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The expected length of the middle piece

probabilityexpectationsymmetryorder-statistics

A stick of length 1 is broken at two points chosen independently and uniformly at random. This produces three pieces: a left piece, a middle piece (between the two breaks) and a right piece.

  1. What is the expected length of the middle piece?
  2. What is the expected length of the shortest of the three pieces? Is that the same question?
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For part 1, think about whether there is anything special about the middle piece compared with the left and right pieces, once you consider all the ways the two random points can fall.

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