Is the middle piece the longest?

Question

A stick of length 1 is broken at two points chosen independently and uniformly at random, giving three pieces: a left piece, a middle piece, and a right piece. What is the probability that the middle piece is the longest?

Accepted Answer

Let the break points be X, Y \sim U(0,1). After sorting them, the three pieces have lengths equal to the three spacings of two uniform points on [0,1]. A key fact: these three spacings are exchangeable, their joint distribution is completely symmetric under relabeling (formally, they are uniform on the simplex, a Dirichlet(1,1,1)). Intuitively, nothing about the uniform mechanism favors one gap over another. If the three lengths are exchangeable, then each is equally likely to be the largest. There are three pieces, so P(	ext{middle piece is longest}) = \frac{1}{3}. The "middle" label refers…

Is the middle piece the longest?

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