Question

On a line segment of length L two points are taken at random, find the probability that the distance between them is $l,$ where $l<1$

Answer 1

Let $AB$ be the line segment
Let $C$ and $D$ be any two points on $AB$ so that $AC=x$ and $CD=y.$ Then $x+y<L,y>l$
$\therefore$ sample space is represented by the region enclosed by $△OPQ.$
Area of $△OPQ=\frac{1}{2}{L}^2$
The event is represented by the region, bounded by the $△RSQ$
Area of $△RSQ=\frac{1}{2}{(L-l)}^2$
$\therefore$ probability of the event$={\left(\frac{L-l}{L}\right)}^2$