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 and 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 , and ,
BUT here I'm assuming which is really the order stats.
I'm trying to figure the logic here.
We need
The joint density of these two is
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