what's the probability a 1 meter steel broke into 3 pieces that forms a triangle
The 1 meter length is unimportant.
I assume we're making two cuts uniformly distributed along the bar.
So let $\displaystyle X_1$ and $\displaystyle X_2$ be iid U(0,1) rvs.
I checked the three case and each case we want the largest cut to be less than .5.
The cuts are $\displaystyle X_1$, $\displaystyle X_2-X_1$ and $\displaystyle 1-X_2$,
BUT here I'm assuming $\displaystyle X_2>X_1$ which is really the order stats.
I'm trying to figure the logic here.
We need $\displaystyle P(max\{X_1,X_2-X_1,1-X_2\}<.5)$
The joint density of these two is $\displaystyle f(x_1,x_2)=2I(0<x_1<x_2<1)$
It will depend on how the stick is broken.
Read the attachment and these:
Stick Broken Into Three Pieces. Solution in Trilinear Coordinates from Interactive Mathematics Miscellany and Puzzles
Random Triangles
Problem 8: Random Triangles and an Introduction to Density
Mathematical Recreations and Essays - Google Book Search