Probability that a stick randomly broken in two places can form a triangle
The three triangle inequalities are
\begin{align} x + y &> 1-x-y \\ x + (1-x-y) &> y \\ y + (1-x-y) &> x \\ \end{align}
Your problem is that in picking the smaller number first from a uniform distribution, it's going to end up being bigger than it would if you had just picked two random numbers and taken the smaller one. (You'll end up with an average value of $1/2$ for the smaller instead of $1/3$ like you actually want.) Now when you pick $y$ on $[0, 1-x]$, you're making it smaller than it should be (ending up with average value of $1/4$). To understand this unequal distribution, we can substitute $y (1-x)$ for $y$ in the original inequalities and we'll see the proper distribution.
(Note that the $y$-axis of the graph doesn't really go from $0$ to $1$; instead the top represents the line $y=1-x$. I'm showing it as a square because that's how the probabilities you were calculating were being generated.) Now the probability you're measuring is the area of the strangely-shaped region on the left, which is $$\int_0^{1/2}\frac1{2-2x}-\frac{2x-1}{2x-2}\,dx=\ln 2-\frac12\approx0.19314$$ I believe that's the answer you calculated.
FYI: This question was included in a Martin Gardner 'Mathematical Games' article for Scientific American some years ago. He showed that there were 2 ways of randomly choosing the 2 'break' points:
- choose two random numbers from 0 to 1, or
- choose one random number, break the stick at that point, then choose one of the two shortened sticks at random, and break it at a random point.
The two methods give different answers.