Introduction to Probability · Problem 2.2.13
Combinatorics — Grinstead & Snell
Mathematics — Probability · continuous-density-functions
Problem
Take a stick of unit length and break it into two pieces, choosing the break point at random. Now break the longer of the two pieces at a random point. What is the probability that the three pieces can be used to form a triangle?Intuitive explanation
The key insight is that three lengths form a triangle if and only if the sum of any two sides exceeds the third—the triangle inequality. After two random breaks, you'll have three pieces with specific length constraints based on where each break occurred. Rather than tracking all possible outcomes, focus on the geometric region in your sample space where all three triangle inequalities are simultaneously satisfied, then compute its probability as a ratio of areas.
Intuitive explanation
The key insight is that three lengths form a triangle if and only if the sum of any two sides exceeds the third—the triangle inequality. After two random breaks, you'll have three pieces with specific length constraints based on where each break occurred. Rather than tracking all possible outcomes, focus on the geometric region in your sample space where all three triangle inequalities are simultaneously satisfied, then compute its probability as a ratio of areas.