Is the middle piece the longest?

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 the longest of the three?

Accepted Answer

Two uniform cuts split the wire into three pieces whose lengths are the three gaps. By symmetry these three gaps are exchangeable: relabeling which is "left," "middle," or "right" does not change the distribution of the collection of lengths. So each of the three pieces is equally likely to be the longest. Ignoring ties (which have probability 0), P(	ext{middle is longest}) = \frac13 . Takeaway. The three pieces from two uniform cuts are interchangeable, so each is the longest with probability 	frac13 and, by the same argument, the shortest with probability 	frac13. Spot the symmetry befor…

Is the middle piece the longest?

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