Random cuts make a triangle?

On 12/5/15, @mburnsmath posted the following classical problem: Break a piece of spaghetti into two pieces so that they are different lengths. Take the longer of the two pieces and break it into two pieces that are different elngths. This leaves you with three pieces of spaghetti. What is the probability that the three pieces can form a triangle? This is an attempt to show why the solution is greater than 1/3. Not seeing red below? GeoGebra is not cooperating. View a better image.
The proportion of good 2nd cuts to bad 2nd cuts in any vertical slice is always maintained. A workaround to see red in the GeoGebra file is to download it, open "Object Properties" for the red areas called "LeftRed" and "RightRed", go to the "Advanced" tab, then deselect and reselect "Graphics1".