A short right 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 rightmost piece is shorter than 	frac13?

Accepted Answer

Let the cuts be X, Y \sim U(0, 1). The rightmost piece runs from the farther cut to 1, so its length is 1 - \max(X, Y). We want 1 - \max(X, Y) < 	frac13, i.e. \max(X, Y) 	frac23, the complement of both cuts being at most 	frac23: P\left(\max(X, Y) \le 	frac23ight) = \left(	frac23ight)^2 = \frac{4}{9} , so P\left(	ext{right piece} < 	frac13ight) = P\left(\max 	frac23ight) = 1 - \frac{4}{9} = \frac{5}{9} . Takeaway. The right piece is 1 - \max(X, Y). A short last piece means at least one cut lands near the far end, which is why the answer exceeds a half.

A short right piece from two cuts

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