Stick Breaking Triangle Inequality Probability

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on probability modeling, specifically regarding Geometric Probability.

Your Task
A stick of unit length $L = 1$ is broken at two points $X, Y \sim \text{Uniform}(0, 1)$ chosen uniformly and independently at random along its length, creating three segments $a$, $b$, and $c$. What is the probability that the three resulting segments can form a valid triangle satisfying the triangle inequalities?

Constraints

• Samples are independent and identically distributed on their respective support.
• Boundary conditions are defined strictly within standard real intervals.