circle of radius 1.

We see now the area of $\large R$ is $\frac{\pi}{4}$ , where the $r= radius=1$ and the area of the circle is $\pi*r^{2}$