What is the probability for a wood stick of real number length breaking in three piece that can forming precisely a triangle? What is the probability for a wood stick of any real number length breaking in three piece that can forming precisely a triangle without any extra segment extend  from the side of the triangle?
My friend told me it can be solve in a calculus way and a algebra-precalculus way, what is the calculus way?
I found a relevant of this problem, but it is not the same to me : Broken stick probability problem
Last but not least: happy puzzle solving!

Picking two numbers independently that are uniformly distributed in $[0,1]$ is the same as picking a uniformly distributed point in the unit square.  Call the smaller one $x$ and the bigger one $y$.  The graph of $0<x<y<1$ is a triangular region and we are choosing a point within it.  The lengths of the three sides will then be $x$, $y-x$, and $1-y$.  We need three instances of the triangle inequality:
\begin{align}
x + (y-x) & > 1-y, \\
x + (1-y) & > y-x, \\
(y-x)+(1-y) & > x.
\end{align}
Draw the graph of the set of inequalities above within the aforementioned triangular region and you'll see the answer.
