A long middle piece from two cuts

Question

A wire of length 1 is cut at two points chosen independently and uniformly at random along it, giving a left, a middle, and a right piece. What is the probability the middle piece is longer than 	frac12?

Accepted Answer

Let the two cut positions be X and Y, independent and uniform on [0, 1], plotted in the unit square where probability equals area. The middle piece is the gap between the cuts, of length X - Y. We want X - Y 	frac12. In the square, that is the two corner triangles far from the diagonal: X - Y 	frac12 and Y - X 	frac12. Each is a right triangle with legs 1 - 	frac12 = 	frac12: P\left(X - Y 	frac12ight) = 2 \cdot 	frac12 \left(	frac12ight)^2 = \left(	frac12ight)^2 = \frac14 . Takeaway. The middle piece length is X - Y, and P(X - Y c) = (1 - c)^2 for two uniform cuts. The two "far…

A long middle piece from two cuts

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