Solution
The Longest of Four Pieces
Show the problem again
A stick of length 1 is broken at three points chosen independently and uniformly at random, producing four pieces. What is the expected length of the longest piece? Give a reduced fraction.
Worked solution
The answer is 25/48. Represent the four piece lengths as (E₁, E₂, E₃, E₄)/S, where the Eᵢ are independent exponential(1) variables and S is their sum. This is a standard fact: uniform spacings have the same joint law as normalised exponentials, and the normalised vector is independent of S. The maximum of four independent exponential(1) variables has mean 1 + 1/2 + 1/3 + 1/4 = 25/12: the first of the four to finish takes on average 1/4, then the next of the remaining three takes 1/3, and so on, by memorylessness. Since max Eᵢ = S · (max piece) with the two factors independent, E[max Eᵢ] = E[S]·E[max piece] = 4·E[max piece]. Hence E[max piece] = (25/12)/4 = 25/48 ≈ 0.521. The same argument gives, for n pieces, an expected longest piece of (1/n)(1 + 1/2 + ... + 1/n).
Source: Classic uniform-spacings result in wide circulation; no traceable original source. Statement written for AxiomIQ.