Can the three stick pieces form a triangle

Question

A stick of length 1 is broken at two points chosen independently and uniformly at random, producing three pieces. What is the probability the three pieces can form a (non-degenerate) triangle?

Accepted Answer

The triangle condition. Three lengths form a triangle iff each is strictly less than the sum of the other two, equivalently each is less than half the total. With total 1, a triangle forms iff every piece is less than 	frac12. So the triangle fails iff some one piece exceeds 	frac12. Name the pieces. Let the two independent uniform break points be X and Y on [0,1]. Sorting them, the three pieces are a = \min(X, Y), \qquad b = X - Y, \qquad c = 1 - \max(X, Y). The three failure events are disjoint. At most one piece can exceed 	frac12, since two pieces each over 	frac12 would sum to more t…

Can the three stick pieces form a triangle

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