A long left piece from two cuts

Question

A wire of length 1 is cut at two independent uniformly random points, giving a left, a middle, and a right piece. What is the probability the leftmost piece is longer than 	frac13?

Accepted Answer

Let the cuts be X, Y \sim U(0, 1). The leftmost piece runs from 0 to the nearer cut, so its length is \min(X, Y). The left piece exceeds 	frac13 exactly when both cuts are beyond 	frac13: \min(X, Y) 	frac13 \iff X 	frac13 	ext{ and } Y 	frac13 . In the unit square that is a corner square of side 1 - 	frac13 = 	frac23: P = \left(	frac23ight)^2 = \frac{4}{9} . Takeaway. The left piece is \min(X, Y), and P(\min c) = (1 - c)^2. A long first piece means the cutting starts late, both cuts past c.

A long left piece from two cuts

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