Can the three pieces form a triangle?

Question

Break a stick of length 1 at two points chosen independently and uniformly along its length, producing three pieces. What is the probability the three pieces can be arranged into a triangle?

Accepted Answer

Three lengths form a triangle exactly when each piece is shorter than the sum of the other two, equivalently, no single piece is longer than 	frac12 (if one piece exceeds half the stick, the other two together can't reach across it). Let the two break points be X and Y, independent and uniform on [0, 1]. Represent the outcome as a point (X, Y) in the unit square (all outcomes equally likely). We need to find the fraction of the square where all three pieces are under 	frac12. Consider the case X < Y (the lower triangle of the square, probability 	frac12; the case X Y is symmetric). The thr…

Can the three pieces form a triangle?

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