Question

If three lines are chosen at random, prove that they are just likely as not to denote the sides of a possible triangle.

Answer 1

Of three lines one must be equal to or greater than each of the other two; denote its length by $l$. Then all we know of the other two lines is that the length of each lies between $0$ and $l$. But if each of two lines is known to be of random length between $0$ and $l$, it is an even chance that their sum is greater than $l$. Thus the required results follows.