Break a stick, can the pieces form a triangle?

Question

You break a stick of length 1 at two points chosen independently and uniformly along it. What is the probability that the three resulting pieces can form a triangle?

Accepted Answer

Let the two cuts fall at x, y \in [0,1], independent and uniform, so the pair (x,y) is uniform over the unit square (total area 1). The three pieces form a triangle if and only if no piece is \ge 	frac12, by the triangle inequality (the longest side must be less than the sum of the other two, which for total length 1 means the longest is less than 	frac12). Assume first that x < y; the pieces are x, y - x, and 1 - y. The triangle conditions are x < 	frac12, \qquad y - x < 	frac12, \qquad 1 - y < 	frac12 \ (	ext{i.e. } y 	frac12). In the triangle \{0 < x < y < 1\} (which has area 	frac…

Break a stick, can the pieces form a triangle?

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