Expected length of the longest and shortest piece

Question

A stick of length 1 is broken at two points chosen independently and uniformly at random, giving three pieces. What is the expected length of the longest piece? And of the shortest?

Accepted Answer

The three pieces are the spacings of two uniform points, exchangeable with mean 	frac13 each. Their extremes are not 	frac13, though. Shortest piece. The probability all three pieces exceed a is (1 - 3a)^2 for a \le 	frac13, so E[	ext{shortest}] = \int0^{1/3} P(	ext{shortest} a)\,da = \int0^{1/3}(1 - 3a)^2\,da = \left[-\frac{(1-3a)^3}{9}ight]0^{1/3} = \frac{1}{9} \approx 0.111. Longest piece. Use inclusion-exclusion on the event "some piece exceeds t." For exchangeable spacings, P(	ext{longest} t) = 3(1-t)+^2 - 3(1-2t)+^2 + (1-3t)+^2. Integrating each term from 0 to 1: \int0^1 3(1-t)^…

Expected length of the longest and shortest piece

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