# probability of a randomly 3 pieces broken stick s.t sum of 2 piece have greater length than the other

A stick is randomly broken into 3 pieces . Determine the probability that the sum of the lengths of any 2 pieces is greater than the length of the third piece.

• As I understand the problem, the probability is 1, since you can choose the largest two pieces to begin with. – Marc Apr 12 '16 at 11:49
• the question probably meant any random 2 pieces chosen – user330785 Apr 12 '16 at 11:51
• This is a well known problem...you can read about the usual formulation here or for some variants here – lulu Apr 12 '16 at 11:55

Geometric solution (non-strict): let $x$, $y$ and $z$ be lengths of pieces. Then in space with $xyz$ axes you have a triangle ($x+y+z=L, x\ge 0, y\ge 0, z \ge 0$). Total area of triangle corresponds to probability 1. Subsets of fitting pieces also correspond to a triangle, with area $1/4$ of the original one.