Four pieces, expected shortest and longest

Question

A stick of length 1 is broken at three points chosen independently and uniformly at random, producing four pieces. What are the expected lengths of the shortest and the longest of the four pieces?

Accepted Answer

With 3 breaks we get n = 4 pieces (spacings). Two general facts about uniform spacings: Shortest piece. All four pieces exceed t iff the three break points avoid the "forbidden" structure of total length 4t; by the standard shrink-and-rescale argument, P(L{\min} t) = (1 - 4t)^3, \qquad 0 \le t \le 	frac14. E[L{\min}] = \int0^{1/4} (1 - 4t)^3\,dt = \left[-\frac{(1-4t)^4}{16}ight]0^{1/4} = \frac{1}{16}. This matches the general formula E[L{\min}] = 	frac{1}{n^2} = 	frac{1}{16}. Longest piece. By inclusion-exclusion on the four events \{Li t\} (each piece has survival (1-t)^3): P(L{\max} t)…

Four pieces, expected shortest and longest

Your answer

More Probability questions