Shorter piece of a randomly broken stick

Question

A stick of unit length is snapped at a single point chosen uniformly at random.

(a) What is the expected length of the shorter piece?

(b) What is the expected value of the ratio of the shorter piece's length to the longer piece's length?

Short answer

(a) $\tfrac14$ ; (b) $2\ln2-1\approx0.386$ . The shorter piece is uniform on $[0,\tfrac12]$ , so its mean is $\tfrac14$ . For the ratio, if the break is at $x\in[\tfrac12,1]$ the ratio is $\tfrac{1-x}{x}$ , and $2\int_{1/2}^1\left(\tfrac1x-1\right)dx=2\ln2-1$ , not the naive $\tfrac13$ .

Adapted from OpenQuant; question and answer rewritten by SelfMadeQuant.com.

See the hint and the full worked solution in the question bank (free account).

Open the worked solution

Lessons behind this question

Sample Spaces, Events, and the Probability Axioms
Continuous Random Variables and Probability Density Functions
Cumulative Distribution Functions

Start with a free lesson: Normal Distribution is part of a free topic. Create a free account, no card needed, to read it in full with worked solutions.

Read it free