A short middle 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 middle piece is shorter than 	frac14?

Accepted Answer

Let the cuts be X, Y \sim U(0, 1) in the unit square; the middle piece has length X - Y. It is easier to subtract the complement X - Y \ge 	frac14, the two corner triangles with legs 1 - 	frac14 = 	frac34: P\left(X - Y \ge 	frac14ight) = \left(	frac34ight)^2 = \frac{9}{16} . Therefore P\left(X - Y < 	frac14ight) = 1 - \frac{9}{16} = \frac{7}{16} . Takeaway. P(X - Y < c) = 1 - (1 - c)^2 = 2c - c^2. A short middle piece corresponds to the two cuts landing close together, the band hugging the diagonal.

A short middle piece from two cuts

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