Probability of area in a region being less than S

Question

Consider rectangular coordinate system and point $L(X,Y)$ which is randomly chosen among all points in the area $A$ which is defined in the following manner:

$A=\{(x,y)\,|\,x\in[-10,10],y\in[-5,5]\}$.

What is the probability $P$ that the area of a rectangle that is defined by points $(0,0)$ and $(X,Y)$ will be greater than $20$? Can it be generalized for any values of the region and given area?

Answer 1

Hint: Can you draw a region of points $(x,y)$ such that the area of the rectangle from $(0,0)$ to $(x,y)$ is $>20$?

Answer 2

I have given the image of the figure where the area is greater than 20 and the total rectangle A. Thus the probability is $\boxed{\dfrac{\left(30-20ln(\frac{10}{4})\right)}{50}} = 0.2335$

Generalizations:

$A=\{(x,y)\,|\,x\in[-x_0,x_0],y\in[-y_0,y_0]\}$.

Let the area $20$ be "a"

Then area of (0,0) and (x,y) $= 4\times \left[y_0\times(x_0-\frac{a}{y_0})-aln(\frac{x_0y_0}{a})\right]$ =C

Thus the probability $= \frac{C}{4x_0y_0}$